2-7 feb 2003samsi multiscale workshop1 nonstationary spatial covariance modeling via spatial...

2-7 Feb 2003 SAMSI Multiscale Workshop 1

Nonstationary spatial covariance modeling via spatial deformation

& extensions to covariates and models for dynamic atmospheric processes

Paul D. Sampson

(w/ Doris Damian, Peter Guttorp, Tilmann Gneiting)

University of Washington and

National Research Center for Statistics and the Environment


A common framework for spatio-temporal modeling:

Response(x,t) = Trend(x,t) + Residual(x,t)

Some (obvious) observations on issues of scale: the practical definition and estimation of Trend and Residual depend on:

Understanding of the spatio-temporal processes of interest

Scientific questions to be addressed by analysis and modeling

Details of available spatio-temporal sampling data—inference at ``small’’ spatial and/or temporal scales generally requires sampling and modeling at corresponding scales.


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 fields—empirical grid cell estimates vs. model predictions—into 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 100’s of monitoring sites across the country for spatial prediction at target locations.


Languedoc-Roussillon Precipitation Sites


SEASONAL (Apr-Oct) ozone sampling sites with 75% complete data each year, 1992-94, N = 257

(target counties highlighted)


ANNUAL ozone sampling sites with 75% complete data each year, 1992-94, N = 407



Objectives for approaches to nonstationary spatial covariance modeling.

Characterizing spatially varying, locally (stationary) anisotropic structure.

Scientific understanding/representation of covariance structure—not just a method of providing covariances for kriging.

Capable of:

reflecting effects of known explanatory environmental processes such as transport/wind, topography, point sources

modeling effects of known explanatory environmental processes


Objectives (cont.)

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

Application in context of dynamic models for space-time structure

Application to “large” problems/data sets

Diagnostics for local and large-scale correlation structure:

o is the spatial structure “right”

o is the nature/degree 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 References:1. Basis function methods (Nychka, Wikle, …)

2. Kernel/smoothing methods (Fuentes, …)

3. Process convolution models (Higdon, …)

4. Parametric models

5. Spatial deformation models

• Sampson & Guttorp, 1989, 1992, 1994 • Meiring, Monestiez, Sampson & Guttorp, 1997. “Developments...

Geostatistics Wollongong '96.

• Mardia & Goodall, 1993. in Multivariate Environmental Statistics.

• Smith, 1996. Estimating nonstationary spatial correlations.

UNC preprint.


4. Spatial deformation models (cont.)

• Perrin & Meiring, 1999. Identifiability J Appl Prob. • Perrin & Senoussi, 1998. Reducing nonstationary random fields to

stationarity or isotropy, Stat & Prob Letters. • Perrin & Monestiez, 1998. Parametric radial basis deformations.

GeoENV-II. • Iovleff & Perrin, 2000. Simulated annealing. (JSM 2000)

• Schmidt & O'Hagan, 2000. Bayesian inference. (JSM 2000)

• Damian, Sampson & Guttorp, 2000. Bayesian estimation of semiparametric nonstationary covariance structures. Environmetrics.

• Damian, Sampson & Guttorp, 2003. Variance modeling for nonstationary spatial processes with temporal replications. JGR (submitted)

• Related recent developments in the atmospheric science literature:


Spatial Deformation ModelDamian et al., 2000 (Environmetrics), 2003 (JGR)

1 2, , ,tZ x t x t x H x x t

( , ) spatio-temporal trendparametric in time; mv spatial process

x t

( ) temporal variance at ,log-normal spatial process

x x

2( , )

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

independent of t

x tN H x

( )( ( ), ( )) 1.

ndmean 0, var 1, 2 -order cont. spatial processCov

t

t t x y

H xH x H y


( )( ( ), ( )) 1.

ndmean 0, var 1, 2 -order cont. spatial processCov

t

t t x y

H xH x H y

( ), ( ) ( ) ( )

:

( )

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

i.e. 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.)


A deformation of the unit square: (b) true, (c) estimated


Biorthogonal grids for a deformation of the unit square,(a) true, (b) estimated


The spatial deformation f encodes the nonstationarity: spatially varying local anisotropy.

We model this in terms of observation sites as a pair of thin-plate splines:

Model (cont.)

1 2, , , Nx x x

Tf x c x x A W

c xA

T xW

1

N

x x

x

x x

2log 0

0 0

h h hh

h

Linear part: global/large scale anisotropy 2 1 2 2, c A

Non-linear part, decomposable into components of varying spatial scale: 2 1, ( ) N Nx W

2 2:{ , , }, , , , :{ , , }f c A W Model parameters:


Likelihood: given observation vectors Z1,…,ZN 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 TZ

Z Z

Z

Z

Σ

Σ Σ

Σ

S Σ

S L

2

,1 ,i j i j

ij

j

i ji j N

i j

Integrating out a flat prior on the (constant) mean

1 2 1

1

| ,( 1)

exp2

| , d trT T

Z

S Σ SΣ ΣS Σ


Posterior:

2

112

2 2

2 2 2

1

( )

, , , , ,

1exp (log )

, : , , , , , , ,

, , ,

(l g

,

)2

,

o

,

Log-normal variance field

Full posterior is:

:

ix

c

c

f

c c

A W

Σ 1

A W S

A WS A W

Σ 1

Σ

1 1 2 2

, :

1exp ( ) , ( )

2

diffuse normal prior on 2 free params (4 constr)

"bending energy" prior on space orthogonal to linear

ij i j

c

I x x

W V V

A

W W SW W SW S


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 and/or unobserved sites,

(c) the observation process at unobserved sites.


Application to Languedoc-Roussillon Precipitation Data

• 108 altitude-adjusted, 10-day aggregate precip records 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

19

22

25

33

41

45

53


Correlation vs Distance in G-plane and D-plane


Equi-correlation (0.9) contours D-plane (a) and G-plane (b)


Estimated (•) and predicted (◦) variances, , vs. observed temporal variances, with one predictive std dev bars.

0( )x


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


Assessment of (10-day aggregate) precipitation predictions at validation sites (pred interval and obs)


SEASONAL (Apr-Oct) ozone sampling sites with 75% complete data each year, 1992-94, N = 257



Spatio-temporal trend modeling:

• Standardize (center and scale) the data for each of N=750 sites over 3 years and assemble N x (3*364) data matrix (“tricks” to deal with missing data).

• Compute SVD and consider first 3-5 right singular vectors (EOFs); smooth these w.r.t. time.

• Use these temporal trend components as basis functions for estimation of site-specific trends by regression.

( , )x t


dates92to94

Tota

l.svd

$sv

d$

v[,

j]

-0.0

50

.05

01/01/1992 01/01/1993 01/01/1994

dates92to94

Tota

l.svd

$sv

d$

v[,

j]

-0.1

00

.00

.10

01/01/1992 01/01/1993 01/01/1994

dates92to94

Tota

l.svd

$sv

d$

v[,

j]

-0.1

00

.0

01/01/1992 01/01/1993 01/01/1994

dates92to94

Tota

l.svd

$sv

d$

v[,

j]

-0.1

00

.00

.10

01/01/1992 01/01/1993 01/01/1994

dates92to94

Tota

l.svd

$sv

d$

v[,

j]

-0.1

00

.00

.10

01/01/1992 01/01/1993 01/01/1994

dates92to94

Tota

l.svd

$sv

d$

v[,

j]

-0.1

00

.00

.10

01/01/1992 01/01/1993 01/01/1994

Annual Trend Components fitted to all sites


Region 6: Southern Calif (annual)


Region 6 : S Calif Starplot of temporal trend coefficients

60730001

60731001

60379002

60710012

6111000761113001


1992-1994

sqrt

(ma

x 8

hr

O3

)

0.0

0.1

0.2

0.3

0.4

01/01/1992 07/01/1993

60730001

1992-1994

sqrt

(ma

x 8

hr

O3

)

0.0

0.1

0.2

0.3

0.4

01/01/1992 07/01/1993

60731001

1992-1994sq

rt(m

ax

8h

r O

3)

0.0

0.1

0.2

0.3

0.4

01/01/1992 07/01/1993

60379002

1992-1994

sqrt

(ma

x 8

hr

O3

)

0.0

0.1

0.2

0.3

0.4

01/01/1992 07/01/1993

60710012

1992-1994

sqrt

(ma

x 8

hr

O3

)

0.0

0.1

0.2

0.3

0.4

01/01/1993 07/01/1994

61110007

1992-1994

sqrt

(ma

x 8

hr

O3

)

0.0

0.1

0.2

0.3

0.4

01/01/1992 07/01/1993

61113001


Region 6 : S Calif Circle radii proportional to (detrended) site variances


N= 55 , S Calif monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Tue Feb 04 18:28:03 PST 2003


N=55, S Calif: 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Feb 04 18:30:29 PST 2003


Region 6 S Calif

Geographic Distance (km)

Co

vari

an

ce

0 100 200 300 400 500

0.0

0.0

00

50

.00

10

0.0

01

50

.00

20

Region 6 S Calif

D-plane Distance

Co

vari

an

ce

0 100 200 300 400

0.0

0.0

00

50

.00

10

0.0

01

50

.00

20


Region 6 S Calif


Co

rre

latio

n

0 100 200 300 400 500

0.0

0.2

0.4

0.6

0.8

1.0

Region 6 S Calif

D-plane Distance

Co

rre

latio

n

0 100 200 300 400

0.0

0.2

0.4

0.6

0.8

1.0


Region 1: Northeast (seasonal)


Region 1 : Northeast Starplot of temporal trend coefficients

510410004516500004

90110008

250010002360530006

360430005


1992-1994

sqrt

(ma

x 8

hr

O3

)

0.0

0.1

0.2

0.3

0.4

01/01/1993 07/01/1994

510410004

1992-1994

sqrt

(ma

x 8

hr

O3

)

0.0

0.1

0.2

0.3

0.4

01/01/1993 07/01/1994

516500004

1992-1994

sqrt

(ma

x 8

hr

O3

)

0.0

0.1

0.2

0.3

0.4

01/01/1993 07/01/1994

90110008

1992-1994

sqrt

(ma

x 8

hr

O3

)

0.0

0.1

0.2

0.3

0.4

01/01/1993 07/01/1994

250010002

1992-1994

sqrt

(ma

x 8

hr

O3

)

0.0

0.1

0.2

0.3

0.4

01/01/1992 07/01/1993

360530006

1992-1994

sqrt

(ma

x 8

hr

O3

)

0.0

0.1

0.2

0.3

0.4

01/01/1993 07/01/1994

360430005


Region 1 : Northeast Circle radii proportional to (detrended) site variances


62 Northeast monitoring sites and their representation in a deformed coordinate system reflecting spatial covariance

Thu Jan 30 20:50:30 PST 2003


N=62, Northeast: 4 samples from the posterior distribution of deformations reflecting spatial covarianceTue Feb 04 18:34:34 PST 2003


Region 1 Northeast


Co

vari

an

ce

0 200 400 600 800 1000 1200

0.0

0.0

00

50

.00

10

0.0

01

5

Region 1 Northeast

D-plane Distance

Co

vari

an

ce

0 200 400 600 800 1000 1200

0.0

0.0

00

50

.00

10

0.0

01

5


Region 1 Northeast


Co

rre

latio

n

0 200 400 600 800 1000 1200

0.0

0.2

0.4

0.6

0.8

1.0

Region 1 Northeast

D-plane Distance

Co

rre

latio

n

0 200 400 600 800 1000 1200

0.0

0.2

0.4

0.6

0.8

1.0


Work in Progress

• Guidelines for MCMC estimation software, including better calibration of priors for spatial deformation and specification of MCMC proposal distributions.

• Incorporation of the spatio-temporal trend model in the software for spatial prediction.

• Incorporation of (simple) temporal autocorrelation structure.

• More flexible deformation modeling for nonstationary structure at multiple scales: “principal warp” decomposition of thin-plate splines and mappings R2→ Rk.

• Explicit modeling/incorporation of covariate effects, spatial (e.g. topography) and spatio-temporal (e.g. wind).


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

• Desroziers, 1999, 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 and/or rotating the computational coordinates via a ‘diffusion’ tensor.

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


Incorporating covariates

• 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 Where X’s represent differences between the two sites in elevation, slope, tree cover, aspect,

Alternative: deform R2 into subspace of R6.

• Riishojgaard, 1998, “flow-dependent” 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 s

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

• 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 V 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 barotropic model; 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.

2-7 feb 2003samsi multiscale workshop1 nonstationary spatial covariance modeling via spatial...

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