Nonstationary spatial covariance modeling

NRCSE

My experience

1 Hourly ozone monitoring data at 100 sites in the San Joaquin Valley CA for assessment of photochemical model predictions

Accounting for sub-grid scale spatial variability in point monitoring observations to assess grid average photochemical predictions

Decomposing spatial difference fieldsmdashempirical grid cell estimates vs model predictionsmdashinto components at different spatial scales

Comparing empirical spatial covariance structure with model spatial covariance structure at different spatial scales

2 10-day aggregate precipitation in Languedoc-Roussillon France

3 Daily ozone monitoring summaries (max 8-hr ave) from 100rsquos of monitoring sites across the country for spatial prediction at target locations

4 Daily and 2-week average NOx and PM25 modeling for monitoring sites for MESA Air project

Perspective 1st and 2nd order stationarity is almost never a realistic assumption for any environmental monitoring data except at small spatial scales

Objectives for approaches to nonstationary spatial covariance modeling

Characterizing spatially varying locally (stationary) anisotropic structure

Scientific understandingrepresentation of covariance structuremdashnot just a method of providing covariances for kriging

Capable of

reflecting effects of known explanatory environmental processes such as transportwind topography point sources

modeling effects of known explanatory environmental processes

Objectives (cont)

Application to purely spatial problems andor problems with data sampled irregularly in space and time

Application in context of dynamic models for space-time structure

Application to ldquolargerdquo problemsdata sets

Diagnostics for local and large-scale correlation structure

o is the spatial structure ldquorightrdquo

o is the naturedegree of nonstationarity (smoothness) right

Evaluation of uncertainty in estimation (interpolation) of spatial covariance structure

Incorporation in an approach to spatial estimation accounting for uncertainty in estimation of (parameters of) spatial covariance structure

Selected Methods and References1 Basis function methods (Nychka Wikle hellip)

2 Kernelsmoothing methods (Fuentes hellip)

3 Process convolution models (Higdon Swall amp Kern hellip Paciorek amp Schervish)

4 Parametric models

5 Spatial deformation modelsbull Sampson amp Guttorp 1989 1992 1994 bull Meiring Monestiez Sampson amp Guttorp 1997

ldquoDevelopmentsrdquo Geostatistics Wollongong 96bull Mardia amp Goodall 1993 in Multivariate Environmental

Statistics bull Smith 1996 Estimating nonstationary spatial

correlations UNC preprint

4 Spatial deformation models (cont)bull Perrin amp Meiring 1999 Identifiability J Appl Prob bull Perrin amp Senoussi 1998 Reducing nonstationary

random fields to stationarity or isotropy Stat amp ProbLetters

bull Perrin amp Monestiez 1998 Parametric radial basis deformations GeoENV-II

bull Iovleff amp Perrin 2004 Simulated annealing J Comp Graph Stat

bull Schmidt amp OHagan 2003 Bayesian inference for non-stationary spatial covariance structure via spatial deformations JRSS-B

bull Damian Sampson amp Guttorp 2000 Bayesian estimation of semiparametric nonstationary covariance structures Environmetrics

bull Damian Sampson amp Guttorp 2003 Variance modeling for nonstationary spatial processes with temporal replications Journal of Geophysical Research

bull Related recent developments in the atmospheric science literature

The spherical correlation

Corresponding variogram

ρ(v) =1minus 15v + 05 v

φ( )3 h lt φ

0 otherwise

( )φ φ

στ + minus le le φ

τ + σ gt φ

22 3

2 2

3 ( ) 02

t t t

t

nugget

sill range

Review Descriptive characteristics of (stationary) spatial covariance expressed in a variogram

Spatial continuity (roughness) of data set characterized by initial slope or range of variogram

Correlation vs Distance for Ontario Ozone Data

Apparent anisotropy

Nonstationary spatial covariance

Basic idea the parameters of a local variogram model---nugget range sill and anisotropy---vary spatially

Look at some pictures of applications from recent methodology publications

Swall amp Higdon Process convolution approachSoil contamination example --- Piazza Rd site

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish 2006 ndashColorado 1981 annual precip (log)

Paciorek amp Schervish 2006 ndashkernels (ellipses of constant Gaussian density) representing estimated correlation structure

Pintore amp Holmes 2005 Spatially adaptive non-stationary covariance functions via spatially adaptive spectra

Nott amp Dunsmuire 2002 Biometrika Fig 2 Sydney wind pattern data Contours of equal estimated correlation with two different fixed sites shown by open squares (a) location 33middot85degS 151middot22degE and (b) location 33middot74degS 149middot88degE The sites marked by dots show locations of the 45 monitored sites

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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bull bull

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bull

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bull

bull

bull

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bull

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bullbull

bull

bull

bull

bull

distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

08

10 bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull bull

bull

bull

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bullbull

bullbull

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bullbull

bullbull

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bullbull

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bullbull

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bullbull

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bull

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bull

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bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

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bull

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bull

bull

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bull bullbull bull

bull

bullbull

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bull

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bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bullbull

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bull

bull

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bull

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bullbull

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bull bull

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bull

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bull

bull

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bull

bull

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bull

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bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

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bull

bull

bull

bull

bull

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bull

bull bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull bullbull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bullbull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bullbull

bull

bullbull

bull

bull

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bull

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bull

bull

bullbull

bull

bull

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bull

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bull

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bull

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bull

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bull bullbull

bullbullbull bullbull

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bull

bullbullbull

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bull

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bull

bull

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bull

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bull bullbull bullbull

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bullbullbull

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bull bullbull

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bull

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bull

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bull

bull

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bull

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bull

bull

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bull bull

bull

bull

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bull

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bull bull

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bull bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

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bull

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bullbullbull

bull

bull

bull

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bullbullbull

bull

bull

bull

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bull

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bull

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bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

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bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

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bull

bull

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bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

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bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

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bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

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bull

bull

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bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

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bull

bullbull

bull

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bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

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bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

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bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

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bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

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bull

bull

bull bullbull

bull

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bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

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bullbullbull

bull

bull

bull

bull

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bull

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bull

bull

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bull

bull

bullbullbull

bull

bull

bull

bull

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bull

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bull

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bull

bull

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bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

My experience

1 Hourly ozone monitoring data at 100 sites in the San Joaquin Valley CA for assessment of photochemical model predictions

Accounting for sub-grid scale spatial variability in point monitoring observations to assess grid average photochemical predictions

Decomposing spatial difference fieldsmdashempirical grid cell estimates vs model predictionsmdashinto components at different spatial scales

Comparing empirical spatial covariance structure with model spatial covariance structure at different spatial scales

2 10-day aggregate precipitation in Languedoc-Roussillon France

3 Daily ozone monitoring summaries (max 8-hr ave) from 100rsquos of monitoring sites across the country for spatial prediction at target locations

4 Daily and 2-week average NOx and PM25 modeling for monitoring sites for MESA Air project

Perspective 1st and 2nd order stationarity is almost never a realistic assumption for any environmental monitoring data except at small spatial scales

Objectives for approaches to nonstationary spatial covariance modeling

Characterizing spatially varying locally (stationary) anisotropic structure

Scientific understandingrepresentation of covariance structuremdashnot just a method of providing covariances for kriging

Capable of

reflecting effects of known explanatory environmental processes such as transportwind topography point sources

modeling effects of known explanatory environmental processes

Objectives (cont)

Application to purely spatial problems andor problems with data sampled irregularly in space and time

Application in context of dynamic models for space-time structure

Application to ldquolargerdquo problemsdata sets

Diagnostics for local and large-scale correlation structure

o is the spatial structure ldquorightrdquo

o is the naturedegree of nonstationarity (smoothness) right

Evaluation of uncertainty in estimation (interpolation) of spatial covariance structure

Incorporation in an approach to spatial estimation accounting for uncertainty in estimation of (parameters of) spatial covariance structure

Selected Methods and References1 Basis function methods (Nychka Wikle hellip)

2 Kernelsmoothing methods (Fuentes hellip)

3 Process convolution models (Higdon Swall amp Kern hellip Paciorek amp Schervish)

4 Parametric models

5 Spatial deformation modelsbull Sampson amp Guttorp 1989 1992 1994 bull Meiring Monestiez Sampson amp Guttorp 1997

ldquoDevelopmentsrdquo Geostatistics Wollongong 96bull Mardia amp Goodall 1993 in Multivariate Environmental

Statistics bull Smith 1996 Estimating nonstationary spatial

correlations UNC preprint

4 Spatial deformation models (cont)bull Perrin amp Meiring 1999 Identifiability J Appl Prob bull Perrin amp Senoussi 1998 Reducing nonstationary

random fields to stationarity or isotropy Stat amp ProbLetters

bull Perrin amp Monestiez 1998 Parametric radial basis deformations GeoENV-II

bull Iovleff amp Perrin 2004 Simulated annealing J Comp Graph Stat

bull Schmidt amp OHagan 2003 Bayesian inference for non-stationary spatial covariance structure via spatial deformations JRSS-B

bull Damian Sampson amp Guttorp 2000 Bayesian estimation of semiparametric nonstationary covariance structures Environmetrics

bull Damian Sampson amp Guttorp 2003 Variance modeling for nonstationary spatial processes with temporal replications Journal of Geophysical Research

bull Related recent developments in the atmospheric science literature

The spherical correlation

Corresponding variogram

ρ(v) =1minus 15v + 05 v

φ( )3 h lt φ

0 otherwise

( )φ φ

στ + minus le le φ

τ + σ gt φ

22 3

2 2

3 ( ) 02

t t t

t

nugget

sill range

Review Descriptive characteristics of (stationary) spatial covariance expressed in a variogram

Spatial continuity (roughness) of data set characterized by initial slope or range of variogram

Correlation vs Distance for Ontario Ozone Data

Apparent anisotropy

Nonstationary spatial covariance

Basic idea the parameters of a local variogram model---nugget range sill and anisotropy---vary spatially

Look at some pictures of applications from recent methodology publications

Swall amp Higdon Process convolution approachSoil contamination example --- Piazza Rd site

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish 2006 ndashColorado 1981 annual precip (log)

Paciorek amp Schervish 2006 ndashkernels (ellipses of constant Gaussian density) representing estimated correlation structure

Pintore amp Holmes 2005 Spatially adaptive non-stationary covariance functions via spatially adaptive spectra

Nott amp Dunsmuire 2002 Biometrika Fig 2 Sydney wind pattern data Contours of equal estimated correlation with two different fixed sites shown by open squares (a) location 33middot85degS 151middot22degE and (b) location 33middot74degS 149middot88degE The sites marked by dots show locations of the 45 monitored sites

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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bull

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bull

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bull

bull

bull

bull

distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

08

10 bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

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bull

bull

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bullbull

bullbull

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bullbull

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bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

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bull

bull bullbull

bull

bull

bull

bull

bull

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bull

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bull bullbull bull

bull

bullbull

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bull

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bullbull

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bull

bull

bull

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bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

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bullbull

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bull

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bull

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bull

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bull

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bull

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bull

bull

bull

bull

bull

bullbull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull bull

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bull

bull

bullbull

bullbull

bullbull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

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bull

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bull

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bull

bull

bullbull

bull

bull bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull bullbull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bullbull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

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bull

bull

bullbull

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bullbullbull bullbull

bullbull

bull

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bull bullbull bullbull

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bull

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bull

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bull

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bull bullbull

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bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

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bull

bull

bull

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bull

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bull

bull

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bull

bull

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bull

bull

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bull

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bull

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bull

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bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

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bull

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bull

bull

bull

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bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

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bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

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bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

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bullbullbull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

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bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

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bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

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bull

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bull

bull

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bull

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bull

bull

bull

bull

bull

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bull

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bull

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bull

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bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

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bull

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bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

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bullbullbull

bull

bull

bull

bull

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bull

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bull

bull

bull

bull

bull

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bull

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bull

bull

bull

bull

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bull bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

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bull

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bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

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bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

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bull

bull

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bull

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bull

bull

bull

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bull

bull

bull

bull

bull

bull

bullbull

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bull

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bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Perspective 1st and 2nd order stationarity is almost never a realistic assumption for any environmental monitoring data except at small spatial scales

Objectives for approaches to nonstationary spatial covariance modeling

Characterizing spatially varying locally (stationary) anisotropic structure

Scientific understandingrepresentation of covariance structuremdashnot just a method of providing covariances for kriging

Capable of

reflecting effects of known explanatory environmental processes such as transportwind topography point sources

modeling effects of known explanatory environmental processes

Objectives (cont)

Application to purely spatial problems andor problems with data sampled irregularly in space and time

Application in context of dynamic models for space-time structure

Application to ldquolargerdquo problemsdata sets

Diagnostics for local and large-scale correlation structure

o is the spatial structure ldquorightrdquo

o is the naturedegree of nonstationarity (smoothness) right

Evaluation of uncertainty in estimation (interpolation) of spatial covariance structure

Incorporation in an approach to spatial estimation accounting for uncertainty in estimation of (parameters of) spatial covariance structure

Selected Methods and References1 Basis function methods (Nychka Wikle hellip)

2 Kernelsmoothing methods (Fuentes hellip)

3 Process convolution models (Higdon Swall amp Kern hellip Paciorek amp Schervish)

4 Parametric models

5 Spatial deformation modelsbull Sampson amp Guttorp 1989 1992 1994 bull Meiring Monestiez Sampson amp Guttorp 1997

ldquoDevelopmentsrdquo Geostatistics Wollongong 96bull Mardia amp Goodall 1993 in Multivariate Environmental

Statistics bull Smith 1996 Estimating nonstationary spatial

correlations UNC preprint

4 Spatial deformation models (cont)bull Perrin amp Meiring 1999 Identifiability J Appl Prob bull Perrin amp Senoussi 1998 Reducing nonstationary

random fields to stationarity or isotropy Stat amp ProbLetters

bull Perrin amp Monestiez 1998 Parametric radial basis deformations GeoENV-II

bull Iovleff amp Perrin 2004 Simulated annealing J Comp Graph Stat

bull Schmidt amp OHagan 2003 Bayesian inference for non-stationary spatial covariance structure via spatial deformations JRSS-B

bull Damian Sampson amp Guttorp 2000 Bayesian estimation of semiparametric nonstationary covariance structures Environmetrics

bull Damian Sampson amp Guttorp 2003 Variance modeling for nonstationary spatial processes with temporal replications Journal of Geophysical Research

bull Related recent developments in the atmospheric science literature

The spherical correlation

Corresponding variogram

ρ(v) =1minus 15v + 05 v

φ( )3 h lt φ

0 otherwise

( )φ φ

στ + minus le le φ

τ + σ gt φ

22 3

2 2

3 ( ) 02

t t t

t

nugget

sill range

Review Descriptive characteristics of (stationary) spatial covariance expressed in a variogram

Spatial continuity (roughness) of data set characterized by initial slope or range of variogram

Correlation vs Distance for Ontario Ozone Data

Apparent anisotropy

Nonstationary spatial covariance

Basic idea the parameters of a local variogram model---nugget range sill and anisotropy---vary spatially

Look at some pictures of applications from recent methodology publications

Swall amp Higdon Process convolution approachSoil contamination example --- Piazza Rd site

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish 2006 ndashColorado 1981 annual precip (log)

Paciorek amp Schervish 2006 ndashkernels (ellipses of constant Gaussian density) representing estimated correlation structure

Pintore amp Holmes 2005 Spatially adaptive non-stationary covariance functions via spatially adaptive spectra

Nott amp Dunsmuire 2002 Biometrika Fig 2 Sydney wind pattern data Contours of equal estimated correlation with two different fixed sites shown by open squares (a) location 33middot85degS 151middot22degE and (b) location 33middot74degS 149middot88degE The sites marked by dots show locations of the 45 monitored sites

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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bull bullbull bullbull

bull

bull

bullbull

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bull

bull

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bull

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bull

bull

bull

bull

distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

08

10 bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

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bull

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bullbull

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bullbull

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bullbull

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bull

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bullbull

bull

bull

bullbull

bull

bull

bull

bull

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bull

bull

bull bullbull

bull

bull

bull

bull

bull

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bull

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bull bullbull bull

bull

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bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bullbull

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bull

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bull bull

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bull bull

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bull

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bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull bull

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bull

bull

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bullbull

bullbull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

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bull

bull

bull

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bull

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bull

bull bull

bull

bullbull

bull

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bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull bullbull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bullbull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bullbull

bull

bullbull

bull

bull

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bull

bull

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bull

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bullbullbull bullbull

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bull bullbull bullbull

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bull

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bull

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bull

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bull

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bull bullbull

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bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

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bullbullbull

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bull

bull

bull

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bull

bull

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bull

bull

bull

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bullbull

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bull

bull

bull

bull

bull

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bull

bull

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bull

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bull

bull

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bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

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bull

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bull

bull

bull

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bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

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bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

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bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

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bull

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bull

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bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

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bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

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bull

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bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

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bull

bull

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bull

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bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

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bull

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bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

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bull

bull

bull

bull

bull

bullbull

bullbull

bull

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bull

bull

bull

bull

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bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

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bull

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bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

bull bullbull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

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bullbullbull

bull

bull

bull

bull

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bull

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bull

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bull

bull

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bull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Objectives (cont)

Application to purely spatial problems andor problems with data sampled irregularly in space and time

Application in context of dynamic models for space-time structure

Application to ldquolargerdquo problemsdata sets

Diagnostics for local and large-scale correlation structure

o is the spatial structure ldquorightrdquo

o is the naturedegree of nonstationarity (smoothness) right

Evaluation of uncertainty in estimation (interpolation) of spatial covariance structure

Incorporation in an approach to spatial estimation accounting for uncertainty in estimation of (parameters of) spatial covariance structure

Selected Methods and References1 Basis function methods (Nychka Wikle hellip)

2 Kernelsmoothing methods (Fuentes hellip)

3 Process convolution models (Higdon Swall amp Kern hellip Paciorek amp Schervish)

4 Parametric models

5 Spatial deformation modelsbull Sampson amp Guttorp 1989 1992 1994 bull Meiring Monestiez Sampson amp Guttorp 1997

ldquoDevelopmentsrdquo Geostatistics Wollongong 96bull Mardia amp Goodall 1993 in Multivariate Environmental

Statistics bull Smith 1996 Estimating nonstationary spatial

correlations UNC preprint

4 Spatial deformation models (cont)bull Perrin amp Meiring 1999 Identifiability J Appl Prob bull Perrin amp Senoussi 1998 Reducing nonstationary

random fields to stationarity or isotropy Stat amp ProbLetters

bull Perrin amp Monestiez 1998 Parametric radial basis deformations GeoENV-II

bull Iovleff amp Perrin 2004 Simulated annealing J Comp Graph Stat

bull Schmidt amp OHagan 2003 Bayesian inference for non-stationary spatial covariance structure via spatial deformations JRSS-B

bull Damian Sampson amp Guttorp 2000 Bayesian estimation of semiparametric nonstationary covariance structures Environmetrics

bull Damian Sampson amp Guttorp 2003 Variance modeling for nonstationary spatial processes with temporal replications Journal of Geophysical Research

bull Related recent developments in the atmospheric science literature

The spherical correlation

Corresponding variogram

ρ(v) =1minus 15v + 05 v

φ( )3 h lt φ

0 otherwise

( )φ φ

στ + minus le le φ

τ + σ gt φ

22 3

2 2

3 ( ) 02

t t t

t

nugget

sill range

Review Descriptive characteristics of (stationary) spatial covariance expressed in a variogram

Spatial continuity (roughness) of data set characterized by initial slope or range of variogram

Correlation vs Distance for Ontario Ozone Data

Apparent anisotropy

Nonstationary spatial covariance

Basic idea the parameters of a local variogram model---nugget range sill and anisotropy---vary spatially

Look at some pictures of applications from recent methodology publications

Swall amp Higdon Process convolution approachSoil contamination example --- Piazza Rd site

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish 2006 ndashColorado 1981 annual precip (log)

Paciorek amp Schervish 2006 ndashkernels (ellipses of constant Gaussian density) representing estimated correlation structure

Pintore amp Holmes 2005 Spatially adaptive non-stationary covariance functions via spatially adaptive spectra

Nott amp Dunsmuire 2002 Biometrika Fig 2 Sydney wind pattern data Contours of equal estimated correlation with two different fixed sites shown by open squares (a) location 33middot85degS 151middot22degE and (b) location 33middot74degS 149middot88degE The sites marked by dots show locations of the 45 monitored sites

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

08

10 bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

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bull

bull

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bullbull

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bullbull

bullbull

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bullbull

bullbull

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bullbull

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bull

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bull

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bull

bull

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bull

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bull

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bull

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bull

bullbull

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bullbullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

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bullbull

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bull

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bull

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bullbull

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bull

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bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

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bull

bull

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bullbull

bullbull

bull

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bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

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bull

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bull

bull

bull

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bull

bull

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bull

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bull

bull

bull

bull

bull

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bull

bull bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull bullbull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bullbull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

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bull

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bull

bull

bull

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bull

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bull

bull

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bull

bull

bull

bull

bull

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bull

bull

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bull bullbull

bullbullbull bullbull

bullbull

bull

bull

bullbullbull

bull

bull

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bull bullbull bullbull

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bull

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bullbullbull

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bullbull bull

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bull bullbull

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bull

bull

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bull

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bull

bull

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bull

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bull

bull

bull

bull

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bull

bullbullbull

bull

bullbull

bull

bull

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bull

bull

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bull

bull

bullbull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

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bull

bull

bull

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bullbull

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bull

bull

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bull

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bull

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bull

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bull

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bull

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bull

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bull

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bull

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bull bull

bull

bull

bull bullbullbull

bull

bull

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bull

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bull

bull

bull

bull

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bull

bull

bull

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bull

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bull

bull

bull bull

bullbull

bull bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

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bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

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bull

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bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Selected Methods and References1 Basis function methods (Nychka Wikle hellip)

2 Kernelsmoothing methods (Fuentes hellip)

3 Process convolution models (Higdon Swall amp Kern hellip Paciorek amp Schervish)

4 Parametric models

5 Spatial deformation modelsbull Sampson amp Guttorp 1989 1992 1994 bull Meiring Monestiez Sampson amp Guttorp 1997

ldquoDevelopmentsrdquo Geostatistics Wollongong 96bull Mardia amp Goodall 1993 in Multivariate Environmental

Statistics bull Smith 1996 Estimating nonstationary spatial

correlations UNC preprint

4 Spatial deformation models (cont)bull Perrin amp Meiring 1999 Identifiability J Appl Prob bull Perrin amp Senoussi 1998 Reducing nonstationary

random fields to stationarity or isotropy Stat amp ProbLetters

bull Perrin amp Monestiez 1998 Parametric radial basis deformations GeoENV-II

bull Iovleff amp Perrin 2004 Simulated annealing J Comp Graph Stat

bull Schmidt amp OHagan 2003 Bayesian inference for non-stationary spatial covariance structure via spatial deformations JRSS-B

bull Damian Sampson amp Guttorp 2000 Bayesian estimation of semiparametric nonstationary covariance structures Environmetrics

bull Damian Sampson amp Guttorp 2003 Variance modeling for nonstationary spatial processes with temporal replications Journal of Geophysical Research

bull Related recent developments in the atmospheric science literature

The spherical correlation

Corresponding variogram

ρ(v) =1minus 15v + 05 v

φ( )3 h lt φ

0 otherwise

( )φ φ

στ + minus le le φ

τ + σ gt φ

22 3

2 2

3 ( ) 02

t t t

t

nugget

sill range

Review Descriptive characteristics of (stationary) spatial covariance expressed in a variogram

Spatial continuity (roughness) of data set characterized by initial slope or range of variogram

Correlation vs Distance for Ontario Ozone Data

Apparent anisotropy

Nonstationary spatial covariance

Basic idea the parameters of a local variogram model---nugget range sill and anisotropy---vary spatially

Look at some pictures of applications from recent methodology publications

Swall amp Higdon Process convolution approachSoil contamination example --- Piazza Rd site

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish 2006 ndashColorado 1981 annual precip (log)

Paciorek amp Schervish 2006 ndashkernels (ellipses of constant Gaussian density) representing estimated correlation structure

Pintore amp Holmes 2005 Spatially adaptive non-stationary covariance functions via spatially adaptive spectra

Nott amp Dunsmuire 2002 Biometrika Fig 2 Sydney wind pattern data Contours of equal estimated correlation with two different fixed sites shown by open squares (a) location 33middot85degS 151middot22degE and (b) location 33middot74degS 149middot88degE The sites marked by dots show locations of the 45 monitored sites

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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bull

bull

bull bull

bull

bull bullbull bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

08

10 bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bullbull

bullbull

bullbull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull bullbull bull

bull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bullbull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bullbullbullbull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull bullbull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bullbull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbullbull

bull

bullbull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bullbullbull bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull bullbull bullbull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bullbull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

4 Spatial deformation models (cont)bull Perrin amp Meiring 1999 Identifiability J Appl Prob bull Perrin amp Senoussi 1998 Reducing nonstationary

random fields to stationarity or isotropy Stat amp ProbLetters

bull Perrin amp Monestiez 1998 Parametric radial basis deformations GeoENV-II

bull Iovleff amp Perrin 2004 Simulated annealing J Comp Graph Stat

bull Schmidt amp OHagan 2003 Bayesian inference for non-stationary spatial covariance structure via spatial deformations JRSS-B

bull Damian Sampson amp Guttorp 2000 Bayesian estimation of semiparametric nonstationary covariance structures Environmetrics

bull Damian Sampson amp Guttorp 2003 Variance modeling for nonstationary spatial processes with temporal replications Journal of Geophysical Research

bull Related recent developments in the atmospheric science literature

The spherical correlation

Corresponding variogram

ρ(v) =1minus 15v + 05 v

φ( )3 h lt φ

0 otherwise

( )φ φ

στ + minus le le φ

τ + σ gt φ

22 3

2 2

3 ( ) 02

t t t

t

nugget

sill range

Review Descriptive characteristics of (stationary) spatial covariance expressed in a variogram

Spatial continuity (roughness) of data set characterized by initial slope or range of variogram

Correlation vs Distance for Ontario Ozone Data

Apparent anisotropy

Nonstationary spatial covariance

Basic idea the parameters of a local variogram model---nugget range sill and anisotropy---vary spatially

Look at some pictures of applications from recent methodology publications

Swall amp Higdon Process convolution approachSoil contamination example --- Piazza Rd site

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish 2006 ndashColorado 1981 annual precip (log)

Paciorek amp Schervish 2006 ndashkernels (ellipses of constant Gaussian density) representing estimated correlation structure

Pintore amp Holmes 2005 Spatially adaptive non-stationary covariance functions via spatially adaptive spectra

Nott amp Dunsmuire 2002 Biometrika Fig 2 Sydney wind pattern data Contours of equal estimated correlation with two different fixed sites shown by open squares (a) location 33middot85degS 151middot22degE and (b) location 33middot74degS 149middot88degE The sites marked by dots show locations of the 45 monitored sites

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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bullbull

bull

bull

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bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bullbull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbull bull

bull

bull

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bull

bullbull

bull

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bull

bull

bull

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bull

bull

bullbull

bullbullbull

bull

bullbullbullbull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull bullbull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bullbull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbullbull

bull

bullbull

bull

bull

bull

bull bull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bullbullbull bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull bullbull bullbull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bullbull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

The spherical correlation

Corresponding variogram

ρ(v) =1minus 15v + 05 v

φ( )3 h lt φ

0 otherwise

( )φ φ

στ + minus le le φ

τ + σ gt φ

22 3

2 2

3 ( ) 02

t t t

t

nugget

sill range

Review Descriptive characteristics of (stationary) spatial covariance expressed in a variogram

Spatial continuity (roughness) of data set characterized by initial slope or range of variogram

Correlation vs Distance for Ontario Ozone Data

Apparent anisotropy

Nonstationary spatial covariance

Basic idea the parameters of a local variogram model---nugget range sill and anisotropy---vary spatially

Look at some pictures of applications from recent methodology publications

Swall amp Higdon Process convolution approachSoil contamination example --- Piazza Rd site

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish 2006 ndashColorado 1981 annual precip (log)

Paciorek amp Schervish 2006 ndashkernels (ellipses of constant Gaussian density) representing estimated correlation structure

Pintore amp Holmes 2005 Spatially adaptive non-stationary covariance functions via spatially adaptive spectra

Nott amp Dunsmuire 2002 Biometrika Fig 2 Sydney wind pattern data Contours of equal estimated correlation with two different fixed sites shown by open squares (a) location 33middot85degS 151middot22degE and (b) location 33middot74degS 149middot88degE The sites marked by dots show locations of the 45 monitored sites

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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bull

bull

bull

bull

bullbull

bull

bull

bull bullbull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bullbullbullbull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull bullbull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bullbull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbullbull

bull

bullbull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bullbullbull bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull bullbull bullbull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bullbull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Spatial continuity (roughness) of data set characterized by initial slope or range of variogram

Correlation vs Distance for Ontario Ozone Data

Apparent anisotropy

Nonstationary spatial covariance

Basic idea the parameters of a local variogram model---nugget range sill and anisotropy---vary spatially

Look at some pictures of applications from recent methodology publications

Swall amp Higdon Process convolution approachSoil contamination example --- Piazza Rd site

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish 2006 ndashColorado 1981 annual precip (log)

Paciorek amp Schervish 2006 ndashkernels (ellipses of constant Gaussian density) representing estimated correlation structure

Pintore amp Holmes 2005 Spatially adaptive non-stationary covariance functions via spatially adaptive spectra

Nott amp Dunsmuire 2002 Biometrika Fig 2 Sydney wind pattern data Contours of equal estimated correlation with two different fixed sites shown by open squares (a) location 33middot85degS 151middot22degE and (b) location 33middot74degS 149middot88degE The sites marked by dots show locations of the 45 monitored sites

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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bull

bull

bull bullbull

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bull

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bull

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bull

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bull

bull

bull bullbullbull

bull

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bull

bull bull

bull

bull

bull

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bull

bull

bull

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bull

bull

bull bull

bull

bull

bullbull bullbull

bull

bull

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bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bullbullbull bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull bullbull bullbull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bullbull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Correlation vs Distance for Ontario Ozone Data

Apparent anisotropy

Nonstationary spatial covariance

Basic idea the parameters of a local variogram model---nugget range sill and anisotropy---vary spatially

Look at some pictures of applications from recent methodology publications

Swall amp Higdon Process convolution approachSoil contamination example --- Piazza Rd site

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish 2006 ndashColorado 1981 annual precip (log)

Paciorek amp Schervish 2006 ndashkernels (ellipses of constant Gaussian density) representing estimated correlation structure

Pintore amp Holmes 2005 Spatially adaptive non-stationary covariance functions via spatially adaptive spectra

Nott amp Dunsmuire 2002 Biometrika Fig 2 Sydney wind pattern data Contours of equal estimated correlation with two different fixed sites shown by open squares (a) location 33middot85degS 151middot22degE and (b) location 33middot74degS 149middot88degE The sites marked by dots show locations of the 45 monitored sites

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Nonstationary spatial covariance

Basic idea the parameters of a local variogram model---nugget range sill and anisotropy---vary spatially

Look at some pictures of applications from recent methodology publications

Swall amp Higdon Process convolution approachSoil contamination example --- Piazza Rd site

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish 2006 ndashColorado 1981 annual precip (log)

Paciorek amp Schervish 2006 ndashkernels (ellipses of constant Gaussian density) representing estimated correlation structure

Pintore amp Holmes 2005 Spatially adaptive non-stationary covariance functions via spatially adaptive spectra

Nott amp Dunsmuire 2002 Biometrika Fig 2 Sydney wind pattern data Contours of equal estimated correlation with two different fixed sites shown by open squares (a) location 33middot85degS 151middot22degE and (b) location 33middot74degS 149middot88degE The sites marked by dots show locations of the 45 monitored sites

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

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( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

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[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

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Σ

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S

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Σ

1

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1ix

c

c

c

c

f

θ

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( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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0 500 1000 1500 2000 2500 3000

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distance

co

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0 500 1000 1500 2000

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Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

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(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

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pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Swall amp Higdon Process convolution approachSoil contamination example --- Piazza Rd site

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish 2006 ndashColorado 1981 annual precip (log)

Paciorek amp Schervish 2006 ndashkernels (ellipses of constant Gaussian density) representing estimated correlation structure

Pintore amp Holmes 2005 Spatially adaptive non-stationary covariance functions via spatially adaptive spectra

Nott amp Dunsmuire 2002 Biometrika Fig 2 Sydney wind pattern data Contours of equal estimated correlation with two different fixed sites shown by open squares (a) location 33middot85degS 151middot22degE and (b) location 33middot74degS 149middot88degE The sites marked by dots show locations of the 45 monitored sites

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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distance

corr

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0 500 1000 1500 2000 2500 3000

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distance

co

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0 500 1000 1500 2000

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Equi-correlation (09) contours D-plane (a) and G-plane (b)

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(a)

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0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

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bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish 2006 ndashColorado 1981 annual precip (log)

Paciorek amp Schervish 2006 ndashkernels (ellipses of constant Gaussian density) representing estimated correlation structure

Pintore amp Holmes 2005 Spatially adaptive non-stationary covariance functions via spatially adaptive spectra

Nott amp Dunsmuire 2002 Biometrika Fig 2 Sydney wind pattern data Contours of equal estimated correlation with two different fixed sites shown by open squares (a) location 33middot85degS 151middot22degE and (b) location 33middot74degS 149middot88degE The sites marked by dots show locations of the 45 monitored sites

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

bull

bull

bull

bull

bull

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bull

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bull bullbullbull

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bull bull bullbull

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bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Paciorek amp Schervish 2006 ndashColorado 1981 annual precip (log)

Paciorek amp Schervish 2006 ndashkernels (ellipses of constant Gaussian density) representing estimated correlation structure

Pintore amp Holmes 2005 Spatially adaptive non-stationary covariance functions via spatially adaptive spectra

Nott amp Dunsmuire 2002 Biometrika Fig 2 Sydney wind pattern data Contours of equal estimated correlation with two different fixed sites shown by open squares (a) location 33middot85degS 151middot22degE and (b) location 33middot74degS 149middot88degE The sites marked by dots show locations of the 45 monitored sites

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

bull

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distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

08

10 bull

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distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

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01

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1

29

3

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6

7

11

13

15

17

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27

3135

38

3943

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5557

59

6163

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7273

74

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(b)

5500 6000 6500 7000 7500 8000 8500 900017

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313538 39

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(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

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observed

pre

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observed

pre

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4

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observed

pre

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02

4

22

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observed

pre

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ted

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02

4

25

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observed

pre

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ted

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02

4

33

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observed

pre

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02

4

41

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observed

pre

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02

4

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observed

pre

dic

ted

-4 -2 0 2 4

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02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

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43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Paciorek amp Schervish 2006 ndashkernels (ellipses of constant Gaussian density) representing estimated correlation structure

Pintore amp Holmes 2005 Spatially adaptive non-stationary covariance functions via spatially adaptive spectra

Nott amp Dunsmuire 2002 Biometrika Fig 2 Sydney wind pattern data Contours of equal estimated correlation with two different fixed sites shown by open squares (a) location 33middot85degS 151middot22degE and (b) location 33middot74degS 149middot88degE The sites marked by dots show locations of the 45 monitored sites

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

bull

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distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

08

10 bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

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distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

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(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

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00

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00

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1

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3

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313538 39

43

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61

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(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

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observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

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observed

pre

dic

ted

-4 -2 0 2 4

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02

4

19

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observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

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observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

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observed

pre

dic

ted

-4 -2 0 2 4

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02

4

33

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observed

pre

dic

ted

-4 -2 0 2 4

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02

4

41

bull

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observed

pre

dic

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02

4

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observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Pintore amp Holmes 2005 Spatially adaptive non-stationary covariance functions via spatially adaptive spectra

Nott amp Dunsmuire 2002 Biometrika Fig 2 Sydney wind pattern data Contours of equal estimated correlation with two different fixed sites shown by open squares (a) location 33middot85degS 151middot22degE and (b) location 33middot74degS 149middot88degE The sites marked by dots show locations of the 45 monitored sites

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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bull

bull

bull

distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

08

10 bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

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bullbull

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bullbull

bullbull

bullbull

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bullbull

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bullbull

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bull

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bullbull

bull

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bull

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bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

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bull bullbull bull

bull

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bull

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bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bullbull

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bull

bull

bull

bull

bull

bull

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bullbull

bullbull

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bullbull

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bull bull

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bull bull

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bull

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bull

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bull

bull

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bull bullbull bull

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bull

bull

bull

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bull

bull

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bull

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bull

bull

bull

bull

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bull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull bullbull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bullbull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull bullbull

bull

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bull

bull

bull

bull

bull

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bull

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bull

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bull

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bull

bull

bull

bull

bull

bull bullbull

bullbullbull bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bullbull

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bull

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bullbull

bull bullbull bullbull

bull

bull

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bullbullbull

bull

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bullbull bull

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bull bullbull

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bull

bull

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bull

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bull

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bull

bull

bull

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bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bull

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bull

bull

bullbull

bull

bull

bull

bull

bull

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bull

bull

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bull

bull

bull

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bull

bull

bull

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bullbull

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bull

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bull

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bull

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bull

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bull bull

bull

bull

bull bullbullbull

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bull

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bull

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bull

bull

bull

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bull

bull

bull

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bull

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bull

bull

bull bull

bullbull

bull bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

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bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

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bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

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bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

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bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

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bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

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bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

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bull

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bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

bull

bullbull

bull

bull

bull

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bull

bull

bullbull

bullbull

bull

bull

bull

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bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Nott amp Dunsmuire 2002 Biometrika Fig 2 Sydney wind pattern data Contours of equal estimated correlation with two different fixed sites shown by open squares (a) location 33middot85degS 151middot22degE and (b) location 33middot74degS 149middot88degE The sites marked by dots show locations of the 45 monitored sites

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bullbull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bullbullbullbull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull bullbull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bullbull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbullbull

bull

bullbull

bull

bull

bull

bull bull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bullbullbull bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull bullbull bullbull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bullbull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Kim Mallock amp Holmes JASA 2005Piecewise Gaussian model for groundwater

permeability data

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbullbull

bull

bullbull

bull

bull

bull

bull bull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bullbullbull bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull bullbull bullbull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bullbull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Deformation-based Nonstationary covariance models

bull P Guttorp and P D Sampson (1994) Methods for estimating heterogeneous spatial covariance functions with environmental applications In G P Patil C R Rao (editors) Handbook of Statistics XII Environmental Statistics 663-690 New York North HollandElsevier bull W Meiring P Guttorp and P D Sampson (1998) Space-time Estimation of Grid-cell Hourly Ozone Levels for Assessment of a Deterministic Model Environmental and Ecological Statistics 5 197-222 bull PD Sampson (2001) Spatial Covariance In Encyclopedia of Environmetricsbull PD Sampson D Damian and P Guttorp (2001) Advances in Modeling and Inference for Environmental Processes with Nonstationary Spatial Covariance In GeoENV 2000 Geostatistics for Environmental Applications P Monestiez D Allard R Froidevaux eds Dordrecht Kluwer pp 17-32bull PD Sampson D Damian P Guttorp and DM Holland (2001) Deformationmdashbased nonstationary spatial covariance modelling and network design In Spatio-Temporal Modelling of Environmental Processes Coleccioacute laquoTreballs DrsquoInformagravetica I Tecnologiaraquo Nuacutem 10 J Mateu and F Montes eds Castellon Spain Universitat Jaume I pp 125-132bull D Damian PD Sampson and P Guttorp (2003) Variance Modeling for Nonstationary Spatial Processes with Temporal Replications Journal of Geophysical Research ndash Atmosphere 108 (D24) bull F Bruno P Guttorp PD Sampson amp D Cocchi (2004) Non-separability of space-time covariance models in environmental studies In The ISI International Conference on Environmental Statistics and Health conference proceedings (Santiago de Compostela July 16-18 2003) a cura di Jorge Mateu David Holland Wenceslao Gonzaacutelez-Manteiga Universidade de Santiago de Compostela Santiago de Compostela 2003 pp 153-161bull John Kent Statistical Methodology for Deformations

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

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D-plane distance

Dis

pers

ion

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00

05

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15

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25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Geometric anisotropy

bull Recall that if we have an isotropic covariance (circular isocorrelation curves)

bull If for a linear transformation A we have geometric anisotropy (elliptical isocorrelation curves)

bull General nonstationary correlation structures are typically locally geometrically anisotropic

( ) ( )C x y C x y= minus

( ) ( )C x y C Ax Ay= minus

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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distance

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0 500 1000 1500 2000 2500 3000

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distance

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0 500 1000 1500 2000

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bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

The deformation idea

In the geometric anisotropic case write

where f(x) = Ax This suggests using a general nonlinear transformation

G-plane rarr D-space

Usually d = 2 or 3We do not want f to fold

Remark Originally introduced as a multidimensional scaling problem find Euclidean representation with intersitedistances monotone in spatial dispersion D(xy)

( ) ( ( ) ( ) )C x y C f x f y= minus

2 df R Rrarr

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

bull

bull

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distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

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10 bull

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distance

co

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latio

n

0 500 1000 1500 2000

04

06

08

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Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

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7273

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(b)

5500 6000 6500 7000 7500 8000 8500 900017

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(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

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observed

pre

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observed

pre

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observed

pre

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4

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observed

pre

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4

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observed

pre

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4

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observed

pre

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02

4

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observed

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observed

pre

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ted

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02

4

53

California ozone

12

3

4

5

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89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

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30

31

32

33

34

35

36

37

3839

40

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43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

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9

10

1112

13

14

15

16

1718

19

20

21

22

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25

26

27

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31

32

3334

35

36

37

38

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43

44

4546

47

48

49

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5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Space-time Model with Spatial DeformationDamian et al 2000 (Environmetrics) 2003 (JGR)

( ) ( ) ( ) ( ) ( )1 2 tZ x t x t x H x x tmicro ν ε= + +

( ) spatio-temporal trendparametric in time mv spatial process

x tmicro

( ) temporal variance at log-normal spatial process

x xν

2( )

(0 ) ( )msmt error and short-scale variation

independent of t

x tN H xε

εσ

( )( ) ( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xC x y H x H y rarr= rarr

2

( ) ( ) ( )( ( ) ( ))( )

Cov x y C x y x yZ x t Z y tx x yε

ν νν σ

ne=

+ =

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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bull bull

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bull bullbull bullbull

bull

bull

bullbull

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bull

bull

bull

bull

bull

bull

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bullbull

bull

bull

bull

bull

distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

08

10 bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull bull

bull

bull

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bullbull

bullbull

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bullbull

bullbull

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bullbull

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bullbull

bull

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bullbull

bull

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bull

bull

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bull

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bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

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bull bullbull bull

bull

bullbull

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bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bullbull

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bull

bull

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bull

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bull bull

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bull bull

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bull

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bull

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bull bullbull bull

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bull

bull

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bull

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bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

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bull

bull

bullbull

bullbull

bullbull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

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bull

bull

bull

bull

bull

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bull

bull bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull bullbull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bullbull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bullbull

bull

bullbull

bull

bull

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bull

bull

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bull

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bullbullbull bullbull

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bull bullbull bullbull

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bullbullbull

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bull

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bull

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bull

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bull

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bull

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bull

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bull bullbull

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bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

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bullbullbull

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bull

bull

bull

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bull

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bullbullbull

bull

bull

bullbull

bull

bull

bull

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bullbull

bull

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bull

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bull

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bull

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bull

bull

bull

bull

bull

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bull

bull

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bull

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bull

bull

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bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

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bull

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bull

bull

bull

bull

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bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

bullbull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

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bull

bull

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bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

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bull

bull

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bull

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bull

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bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

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bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

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bull

bull

bull

bull

bull

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bullbull

bull

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bull

bull

bull

bull

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bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

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bull

bull bullbull

bull

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bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

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bullbullbull

bull

bull

bull

bull

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bull

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bull

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bull

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bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

( )( ( ) ( )) 1

ndmean 0 var 1 2 -order cont spatial processCov

t

t t x y

H xH x H y rarrrarr

( ) ( )( ) ( ) ( ) ( )

( )

smooth bijective(Geographic Deformed plane)

isotropic correlation functionin a known parametric family(exponential power exp Matern)

Cor t t

f G D

H x H y f x

d

f y

θ

θ

ρ

ρ=

rarr

minus

rarr

ie The correlation structure of the spatial process is an (isotropic) function of Euclidean distances between site locations after a bijective transformation of the geographic coordinate system

Model (cont)

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbullbull

bull

bullbull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bullbullbull bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull bullbull bullbull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bullbull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

The spatial deformation f encodes the nonstationarity spatially varying local anisotropyWe model this in terms of observation sites as a pair of thin-plate splines

Model (cont)

1 2 Nx x xhellip

( ) ( )Tf x c x xσ= + +A W

c x+A

( )T xσW

( )( )

( )

1

N

x xx

x x

σσ

σ

minus = minus

( ) ( )2 log 0

0 0

h h hh

hσ

gt==

Linear part globallarge scale anisotropy 2 1 2 2 c times timesA

Non-linear part decomposable into components of varying spatial scale

2 1 ( ) N Nxσtimes timesW

2 2 f c εmicro θ σ ν microθ σAWrArr Model parameters

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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distance

corr

ela

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0 500 1000 1500 2000 2500 3000

04

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distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

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Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

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85

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00

1

29

3

5

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7

11

13

15

17

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27

3135

38

3943

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49 51

5557

59

6163

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7273

74

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(b)

5500 6000 6500 7000 7500 8000 8500 900017

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313538 39

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(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

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bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

ImplementationConsider observations at sites x1 xn Let

be the empirical covariance between sites xiand xj Minimize

where J(f) is a penalty for non-smooth transformations such as the bending energy

c A W

ˆijC

( )( )2

ˆ( ) ( ) ( ) ( )ij ij i ji j

f w C C f x f x J fθ θ λminus minus +sum

2 2 22 2 2

2 2( ) 2f f fJ f dxdyx x y y

part part partpart part part part

= + +

intint

When f is computed as a thin-plate spline the minimization above can be considered in terms of the deformed coordinates or the parameters of the analytic representation of the thin-plate spline

( )i if xξ =

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

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bull

bull

bull

bull bull

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bull

bull

bull

bull

bullbull

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bull

bull

bullbullbullbull

bullbull

bull

bull

bull

bullbull

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bull

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bull

bull

bull

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bull

bullbull

bull

bull bull

bull

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bull

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bull

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bull

bullbullbullbull

bullbullbull

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bull bullbull

bull

bullbull

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bullbull

bull

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bullbull

bull

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bull

bull

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bull

bull

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bull

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bull bull

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bullbullbull

bull

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bull

bull

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bull

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bull bull

bullbullbullbull

bull

bull

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bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

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bullbull

bull

bull

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bull

bullbull

bull

bull bull

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bull

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bull

bull

bullbull

bull

bull

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bull

bull

bull

bull

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bull bull

bull

bull

bull

bullbullbull

bullbull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbullbull

bull bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

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bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbullbullbull

bullbull

bull

bullbull

bullbull

bull

bull

bullbull

bullbull

bull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

bull

bullbull

bull

bull bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

08

10 bull

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distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

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01

85

00

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1

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7

11

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3135

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5557

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6163

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7273

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(b)

5500 6000 6500 7000 7500 8000 8500 900017

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313538 39

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(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

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observed

pre

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observed

pre

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4

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observed

pre

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02

4

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observed

pre

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02

4

25

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observed

pre

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02

4

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observed

pre

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02

4

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observed

pre

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observed

pre

dic

ted

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02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

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43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

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50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

More on the equations of the thin-plate spline

( ) 2 21 2( ) ( ) ( )

( ) 1 1 2

( ) )

1 0 0

minimizing bending energy subject to interpolation constraints

is an equation of the form

where the coefficients satisfy I

T

j i ij

T

T T

f x f x f x

f x i N j

f s c s s

ξ

σ

= rarr

= le le =

= + + (

= =

A W

W W X W

R R

1 2

1 2 1 21 1 0 0 0

1

0 1 0 00 0 0

e the columns and of are vectors in the subspace

spanned by

The system of equations for computation of a thin-plate spline is

N T T T

T

T

W W

X X v v v X v X= isin = = =

=

W

V

S XΞ

X

R

( ) ( ) tr( )

where is with elements

and the bending energy is

T

T

Tij i j

c N N

x x J fσ

times

= = =

W

S

A

S W SWΓ

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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distance

corr

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tion

0 500 1000 1500 2000 2500 3000

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bullbull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull bullbull bull

bull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bullbull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bullbullbullbull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull bullbull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bullbull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbullbull

bull

bullbull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bullbullbull bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull bullbull bullbull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bullbull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

SARMAP

An ozone monitoring exercise in California summer of 1990 collected data on some 130 sites

-150 -100 -50 0

100

150

200

250

6 7 8 9 10 11 12 13 14 15 16 17 18

28

29

30

31

32

33

34

35

36

37

38

39

Sites in this study

Centers of grid cells for photochemical model

Regular sub-grid of points within 1 grid cell

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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ela

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0 500 1000 1500 2000 2500 3000

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distance

co

rre

latio

n

0 500 1000 1500 2000

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Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

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1

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3

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11

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38

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(b)

5500 6000 6500 7000 7500 8000 8500 900017

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(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

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bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

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bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

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bull bull

bull

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bull bullbull

bull

bull

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bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

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observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Transformation

This is for hr 16 in the afternoon

-200 -100 0 50 100 150

010

020

030

040

0

1

2

3

4

5

6

7

8

9

10

11

1213

14

15

16

1718

1920212223

24

252627

28

29

30

31

32

D-plane distance

Dis

pers

ion

0 50 100 150 200 250 300

00

05

10

15

20

25

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

bull

bull

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bullbull

bullbull

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bull bullbull

bull

bull bull

bull

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distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

08

10 bull

bull

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distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

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(b)

5500 6000 6500 7000 7500 8000 8500 900017

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01

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1

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313538 39

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(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

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observed

pre

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4

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observed

pre

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02

4

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observed

pre

dic

ted

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02

4

22

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observed

pre

dic

ted

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02

4

25

bull bull

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observed

pre

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ted

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02

4

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observed

pre

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ted

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02

4

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observed

pre

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02

4

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observed

pre

dic

ted

-4 -2 0 2 4

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02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

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404142

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444546

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63 12

3

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67 89

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20

2122

2324

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63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Fig 7 Precipitation in Southern France -an example of a non-linear deformation

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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distance

corr

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tion

0 500 1000 1500 2000 2500 3000

04

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bullbull

bull

bullbullbull

bullbull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbullbull

bull

bullbull

bull

bull

bull

bull bull

bull

bull

bull

bull

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bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bullbullbull bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull bullbull bullbull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bullbull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

G-plane Equicorrelation Contours

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbullbullbull

bullbull

bull

bull

bull

bullbull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

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bull

bull

bull

bull

bull

bull

bull

bull

bullbull

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bull

bull

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bull

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bull

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bull

bull

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bull

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bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

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observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

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observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

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34

35

36

37

3839

40

4142

43

44 454647

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50

51

52

53

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58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

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50

5152

53

54

55

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58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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bull

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bull

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bull

bull

bull

bull

bull

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bull

bull

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bullbull

bull

bull

bull

bull

distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

08

10 bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull bull

bull

bull

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bullbull

bullbull

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bullbull

bullbull

bullbull

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bullbull

bullbull

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bullbull

bull

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bullbull

bull

bull

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bullbull

bull

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bull

bull

bull

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bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

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bull bullbull bull

bull

bullbull

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bull

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bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bullbull

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bull

bull

bull

bull

bull

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bullbull

bullbull

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bullbull

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bull bull

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bullbull

bull bull

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bull

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bull

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bull

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bull bullbull bull

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bull

bull

bull

bull

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bull

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bull

bull

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bull

bull

bull

bull

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bull

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bull

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bull

bullbull

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bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull bullbull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bullbull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bullbull

bull

bullbull

bull

bull

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bull

bull

bull

bull

bullbull

bull

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bull

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bull

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bull

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bull

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bull

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bullbullbull bullbull

bullbull

bull

bull

bullbullbull

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bull

bull

bull

bull

bullbull

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bull

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bull bullbull bullbull

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bullbullbull

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bullbull bull

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bull bullbull

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bull

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bull

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bull

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bull

bullbullbull

bull

bullbull

bull

bull

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bull

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bull

bull

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bull

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bull bull

bull

bull

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bull

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bull

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bull

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bull bull

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bull bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

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bull

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bullbullbull

bull

bull

bull

bull

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bullbullbull

bull

bull

bull

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bull

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bull

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bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

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bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

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bull

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bull

bull

bull

bull

bull

bull

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bull

bull

bullbull

bull

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bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

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bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

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bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

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bull

bull

bullbull

bull

bullbull

bull

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bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

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bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

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bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

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bullbull

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bull

bull

bull bullbull

bull

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bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

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bullbullbull

bull

bull

bull

bull

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bull

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bull

bull

bullbull

bull

bull

bullbullbull

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bull

bull

bull

bull

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bull

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bull

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bull

bull

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bullbull

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bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn

(2) is differentiable for ugt0then is unique up to a scaling for and a homothetic transformation for (rotation scaling reflection)

( )( ) ( ) ( ) ( ) n nD x y f x f y x y R Rγ= minus isin times1f minusf

( )uγ( )f γ

fγ

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

A Bayesian implementation

Likelihood

Nonlinear part Bending energy Prior

Linear part ndashfix two points in the G-D mapping ndashput a (proper) prior on the remaining two parameters

Posterior computed using Metropolis-Hastings

L(S | Σ) = (2π Σ )minus(Tminus1) 2 exp minusT2

trΣminus1S

p(W) prop exp minus1

2τWi

˜ S Wii=1

2sum

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

bull

bull

bull

bull

bull

bull

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bullbull

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bull bull

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bullbull

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bull

bullbullbullbull

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bullbull

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bull bull

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bullbull

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bull bull bullbull

bullbull

bullbullbull bullbull

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bull bull

bullbullbull

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bull bull

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bull

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bullbull

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bull

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bullbull

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bullbull

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bullbull

bull

bull

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bull

bull bull

bull

bull

bull

bull

bull

bull

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bull

bullbullbullbull

bullbullbull

bullbull

bullbull

bull

bull

bullbull

bullbull

bull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull bullbull

bull

bullbull

bull

bull

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bull

bullbull

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bullbull

bullbull

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bull

bullbull

bull

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bullbull

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bull bull

bullbull

bullbull

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bullbullbull

bull

bullbull

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bullbull

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bull

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bull

bullbull

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bull bull

bullbullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

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bull

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bull

bullbull

bull

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bullbull

bull

bullbull

bull

bull bull

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bullbull

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bullbull

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bull

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bullbull

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bull bull

bull

bull

bull

bullbullbull

bullbull

bull

bull

bullbull

bull bull

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bull

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bull

bull

bull

bull

bull

bull

bull

bull

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bullbullbullbull

bull bull

bullbull

bull

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bull

bull bull

bull

bull

bull

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bull

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bull

bull

bull

bull bull

bull

bull

bull

bullbullbullbull

bullbull

bull

bullbull

bullbull

bull

bull

bullbull

bullbull

bull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

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bullbull

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bull

bullbull

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bull

bull

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bull

bull

bull

bull

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bull

bull

bullbullbullbull

bull bull

bullbull

bull

bull

bull

bull

bull

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distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

08

10 bull

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distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

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1

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3

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7

11

13

15

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27

3135

38

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5557

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6163

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7273

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(b)

5500 6000 6500 7000 7500 8000 8500 900017

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313538 39

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(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

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observed

pre

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observed

pre

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4

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observed

pre

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02

4

22

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observed

pre

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ted

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02

4

25

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observed

pre

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02

4

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observed

pre

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02

4

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observed

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observed

pre

dic

ted

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02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

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12

3

4

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67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

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62

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3

4

5

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10

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16

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20

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22

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26

27

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36

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43

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54

55

56

57

58

5960 61

62

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3

4

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10

1112

1314

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16

1718

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20

21

22

2324

25

26

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2829

30

31

32

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404142

43

444546

47

48

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54

55

56

57

58

5960 61

62

63 12

3

4

5

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10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

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43

444546

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5152

53

54

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5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Likelihood given observation vectors Z1hellipZN of length T

with covariance matrix having elements

[ ]

( ) ( )

21

2 1 1

1 |( )

( 1

)2 exp tr

|

2 2

N

T

Nf Z Z

T T Z

Z Z

Z

Zεmicro θ ν σ

π micro

micro micro

microminus minus minus

= =

minus primeminus minus minus minus

=

=

Σ

Σ Σ

Σ

S Σ

Shellip hellipL

( )2

1 i j i jij

j

i ji j N

i jθ

ε

νν ρ ξ ξσ

ν σ

minus ne= le le+ =

Integrating out a flat prior on the (constant) mean

[ ]

[ ] [ ] ( )1 2 1

1

| ( 1)exp2

| d trT TZ

micro

micro micro micro minus minus minusminus

prop rArr

= prop minus intS Σ SΣ ΣS Σ

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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distance

corr

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0 500 1000 1500 2000 2500 3000

04

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bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

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bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbullbull

bull

bullbull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bullbullbull bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull bullbull bullbull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bullbull

bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bullbullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbullbull

bullbull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Posterior

[ ] [ ][ ][ ]

2

1

2 2

12

2

2 2

1( )

1exp (log ) (

log )2

Log-normal variance

Full posterior is

fiel

d

ε

ε

ν

θ σ ν micro σ θ

θ σ ν micro σ θ

ν micro σ θ

ν micro ν micro

micro σ

minus minus

prop

prop prime

primeminus minus sdot minus sdot

sdot

prod

AW A

AW

Σ

AW S

S

Σ

Σ

1

W

1ix

c

c

c

c

f

θ

[ ]

[ ] 1 1 2 2

( )

1exp ( )2

diffuse normal prior on 2 free linear params (4 constr)

ij i jx x

c

I στ isin times

prime primeprop minusminus +

=W V V

A

W WS SW W SW

the bending energy prior on space orthogonal to linear

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbullbullbull

bullbull

bull

bull

bull

bullbull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull bullbull

bullbull

bullbullbull bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

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bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

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bull bull

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bull

bull

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observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

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50

51

52

53

54

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56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Summary of prior distributions

[ ] ( )

1

2

11 1 2 22

1

2

2

0 1 0

0 0 2

exp

exp( )(0 2) --- (if power exponential)

exp(ε

τ

θθ

σ

isin times

=

prop minus + W V V

Deformation parameters

W S S I

Correlation parameters

Variance parameters

sim

simsim

sim

T T

a s aa N

a s a

W W W W

pthetaU ptheta

)pnugget

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

08

10 bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

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bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

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bull

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bull

bull

bull

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bull

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bull

bull

bull

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bull

bull

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bull

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bull

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bull

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bull

bull bull

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bull

bull bullbull

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bull bull

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bull

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bull

bull

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bull

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bull

bull

bull

bull

bull

bullbullbull

bull

bull bull bullbull

bull

bull

bull

bull

bull

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bull

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bull

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bull

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bull

bull

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bull bullbull

bullbullbull bullbull

bullbull

bull

bull

bullbullbull

bull

bull

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bull

bull

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bull

bull

bull

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bull bullbull bullbull

bull

bull

bull

bull

bullbullbull

bull

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bull

bullbull bull

bull

bull

bull

bull

bull

bull

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bull

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bullbull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

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bull

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bull

bull

bull

bull

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bull

bullbullbull

bull

bullbull

bull

bull

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bull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

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bull

bull

bullbullbull

bullbull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

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bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

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bull

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bull

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bull

bull

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bull

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bull

bull

bull

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bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

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bull bull

bullbull

bull bullbull

bullbull

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bull

bull

bull

bull

bullbull

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bullbull

bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

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bull

bull bull

bullbullbull

bull

bull

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bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

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bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

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bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

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bullbull

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bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

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bull

bull

bullbull

bull

bull

bull

bull

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bullbull

bullbull

bull

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bull

bull

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bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Summary or prior distributions (cont)

[ ] ( ) ( )

( )( )

12 1

2

1

2

2

1 exp log( ) 1) log( ) 1)

is with elements ( ) ( )

( ) exp

( )exp( )

( 2)

T TN

ii

ij i jN N f x f x

d d

palpha pbetapthetat

N pmu psigma

θ

θ

ν micro ν microν

σ σ ρ

ρ θ

σ

θmicro

minus

=

prop minus minus sdot minus sdot

times = minus

= minus

Γ

Variance parameters (cont)-1ν Σ Σ

Σ

simsimsim

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

bull

bull

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bull

bull

bullbullbullbull

bullbull

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bull

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bull

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bull

bull

bull bullbullbull

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bull bull bullbull

bullbull

bullbullbull bullbull

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bull bull

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bull bull

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bullbull

bullbull

bull

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bull

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bull bull

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bull

bullbullbullbull

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bullbull

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observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

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33

34

35

36

37

3839

40

4142

43

44 454647

48

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50

51

52

53

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57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

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48

49

50

5152

53

54

55

56

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58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Computation

Metropolis-Hastings algorithm for sampling from the highly multidimensional posterior

Given estimates of D-plane locations f(xi) the transformation is extrapolated to the whole domain using thin-plate splines (Visualization and diagnostics)

Predictive distributions for

(a) temporal variance at unobserved sites

(b) the spatial covariance for pairs of observed andor unobserved sites

(c) the observation process at unobserved sites

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

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10 bull

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bull bull bullbull

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bull

bull

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bull

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bull

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distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

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bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

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bull

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bull

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bull

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bull

bull

bull

bull

bull

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bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

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bull

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bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

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bull

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bull bull

bull

bull

bull

bull

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bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

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bull

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bull

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bull

bull

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bull

bull

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bull

bull

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bull

bullbull

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bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

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bull

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observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

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observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

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observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Application to Languedoc-Roussillon Precipitation Data

108 altitude-adjusted 10-day aggregate preciprecords at 39 sites (Nov-Dec 1975-1992)Data log-transformed and site-specific means removed (for this analysis)Estimated deformation is non-linear correlation stronger in the NE region weaker in the SW

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

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10 bull

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distance

co

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0 500 1000 1500 2000

04

06

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Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

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Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

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observed

pre

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observed

pre

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observed

pre

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observed

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observed

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observed

pre

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4

53

California ozone

12

3

4

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10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

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30

31

32

33

34

35

36

37

3839

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43

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48

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52

53

54

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57

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5960 61

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9

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14

15

16

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20

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26

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31

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3334

35

36

37

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43

44

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47

48

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5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Languedoc-Roussillon Precipitation Sites

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

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bull

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bull

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bull

bull

bull

bull

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bull

bull

bull

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bull

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bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull bullbullbull

bull

bull

bull

bull

bull

bull

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bull

bull

bull

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bull

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bull

bull

bull

bull

bull

bull

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bull

bull bull

bullbull

bull bullbull

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bull

bull

bull

bull

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bull

bull

bull

bull

distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

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bull

bull bull

bullbullbull

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bull

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bull

bull

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bull

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bull

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bull

bull

bull

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bull bullbullbullbull

bullbull

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bull

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bull

bull

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bull

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bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

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bull bull

bull

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bull

bull

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bull

bull

bull

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bull

bull

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bull

bull

bull

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bull

bull

bull

bull

bull

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bull

bull

bull

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bull

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bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

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bull

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bull

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bull

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bullbull

bullbullbull

bull

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bull

bull

bull

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bull

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bullbull

bull

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bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

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bull bullbull

bull

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bull

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bull

bull

bull

bullbull

bull

bull

bull

bull

bull

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bull

bull

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bull

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bull

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bull

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bull bullbull

bull

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bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

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bullbull

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bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

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bullbullbull

bull

bull

bull

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bull

bull

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bull bull

bull

bull

bull

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bull

bull

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bull

bull

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bull

bull

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bull

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bull

bull

bull

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bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

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bull

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bull

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bull

bull

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bull

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observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

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bull

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observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Estimated deformation of Languedoc-Roussillon region

(a)

9

19

22

25

33

41

4553

(b)

9

1922

25

33

41

45

53

Circled monitoring sites are reserved for model validation

Correlation vs Distance in G-plane and D-plane

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distance

corr

ela

tion

0 500 1000 1500 2000 2500 3000

04

06

08

10 bull

bull

bull

bull

bull

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distance

co

rre

latio

n

0 500 1000 1500 2000

04

06

08

10

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

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1

29

3

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6

7

11

13

15

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2123

27

3135

38

3943

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49 51

5557

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6163

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7273

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(b)

5500 6000 6500 7000 7500 8000 8500 900017

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1

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313538 39

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(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

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observed

pre

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02

4

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observed

pre

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4

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observed

pre

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ted

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02

4

22

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observed

pre

dic

ted

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02

4

25

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observed

pre

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ted

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02

4

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observed

pre

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02

4

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observed

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observed

pre

dic

ted

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02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

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12

3

4

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67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

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43

44

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47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

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43

444546

47

48

49

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53

54

55

56

57

58

5960 61

62

63 12

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4

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67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

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30

31

32

3334

35

36

37

38

39

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43

444546

47

48

49

50

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53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Correlation vs Distance in G-plane and D-plane

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co

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0 500 1000 1500 2000

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Equi-correlation (09) contours D-plane (a) and G-plane (b)

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(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

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bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Equi-correlation (09) contours D-plane (a) and G-plane (b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

11

13

15

17

2123

27

3135

38

3943

47

49 51

5557

59

6163

71

7273

74

75

(b)

5500 6000 6500 7000 7500 8000 8500 900017

00

01

75

00

18

00

01

85

00

19

00

01

95

00

1

29

3

5

6

7

1113

15

17

212327

313538 39

43

47

49

51

5557

59

61

63

71

7273

74

75

(a)

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Estimated (bull) and predicted () variances vs observed temporal variances with one predictive std dev bars

0( )xν

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Assessment of (10-day aggregate) precipitation predictions at validation sites

bullbull

bull

bull

bull

bull

bullbull

bull

bull bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull bullbullbullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bullbull bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

9

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

19

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbullbull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

22

bullbull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull bullbull

bull

bull

bullbull

bull

bullbull

bull bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

25

bull bull

bullbull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bullbullbull

bullbullbull

bullbull

bullbull

bullbull

bullbull

bull

bull

bullbullbull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bull bull

bull

bull

bullbull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

33

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bullbull

bullbullbull

bull

bull

bull

bull

bullbullbull

bull

bullbull

bullbull

bullbull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bullbull

bull

bull

bull

bull

bullbull

bullbull

bullbull

bull bull

bull

bull

bull

bull

bull

bull

bull bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

41

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bull bullbull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

45

bull bull

bull bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull bull

bull

bull

bull

bull

bullbullbull

bull

bull

bull

bull

bullbull

bull

bull

bullbull

bullbull

bullbull

bull

bull

bullbull

bull

bull

bullbullbull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bullbull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bullbull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

bull

observed

pre

dic

ted

-4 -2 0 2 4

-4-2

02

4

53

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

California ozone

12

3

4

5

67

89

10

1112

1314

15

16

1718

19

20

21

22

23

24

25

26

27

2829

30

31

32

33

34

35

36

37

3839

40

4142

43

44 454647

48

49

50

51

52

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 8

9

10

1112

13

14

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

44

4546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

63 Region 6 monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Oct 30 001236 PST 2003

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Region 6 S Calif

Geographic Distance (km)

Cor

rela

tion

0 100 200 300 400 500

00

02

04

06

08

10

Region 6 S Calif

D-plane Distance

Cor

rela

tion

0 100 200 300 400

00

02

04

06

08

10

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Posterior samples

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 8

9

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

12

3

4

5

67 89

10

1112

1314

15

16

1718

19

20

21

22

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63 12

3

4

5

67 89

10

1112

1314

15

16

1718 19

20

2122

2324

25

26

27

2829

30

31

32

3334

35

36

37

38

39

404142

43

444546

47

48

49

50

5152

53

54

55

56

57

58

5960 61

62

63

N=63 S Calif 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Oct 28 221829 PST 2003

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Other approachesHaas 1990 Moving window krigingNott amp Dunsmuir 2002 Biometrikamdash

computationally convenient but hellipHigdon amp Swall 1998 2000 Gaussian

moving averages or ldquoprocess convolutionrdquo model

Fuentes 2002 Kernel averaging of orthogonal locally stationary processes

Kim Mallock amp Holmes 2005 Piecewise Gaussian modeling

Pintore amp Holmes 2005 Fourier and Karhunen-Loeve expansions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Gaussian moving averages

Higdon (1998) Swall (2000)Let ξ be a Brownian motion without drift and This is a Gaussian process with correlogram

Account for nonstationarity by letting the kernel b vary with location

X(s) = b(s minus u)dξ(u)R2int

ρ(d) = b(u)R2int b(u minus d)du

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

DetailsFor Gaussian kernels one can show that the nonstationary covariance takes the simple form

1 21 4 1 42( ) exp( )

2i jNS

i j i i ijC x x QσminusΣ + Σ

= Σ Σ minus

where1

( ) ( )2

i jTij i j i jQ x x x x

minusΣ + Σ = minus minus

And where the kernel matrix is the covariance matrix of the Gaussian kernel centered at

( )i ixΣ = Σ

ix

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Paciorek amp Schervish thm

If an isotropic correlation function is positive definite on for every p = 1 2 then the function

( )sR τ

1 21 4 1 4( ) ( )

2i jNS S

i j i i ijR x x R QminusΣ + Σ

= Σ Σ

pR

is a nonstationary correlation function

The authors use a Matern correlation function The challenge is specifying and estimating a field of smoothly varying kernels as a Gaussian process

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Kernel averagingFuentes (2000) Introduce orthogonal local stationary processes Zk(s) k=1K defined on disjoint subregions Sk and construct

where wk(s) is a weight function related to dist(sSk) Then

A continuous version has

Z(s) = wk (s)Zk (s)k= 1

Ksum

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk= 1

Ksum (s1 minus s2 )

Z(s) = w(x minus s)Zθ (s )int (x)ds

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

Desroziers 1997 A coordinate change for data assimilation in spherical geometry of frontal structures Monthly Weather Review

The main impact of this transformation in the framework of data assimilation is that it enables the use of anisotropic forecast correlations that are flow dependent

Riishojgaard 1998 A direct way of specifying flow-dependent background correlations for meteorological analysis systems Tellus

Weaver and Courtier 2001 Correlation modelling on the sphere using a generalized diffusion equation Quar J Royal Met Soc

Generalization to account for anisotropic correlations are also possible by stretching andor rotating thecomputational coordinates via a lsquodiffusionrsquo tensor

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Some recent atmospheric science literature and proposals for spatio-temporal covariance models

(cont)

Wu et al 2002 3-D variational analysis with spatially inhomogeneous covariances Monthly Weather Review

Purser et al 2003 Numerical aspects of the application of recursive filters to variational statistical analysis Part II Spatially inhomogeneous and anisotropic general covariances Monthly Weather Review

Fu et al 2004 Ocean data assimilation with background error covariance derived from OGCM outputs Advances in Atmospheric Sciences

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Incorporating covariates

bull Carroll and Cressie 1997 geomorphic site attributes in correlation model for snow water equivalent in river basins

1 2 1 2( ) exp( ) c d e fc s s B s s CX DX EX FX= minus minus minus minus minus minus

Where Xrsquos represent differences between the two sites in elevation slope tree cover aspect

Alternative deform R2 into subspace of R6

bull Riishojgaard 1998 ldquoflow-dependentrdquo correlation structures for meteorological analysis systems For z(s)a realization of a random field in Rd

( ) ( ) ( )1 2 1 2 1 1 2 ( ) ( )dc s s s s z s z sϕ ϕ= minus sdot minus

an embedding and deformation of the geographic coordinate space Rd into Rd+1 with a separable stationary correlation model fitted in new coordinate space

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions

Covariance models for dynamic error structures in the context of data assimilation

bull Cox and Isham 1988 with v a velocity vector in R2 a physical model for rainfall leads to space-time covariance function

( )1 2 1 2 2 1 2 1( ) ( ) ( )c s s t t E G s s t t= minus minus minusV V

where G(r) denotes area of intersection of two disks of unit radius with centers a distance r apart

There are variants in the meteorological and hydrological literature depending on tangent line in a barotropicmodel using geostrophic or semigeostropic coordinates or working in a Lagrangian reference frame for convective rainstorms These yield interesting anisotropic and nonstationary correlation models (cf Desroziers 1997) They suggest interesting space-time extensions of current deformation approach and statistical model fitting questions