Spatio-Temporal StatisticsNoel Cressie *

Program in Spatial Statistics and Environmental Statistics

The Ohio State University

Christopher K. Wikle

Department of Statistics

University of Missouri, Columbia

Slides are based on the book, “Statistics for Spatio-Temporal Data”

by Cressie and Wikle, 2011, Wiley, Hoboken, NJ

*ncressie@stat.osu.edu

– p.1/26

Spatio-Temporal Statistics

There is no history without geography (and v.v.). We consider space and timetogether

The dynamical evolution (time dimension) of spatial processes means that we areable to reach more forecefully for the “Why” question. The problems are clearestwhen there is no aggregation ; henceforth consider processes at point-levelsupport

Consider the deterministic 1-D space × time , reaction-diffusion equation:

∂Y (s; t)

∂t= β

∂2Y (s; t)

∂s2− αY (s; t) ;

β is the diffusion coefficient

– p.2/26

Reaction-Diffusion Plots

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e

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Y (s, 0) = I(15 ≤ s ≤ 24)

(a) α = 1, β = 20; (b) α = 0.05, β = 0.05; (c) α = 1, β = 50

– p.3/26

Stochastic Version

Consider the stochastic PDE:

∂Y

∂t− β

∂2Y

∂s2+ αY = δ ,

where {δ(s; t) : s ∈ R, t ≥ 0} is a zero-mean random process. Here we assume whitenoise for δ:

E(δ(s; t)) ≡ 0

cov(δ(s; t), δ(u; r)) = σ2I(s = u, t = r)

– p.4/26

Stochastic Reaction-Diffusion Plots

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Y (s, 0) = I(15 ≤ s ≤ 24)

α = 1 , β = 20

(a) σ = 0.01; (b) σ = 0.1; (c) σ = 1

– p.5/26

Spatio-Temporal Covariance Function

The stochastic reaction-diffusion equation implies a (stationary in space and time)covariance function :

C(h; τ) ≡ cov(Y (s; t), Y (s + h; t + τ))

and correlation function :

ρ(h; τ) ≡ C(h; τ)/C(0; 0)

Heine (1955) Biometrika, gives a closed-form solution for ρ(·; ·)

– p.6/26

Contour Plot of Spatio-Temporal Correlation Function

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τ

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ρ(h; τ) for the stochastic reaction-diffusion equation

– p.7/26

Separability of Spatio-Temporal Covariance Functions

Stochastic PDEs are built from dynamical physical considerations and they implycovariance functions

Covariance functions have to be positive-definite (p-d) . So, specifying classesof spatio-temporal covariance functions to describe the dependence inspatio-temporal data is not all that easy

Suppose the spatial C(1)(h) is p-d and the temporal C(2)(τ) is p-d. Then theseparable class:

C(h; τ) ≡ C(1)(h) · C(2)(τ)

is guaranteed to be p-d

Separability is unusual in dynamical models ; it says that temporal evolutionproceeds independently at each spatial location

– p.8/26

Stochastic Reaction-Diffusion and Separability

If C(h; τ) = C(1)(h) · C(2)(τ),then

C(h; 0) = C(1)(h)C(2)(0)

C(0; τ) = C(1)(0)C(2)(τ) ,

and henceρ(h; τ) =

C(1)(h) · C(2)(τ)

C(0; 0)

=C(h; 0) · C(0; τ)

C(0; 0) · C(0; 0)

= ρ(h; 0) · ρ(0; τ)

Is this true for the stochastic reaction-diffusion equation? Plot

ρ(h; 0) · ρ(0; τ) versus (h, τ)

ρ(h; τ) versus (h, τ)

– p.9/26

Contour Plots of Spatio-Temporal Correlation Functions

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(a) ρ(h; 0) · ρ(0; τ); (b) ρ(h; τ)

The difference in correlation functions is striking . Hence ρ(·; ·) is not separable .Can we see the difference between separability and non-separability in theirrealizations ?

– p.10/26

Non-Separable Realizations in Space-Time

Three realizations of Y (s; t)

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Realizations are generated from a stationary Gaussian process with the

non-separable , reaction-diffusion correlation function, ρ(h; τ)

– p.11/26

Separable Realizations in Space-Time

Three realizations of Y (s; t)

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Realizations are generated from a stationary Gaussian process with separablecorrelation function, ρ(h; 0) · ρ(0; τ)

– p.12/26

Inference on a Hidden Spatio-Temporal Process

We could ignore the dynamics and treat time as another “spatial” dimension. Writethe data as:

Z = (Z(s1; t1), . . . , Z(sn; tn))′ ,

which are observations taken at known space-time “locations”. The data are noisyand not observed at all locations of interest

Assume a hidden (“true”) process {Y (s; t) : s ∈ D ⊂ Rd ; t ≥ 0}, which is not

observable due to measurement error and missingness. Write

Z = Y + ε ,

where E(ε) = 0, var(ε) = σ2εI. We wish to predict Y (s0; t0) from data Z

– p.13/26

Spatio-Temporal Kriging

Predict Y (s0; t0) with the linear predictor λ′Z:

For simplicity, assume E(Y (s; t)) ≡ 0. Then minimize w.r.t. λ, the mean squaredprediction error ,

E(Y (s0; t0) − λ′Z)2 .

This results in the simple kriging predictor :

bY (s0; t0) = c(s0; t0)′Σ−1Z Z ,

where ΣZ ≡ var(Z) and c(s0; t0) = cov(Y (s0; t0),Z)

The simple kriging standard error (s.e.) is:

σk(s0; t0) = {var(Y (s0; t0)) − c(s0; t0)′Σ−1

Z c(s0; t0)}1/2

– p.14/26

Kriging for the Stochastic Reaction-Diffusion Equation

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(a) Full realization; ε = 0

(b) Crosses show {(si; ti)} superimposed on the kriging predictormap, {bY (s0; t0)}

(c) Kriging s.e. map, {σk(s0; t0)}

– p.15/26

Kriging for the Stochastic Reaction-Diffusion Equation, ctd.

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(a) Same full realization; ε = 0

(b) Crosses show different {(si; ti)} superimposed on the kriging predictormap, {bY (s0; t0)}

(c) Kriging s.e. map, {σk(s0; t0)}

– p.16/26

Emphasize the Dynamics

Approximate the differentials in the reaction-diffusion equation:

∂Y

∂t= β

∂2Y

∂s2− αY

with differences :

Y (s; t + ∆t) − Y (s; t)

∆t= β

Y (s + ∆s; t) − 2Y (s; t) + Y (s − ∆s; t)

∆2s

ff− αY (s; t)

Define Yt ≡ (Y (∆s; t), . . . , Y (79 − ∆s; t);YBt ≡ (Y (0; t), Y (79; t))′. Then the

stochastic version of the difference equation is:

Yt+∆t= MYt + MBY

Bt + δt+∆t

,

where MBYBt represents given boundary effects. The difference equation is a good

approximation to the differential equation, provided α∆t < 1 and 2β∆t/∆2s < 1

– p.17/26

Emphasize the Dynamics, ctd.

Importantly,

M =

2666666666664

θ1 θ2 0 . . . 0

θ2 θ1 θ2 . . ....

0 θ2 θ1

. . ....

.... . .

. . . θ2

0 0 . . . θ2 θ1

3777777777775

,

where θ1 = (1 − α∆t − 2β∆t/∆2s), θ2 = β∆t/∆2

s . This can be viewed as thepropagator matrix of a VAR(1) process. The matrix is defined by the dynamics. In otherwords, in a model of spatio-temporal dependence, M has structure (and is sparse ).Conditional on the boundary effects, we see that the lagged covariances are given by,

C(m)Y = MmC

(0)Y ,

where C(m)Y ≡ cov(Yt,Yt+m∆t

); m = 0, 1, 2, . . .

– p.18/26

Comparison of Differential and Difference Equations

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ρ(h,

τ)

h = 0

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ρ(h,

τ)

h = 1

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ρ(h,

τ)

h = 2

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ρ(h,

τ)

h = 3

Spatio-temporal correlations; α = 1, β = 20, ∆s = 1, and ∆t = 0.01

Solid blue line : from differential equations

Red dots : from difference equations

– p.19/26

The Dynamics in the Difference Equation

Think of a spatial process at time t rather than a spatio-temporal process. Call it thevector Yt. Then describe the dynamics by a VAR(1):

Yt = MYt−1 + δt

The choice of M is crucial. Define M ≡ (mij) “spatially” , that is, where the mij

corresponding to nearby locations si and sj are non-zero , and are zero when locationsare far apart

This applies the “First Law of Geography” (Tobler; cf. Fisher and wheat yields) to the

dynamical evolution of the process

– p.20/26

Structure of M

su = s

u = u1

u = u2

t-1 t

u

u

u

uii

i − 2

i − 1

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i − 1

i + 1

i + 2

t-1 t

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General M M defined “spatially”

– p.21/26

Instantaneous Spatial Dependence (ISD)

To capture the process’ behavior at small temporal scales between time t and time t + 1,we need a component of variation that is modeled as instantaneous spatialdependence (ISD) :

Yt = B0Yt + B1Yt−1 + νt ,

where B0 has zero down its diagonal. Model B0 and B1 “spatially” ; see the figurebelow. This implies

Yt = MYt−1 + δt ,

where M = (I − B0)−1B1 and δt = (I − B0)−1νt. What use are B0 and B1? They

have dynamic structure and are sparse !

– p.22/26

ISD in Graphical Form

(a)

ii

i − 2

i − 1

i + 1

i + 2

i − 2

i − 1

i + 1

i + 2

t-1 t

u u

u

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t-1 t

u u

u

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u

u

(a) Graph structure (sparse) showing (b) Equivalent directed graph

relationships that are defined “spatially” structure (non-sparse )

– p.23/26

Nonstationarity

Stationarity can be an unrealistic assumption. Descriptive approaches tospatio-temporal modeling, expressed in terms of covariance functions, almostdemand it.

Dynamical approaches are much more forgiving. Consider the nonstationaryVAR(1) process:

Yt = MtYt−1 + δt

For example,

Mt = f(t) · M ,

where

f(t) =

8>><>>:

1 0 ≤ t ≤ 29

−1 30 ≤ t ≤ 59

1 60 ≤ t ≤ 79 ,

and M is tridiagonal but has different parameters for 0 ≤ s ≤ 19 and for 20 ≤ s ≤ 39

– p.24/26

Realizations for Nonstationary Process

Three realizations of Y (s; t)

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Yt = MtYt−1 + δt

– p.25/26

Space-Time

... the NEXT frontier

(with apologies to Gene Roddenberry and Trekkies)

– p.26/26