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STAT 592A(UW) 526 (UBC-V) 890-4(SFU)

Spatial Statistical Methods

[email protected]/peter/592

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Course content1. Kriging 1. Gaussian regression 2. Simple kriging 3. Ordinary and universal kriging 4. Effect of estimated covariance 5. Bayesian kriging

2. Spatial covariance 1. Isotropic covariance in R2

2. Covariance families 3. Parametric estimation 4. Nonparametric models 5. Fourier analysis 6. Covariance on a sphere

3. Nonstationary structures I: deformations1. Linear deformations2. Thin-plate splines3. Classical estimation4. Bayesian estimation5. Other deformations

4. Markov random fields1. The Markov property2. Hammersley-Clifford3. Ising model4. Gaussian MRF5. Conditional autoregression6. The non-lattice case

5. Nonstationary structures II: linear combinations etc.

1.Moving window kriging2.Integrated white noise3.Spectral methods4.Testing for nonstationarity

5. Wavelet methods

6. Space-time models1.Mean surface2.Separability3.A simple non-separable model4.Stationary space-time processes5.Space-time covariance models6.Testing for separability7.The Le-Zidek approach

7. Statistics, data and deterministic models

1. The kriging approach2. Bayesian hierarchical models3. Bayesian melding4. Data assimilation5. Model approximation

8. Statistics of compositions1. An algebra for compositions2. The logistic normal distribution3. Source apportionment4. Analysis of variance5. A space-time model

9. Wavelet tools1. Basic wavelet theory2. Multiscale analysis 3. Longterm memory models

4. Wavelet analysis of trends

10. Setting air quality standards1.Bayesian model averaging2.Standards as hypothesis tests3.Potential network bias4.Maxima of spatial processes5.Operational evaluation of air quality standards

Programs

RgeoR

fields

spBayes

RandomFields

GMRFLib: a C-library for fast and exact simulation of Gaussian Markov random fields

Course requirements

Submit at least 8 homework problems

(3 can be replaced by an approved project)

Submit at least three lab reports

Every other Thursday will be a lab day.

Virtual lab machine (get Remote Desktop Connection)

Office hours

MTh 10-11 B213 Padelford

Skype name: guttorp

Homework solutions and lab reports must be submitted electronically

Kriging

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Research goals in environmental research

Calculate pollution fields for health effect studies

Assess deterministic models against data

Interpret and set environmental standards

Improve understanding of complicated systems

The geostatistical model

Gaussian process(s)=EZ(s) Var Z(s) < ∞Z is strictly stationary if

Z is weakly stationary if

Z is isotropic if weakly stationary and

Z(s),s ∈D⊆R2

(Z(s1),...,Z(sk))=d(Z(s1+h),...,Z(sk +h))

μ(s)≡μ Cov(Z(s1),Z(s2))=C(s1−s2)

C(s1−s2)=C0(s1−s2 )

The problem

Given observations at n locations

Z(s1),...,Z(sn)

estimate

Z(s0) (the process at an unobserved location)

(an average of the process)

In the environmental context often time series of observations at the locations.

Z(s)dν(s)A∫or

Some history

Regression (Bravais, Galton, Bartlett)

Mining engineers (Krige 1951, Matheron, 60s)

Spatial models (Whittle, 1954)

Forestry (Matérn, 1960)

Objective analysis (Gandin, 1961)

More recent work Cressie (1993), Stein (1999)

QuickTime™ and a decompressor

are needed to see this picture.

A Gaussian formula

If

then

€

X

Y

⎛

⎝ ⎜

⎞

⎠ ⎟~ N

μX

μY

⎛

⎝ ⎜

⎞

⎠ ⎟,

ΣXX ΣXY

ΣYX ΣYY

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟

€

(Y |X) ~ N(μY + ΣYXΣXX−1 (X −μX ),

ΣYY − ΣYXΣXX−1 ΣXY )

Simple kriging

Let X = (Z(s1),...,Z(sn))T, Y = Z(s0), so thatX=1n, Y=,

XX=[C(si-sj)], YY=C(0), and

YX=[C(si-s0)].

Then

This is the best unbiased linear predictor when and C are known (simple kriging).

The prediction variance is

p(X) ≡Z(s0 ) = + C(s i −s0 )[ ]T

C(s i −s j )⎡⎣ ⎤⎦−1

X−1n( )

m1 =C(0) − C(s i −s0 )[ ]T

C(s i −s j )⎡⎣ ⎤⎦−1

C(s i −s0 )[ ]

Some variants

Ordinary kriging (unknown )

where

Universal kriging ( (s)=A(s) for some spatial variable A)

Still optimal for known C.

p(X) ≡Z(s0 ) = + C(s i −s0 )[ ]T

C(s i −s j )⎡⎣ ⎤⎦−1

X−1n( )

= 1nT C(s i −s j )⎡⎣ ⎤⎦

−11n( )

−1

1nT C(s i −s j )⎡⎣ ⎤⎦

−1X

=( A(s i )[ ]T

C(s i −s j )⎡⎣ ⎤⎦−1

A(s i )[ ])−1

A(s i )[ ]T

C(s i −s j )⎡⎣ ⎤⎦−1

X

Universal kriging variance

E Z(s0 ) −Z(s0 )( )2=m1 +

A(s0 ) −[A(s i )T [C(s i −s j )]

−1[C(s i −s0 )]( )T

×( A(s i )[ ]T

C(s i −s j )⎡⎣ ⎤⎦−1

A(s i )[ ])−1

× A(s0 ) −[A(s i )T [C(s i −s j )]

−1[C(s i −s0 )]( )

simple kriging variance

variability due to estimating

Some other kriging variants

Indicator kriging

Block kriging

Co-krigingU

sing a covariate to improve kriging

Disjunctive krigingA

nonlinear version of kriging: expand the

field into CONS and co-krige these

1(Z(s0 ) > c)

Z(s)dsA∫

The (semi)variogram

Intrinsic stationarity

Weaker assumption (C(0) needs not exist)

Kriging predictions can be expressed in terms of the variogram instead of the covariance.

γ( h ) =1

2Var(Z(s + h) − Z(s)) = C(0) − C( h )

Ordinary kriging

where

and kriging variance

)Z(s0 ) = λ iZ(s i )

i=1

n

∑

λT = γ + 11− 1T Γ−1γ

1T Γ−11

⎛

⎝⎜⎞

⎠⎟

T

Γ−1

γ = (γ(s0 − s1),...,γ(s0 − sn ))T

Γ ij = γ(si − s j )

m1(s0 ) =2 λ iγ(s0 −s i )i=1

n

∑ − λ iλ jγ(s i −s j )j=1

n

∑i=1

n

∑

An example

Ozone data from NE USA (median of daily one hour maxima June–August 1974, ppb)

Fitted variogram

€

γ(t) = σe2 + σs

2 1− e− t

θ ⎛ ⎝ ⎜

⎞ ⎠ ⎟

Kriging surface

Kriging standard error

A better combination

Effect of estimated covariance structure

The usual geostatistical method is to consider the covariance known. When it is estimated

• the predictor is not linear

• nor is it optimal

• the “plug-in” estimate of the variability often has too low mean

Let . Is a good estimate of m2() ?

p2 (X) =p(X; (X))

m1((X))

m2 () =E (p2 (X) −)2 m1()

Some results

1. Under Gaussianity, m2 ≥ m1( with equality iff p2(X)=p(X;) a.s.2. Under Gaussianity, if is sufficient, and if the covariance is linear in then

3. An unbiased estimator of m2( is

where is an unbiased estimator of m1().

Em1() =m2 () −2(m2 () −m1())

2m−m1()m

Better prediction variance estimator

(Zimmerman and Cressie, 1992)

(Taylor expansion)

(often approx. unbiased)

A Bayesian prediction analysis takes account of all sources of variability (Le and Zidek, 1992)

Var(Z(s0; )) ≈m1()

+2 tr cov() ⋅cov(∇Z(s0; )⎡⎣

⎤⎦

m2 () ≈m1() + tr Cov ()∂p(X; )

∂⎡⎣⎢

⎤⎦⎥

⎛⎝⎜

⎞⎠⎟

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