ST Book Short

8/13/2019 ST Book Short

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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

*[email protected]

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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

= 2 Y (s ; t )

s 2 Y (s ; t ) ;

is the diffusion coefcient

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Reaction-Diffusion Plots

Space

T i m e

(a)

20 40

10

20

30

40

50

60

70

80

0 0.5 1

Space

(b)

20 40

10

20

30

40

50

60

70

80

0 0.5 1

Space

(c)

20 40

10

20

30

40

50

60

70

80

0 0.5 1

Y (s, 0) = I (15 s 24)

(a) = 1 , = 20 ; (b) = 0 .05, = 0 .05; (c) = 1 , = 50

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Stochastic Version

Consider the stochastic PDE:

Y

t

2 Y

s 2 + Y = ,

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

E ((s ; t )) 0

cov ((s ; t ), (u ; r )) = 2 I (s = u, t = r )

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Stochastic Reaction-Diffusion Plots

Space

T i m e

(a)

20 40

10

20

30

40

50

60

70

80

0 0.5 1

Space

(b)

20 40

10

20

30

40

50

60

70

80

0.5 0 0.5 1

Space

(c)

20 40

10

20

30

40

50

60

70

80

5 0 5

Y (s, 0) = I (15 s 24) = 1 , = 20

(a) = 0 .01; (b) = 0 .1; (c) = 1

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Contour Plot of Spatio-Temporal Correlation Function

0 .10 .1

0 .1

0 .1

0 .2

0 .2

0 .2

0 .3 0 .3

0 . 4

0 .4

0 .5

0 .6 0 .7 0 .8

h

0 1 2 3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(h; ) for the stochastic reaction-diffusion equation

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Separability of Spatio-Temporal Covariance Functions

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

Covariance functions have to be positive-denite (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) ( ) is p-d. Then the

separable class: C (h ; ) C (1) (h ) C (2) ( )

is guaranteed to be p-d

Separability is unusual in dynamical models ; it says that temporal evolution

proceeds independently at each spatial location

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Stochastic Reaction-Diffusion and Separability

If C (h ; ) = C (1) (h) C (2) ( ) ,then

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

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

and hence(h; ) =

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

= C (h ; 0) C (0; )

C (0; 0) C (0; 0)= (h ; 0) (0; )

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

(h; 0) (0; ) versus (h, )

(h; ) versus (h, )

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Contour Plots of Spatio-Temporal Correlation Functions

0 .1

0 .1

0 .1

0 .1 0 .2

0 .2

0 .2

0 .3

0 .3

0 .4

0 .0 .6 0 .7 0 .

h

(a)

0 1 2 3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.10 .1

0 .1

0 . 2

0 .2

0.2

0 .3

0 .3

0 . 4

0 .4

0 .5 0 .6 0 .7 0 .8

h

(b)

0 1 2 3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

(a) (h ; 0) (0; ) ; (b) (h ; )

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

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Non-Separable Realizations in Space-Time

Three realizations of Y (s ; t )

Space

T i m e

(a)

20 40

10

20

30

40

50

60

70

80

2 0 2

Space

(b)

20 40

10

20

30

40

50

60

70

80

2 0 2

Space

(c)

20 40

10

20

30

40

50

60

70

80

2 0 2

Realizations are generated from a stationary Gaussian process with the

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

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Spatio-Temporal Kriging

Predict Y (s 0 ; t0 ) with the linear predictor Z :

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

E (Y (s 0 ; t0 ) Z )2 .

This results in the simple kriging predictor :

bY (s 0 ; t0 ) = c (s 0 ; t0 )

1Z

Z ,

where Z var (Z ) and c (s 0 ; t0 ) = cov(Y (s 0 ; t0 ), Z )

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

k (s 0 ; t0 ) = {var (Y (s 0 ; t0 )) c (s 0 ; t0 ) 1Z c (s 0 ; t0 )}

1/ 2

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Kriging for the Stochastic Reaction-Diffusion Equation

Space

T i m e

(a)

20 40

10

20

30

40

50

60

70

80

2 0 2

Space

(b)

20 40

10

20

30

40

50

60

70

80

2 0 2

Space

(c)

20 40

10

20

30

40

50

60

70

80

0 0.5 1

(a) Full realization; = 0

(b) Crosses show {(s i ; t i )} superimposed on the kriging predictormap, {

bY (s 0 ; t0 )}

(c) Kriging s.e. map, {k (s 0 ; t0 )}

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Kriging for the Stochastic Reaction-Diffusion Equation, ctd.

Space

T i m e

(a)

20 40

10

20

30

40

50

60

70

80

2 0 2

Space

(b)

20 40

10

20

30

40

50

60

70

80

2 0 2

Space

(c)

20 40

10

20

30

40

50

60

70

80

0 0.5 1

(a) Same full realization; = 0

(b) Crosses show different {(s i ; t i )} superimposed on the kriging predictormap, {

bY (s 0 ; t0 )}

(c) Kriging s.e. map, {k (s 0 ; t0 )}

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Emphasize the Dynamics

Approximate the differentials in the reaction-diffusion equation:

Y

t =

2 Y

s 2 Y

with differences :

Y (s ; t + t ) Y (s ; t )

t=

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

2s

ff

Y (s ; t )

Dene Y t (Y ( s ; t ), . . . , Y (79 s ; t ); Y Bt (Y (0; t ), Y (79; t )) . Then the

stochastic version of the difference equation is:

Y t + t = M Y t + M B Y Bt + t + t ,

where M B Y Bt represents given boundary effects. The difference equation is a good

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

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Comparison of Differential and Difference Equations

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

( h

, )

h = 0

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

( h

, )

h = 1

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

( h

, )

h = 2

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

( h

, )

h = 3

Spatio-temporal correlations; = 1 , = 20 , s = 1 , and t = 0 .01

Solid blue line : from differential equations

Red dots : from difference equations

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The Dynamics in the Difference Equation

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

Y t = M Y t 1 + t

The choice of M is crucial. Dene M (m ij ) spatially , that is, where the m ijcorresponding to nearby locations s i and s j are non-zero , and are zero when locations

are far apart

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

dynamical evolution of the process

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Structure of M

su = s

u = u1

u = u2

t-1 t

ii

i 2

i 1

i + 1

i + 2

i 2

i 1

i + 1

i + 2

t-1 t

General M M dened spatially

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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) :

Y t = B0 Y t + B1 Y t 1 + t ,

where B 0 has zero down its diagonal. Model B 0 and B 1 spatially ; see the gurebelow. This implies

Y t = M Y t 1 + t ,

where M = ( I B 0 ) 1 B 1 and t = ( I B 0 ) 1 t . What use are B0 and B1 ? They

have dynamic structure and are sparse !

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ISD i G hi l F


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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

(b)

ii

i 2

i 1

i + 1

i + 2

i 2

i 1

i + 1

i + 2

t-1 t

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

relationships that are dened spatially structure ( non-sparse )

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N i i


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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:

Y t = M t Y t 1 + t

For example,

M t = 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

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R li ti f N t ti P


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Realizations for Nonstationary Process

Three realizations of Y (s ; t )

T

i m e

Space

(a)

20 40

10

20

30

40

50

60

70

80

2 0 2

Space

(b)

20 40

10

20

30

40

50

60

70

80

2 0 2

Space

(c)

20 40

10

20

30

40

50

60

70

80

2 0 2

Y t = M t Y t 1 + t

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Space Time


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Space-Time

... the NEXT frontier

(with apologies to Gene Roddenberry and Trekkies)

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