Nonstationary covariance structures I Deformations

Peter Guttorp amp Paul D Sampson

University of Washington

NRCSE

The spherical correlation

Corresponding variogram

ρ(v) =1minus 15v + 05 v

φ( )3 h lt φ

0 otherwise

⎧⎨⎪

⎩⎪

( )φ φ

στ + minus le leφ

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

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

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 2005 Piecewise Gaussian model for groundwater permeability data

Nonstationary covariance models 1 Deformations

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

The spherical correlation

Corresponding variogram

ρ(v) =1minus 15v + 05 v

φ( )3 h lt φ

0 otherwise

⎧⎨⎪

⎩⎪

( )φ φ

στ + minus le leφ

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

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

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 2005 Piecewise Gaussian model for groundwater permeability data

Nonstationary covariance models 1 Deformations

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

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

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 2005 Piecewise Gaussian model for groundwater permeability data

Nonstationary covariance models 1 Deformations

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

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

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

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 2005 Piecewise Gaussian model for groundwater permeability data

Nonstationary covariance models 1 Deformations

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

Swall amp Higdon Process convolution approachPosterior mean and covariance kernel ellipses

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 2005 Piecewise Gaussian model for groundwater permeability data

Nonstationary covariance models 1 Deformations

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

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 2005 Piecewise Gaussian model for groundwater permeability data

Nonstationary covariance models 1 Deformations

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

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 2005 Piecewise Gaussian model for groundwater permeability data

Nonstationary covariance models 1 Deformations

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

Kim Mallock amp Holmes JASA 2005 Piecewise Gaussian model for groundwater permeability data

Nonstationary covariance models 1 Deformations

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

Nonstationary covariance models 1 Deformations

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

General space-time setup

Z(xt) = (xt) + (x)12E(xt) + (xt) trend + smooth + error

We shall assume that is known or constantt = 1T indexes temporal replicationsE is L2-continuous mean 0 variance 1 independent of the error C(xy) = Cor(E(xt)E(yt))D(xy) = Var(E(xt)-E(yt)) (dispersion)

euro

Cov(Z(xt)Z(yt)) =ν(x)ν(y)C(xy) x ne y

ν(x) + σε2 x = y

⎧ ⎨ ⎩

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

euro

C(xy) = C( x minus y )

euro

C(xy) = C( Ax minus Ay )

The deformation ideaIn 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 intersite distances monotone in spatial dispersion D(xy)

euro

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

2 df R Rrarr

Implementation

Consider observations at sites x1 xn Let be the empirical covariance between sites xi and xj Minimize

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

euro

ˆ C ij

euro

(θf ) a wijˆ C ij minus C(f(xi )f (x j)θ)( )

ijsum

2+ λJ(f )

J (f)=part2f

partx2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

+2part2f

partxparty

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

+part2f

party2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥ dxdyintint

SARMAP

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

Transformation

This is for hr 16 in the afternoon

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) γ(u) is differentiable for ugt0then (fγ) is unique up to a scaling for γand a homothetic transformation for f(rotation scaling reflection)

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

Thin-plate splines

euro

f(s) = c + As + W ˜ σ (s)

Linear part

euro

˜ σ (s) = σ(s minus x1)σ(s minus xn)( )

euro

σ(h) = h 2 log( h )

euro

1 W = 0 X W = 0

euro

J(f) = tr(W ˜ S W)

A Bayesian implementation

Likelihood

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

euro

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

trΣminus1S ⎧ ⎨ ⎩

⎫ ⎬ ⎭

euro

p(W) prop exp minus1

2τWi

˜ S Wii=1

2

sum ⎛

⎝ ⎜

⎞

⎠ ⎟

California ozone

12

3

4

5

6

789

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

444546

47

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

17

18

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

Posterior samples

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 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

44

4546

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

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

444546

47

48

49

50

5152

53

54

55

56

57

58

596061

62

63

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

Other applications

Point process deformation (Jensen amp Nielsen Bernoulli 2000)

Deformation of brain images (Worseley et al 1999)

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions pq the angle between them and Pi the Legendre polynomials

Example ai=(2i+1)ρi

euro

C(pq) = aii= 0

infin

sum Pi (cosγpq )

euro

C(pq) =1minus ρ2

1minus 2ρcos γpq + ρ2 minus 1

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

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

euro

C(xy) = C( x minus y )

euro

C(xy) = C( Ax minus Ay )

The deformation ideaIn 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 intersite distances monotone in spatial dispersion D(xy)

euro

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

2 df R Rrarr

Implementation

Consider observations at sites x1 xn Let be the empirical covariance between sites xi and xj Minimize

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

euro

ˆ C ij

euro

(θf ) a wijˆ C ij minus C(f(xi )f (x j)θ)( )

ijsum

2+ λJ(f )

J (f)=part2f

partx2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

+2part2f

partxparty

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

+part2f

party2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥ dxdyintint

SARMAP

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

Transformation

This is for hr 16 in the afternoon

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) γ(u) is differentiable for ugt0then (fγ) is unique up to a scaling for γand a homothetic transformation for f(rotation scaling reflection)

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

Thin-plate splines

euro

f(s) = c + As + W ˜ σ (s)

Linear part

euro

˜ σ (s) = σ(s minus x1)σ(s minus xn)( )

euro

σ(h) = h 2 log( h )

euro

1 W = 0 X W = 0

euro

J(f) = tr(W ˜ S W)

A Bayesian implementation

Likelihood

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

euro

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

trΣminus1S ⎧ ⎨ ⎩

⎫ ⎬ ⎭

euro

p(W) prop exp minus1

2τWi

˜ S Wii=1

2

sum ⎛

⎝ ⎜

⎞

⎠ ⎟

California ozone

12

3

4

5

6

789

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

444546

47

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

17

18

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

Posterior samples

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 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

44

4546

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

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

444546

47

48

49

50

5152

53

54

55

56

57

58

596061

62

63

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

Other applications

Point process deformation (Jensen amp Nielsen Bernoulli 2000)

Deformation of brain images (Worseley et al 1999)

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions pq the angle between them and Pi the Legendre polynomials

Example ai=(2i+1)ρi

euro

C(pq) = aii= 0

infin

sum Pi (cosγpq )

euro

C(pq) =1minus ρ2

1minus 2ρcos γpq + ρ2 minus 1

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

The deformation ideaIn 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 intersite distances monotone in spatial dispersion D(xy)

euro

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

2 df R Rrarr

Implementation

Consider observations at sites x1 xn Let be the empirical covariance between sites xi and xj Minimize

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

euro

ˆ C ij

euro

(θf ) a wijˆ C ij minus C(f(xi )f (x j)θ)( )

ijsum

2+ λJ(f )

J (f)=part2f

partx2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

+2part2f

partxparty

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

+part2f

party2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥ dxdyintint

SARMAP

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

Transformation

This is for hr 16 in the afternoon

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) γ(u) is differentiable for ugt0then (fγ) is unique up to a scaling for γand a homothetic transformation for f(rotation scaling reflection)

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

Thin-plate splines

euro

f(s) = c + As + W ˜ σ (s)

Linear part

euro

˜ σ (s) = σ(s minus x1)σ(s minus xn)( )

euro

σ(h) = h 2 log( h )

euro

1 W = 0 X W = 0

euro

J(f) = tr(W ˜ S W)

A Bayesian implementation

Likelihood

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

euro

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

trΣminus1S ⎧ ⎨ ⎩

⎫ ⎬ ⎭

euro

p(W) prop exp minus1

2τWi

˜ S Wii=1

2

sum ⎛

⎝ ⎜

⎞

⎠ ⎟

California ozone

12

3

4

5

6

789

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

444546

47

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

17

18

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

Posterior samples

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 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

44

4546

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

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

444546

47

48

49

50

5152

53

54

55

56

57

58

596061

62

63

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

Other applications

Point process deformation (Jensen amp Nielsen Bernoulli 2000)

Deformation of brain images (Worseley et al 1999)

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions pq the angle between them and Pi the Legendre polynomials

Example ai=(2i+1)ρi

euro

C(pq) = aii= 0

infin

sum Pi (cosγpq )

euro

C(pq) =1minus ρ2

1minus 2ρcos γpq + ρ2 minus 1

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

Implementation

Consider observations at sites x1 xn Let be the empirical covariance between sites xi and xj Minimize

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

euro

ˆ C ij

euro

(θf ) a wijˆ C ij minus C(f(xi )f (x j)θ)( )

ijsum

2+ λJ(f )

J (f)=part2f

partx2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

+2part2f

partxparty

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

+part2f

party2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥ dxdyintint

SARMAP

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

Transformation

This is for hr 16 in the afternoon

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) γ(u) is differentiable for ugt0then (fγ) is unique up to a scaling for γand a homothetic transformation for f(rotation scaling reflection)

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

Thin-plate splines

euro

f(s) = c + As + W ˜ σ (s)

Linear part

euro

˜ σ (s) = σ(s minus x1)σ(s minus xn)( )

euro

σ(h) = h 2 log( h )

euro

1 W = 0 X W = 0

euro

J(f) = tr(W ˜ S W)

A Bayesian implementation

Likelihood

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

euro

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

trΣminus1S ⎧ ⎨ ⎩

⎫ ⎬ ⎭

euro

p(W) prop exp minus1

2τWi

˜ S Wii=1

2

sum ⎛

⎝ ⎜

⎞

⎠ ⎟

California ozone

12

3

4

5

6

789

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

444546

47

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

17

18

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

Posterior samples

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 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

44

4546

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

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

444546

47

48

49

50

5152

53

54

55

56

57

58

596061

62

63

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

Other applications

Point process deformation (Jensen amp Nielsen Bernoulli 2000)

Deformation of brain images (Worseley et al 1999)

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions pq the angle between them and Pi the Legendre polynomials

Example ai=(2i+1)ρi

euro

C(pq) = aii= 0

infin

sum Pi (cosγpq )

euro

C(pq) =1minus ρ2

1minus 2ρcos γpq + ρ2 minus 1

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

SARMAP

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

Transformation

This is for hr 16 in the afternoon

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) γ(u) is differentiable for ugt0then (fγ) is unique up to a scaling for γand a homothetic transformation for f(rotation scaling reflection)

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

Thin-plate splines

euro

f(s) = c + As + W ˜ σ (s)

Linear part

euro

˜ σ (s) = σ(s minus x1)σ(s minus xn)( )

euro

σ(h) = h 2 log( h )

euro

1 W = 0 X W = 0

euro

J(f) = tr(W ˜ S W)

A Bayesian implementation

Likelihood

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

euro

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

trΣminus1S ⎧ ⎨ ⎩

⎫ ⎬ ⎭

euro

p(W) prop exp minus1

2τWi

˜ S Wii=1

2

sum ⎛

⎝ ⎜

⎞

⎠ ⎟

California ozone

12

3

4

5

6

789

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

444546

47

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

17

18

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

Posterior samples

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 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

44

4546

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

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

444546

47

48

49

50

5152

53

54

55

56

57

58

596061

62

63

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

Other applications

Point process deformation (Jensen amp Nielsen Bernoulli 2000)

Deformation of brain images (Worseley et al 1999)

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions pq the angle between them and Pi the Legendre polynomials

Example ai=(2i+1)ρi

euro

C(pq) = aii= 0

infin

sum Pi (cosγpq )

euro

C(pq) =1minus ρ2

1minus 2ρcos γpq + ρ2 minus 1

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

Transformation

This is for hr 16 in the afternoon

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) γ(u) is differentiable for ugt0then (fγ) is unique up to a scaling for γand a homothetic transformation for f(rotation scaling reflection)

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

Thin-plate splines

euro

f(s) = c + As + W ˜ σ (s)

Linear part

euro

˜ σ (s) = σ(s minus x1)σ(s minus xn)( )

euro

σ(h) = h 2 log( h )

euro

1 W = 0 X W = 0

euro

J(f) = tr(W ˜ S W)

A Bayesian implementation

Likelihood

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

euro

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

trΣminus1S ⎧ ⎨ ⎩

⎫ ⎬ ⎭

euro

p(W) prop exp minus1

2τWi

˜ S Wii=1

2

sum ⎛

⎝ ⎜

⎞

⎠ ⎟

California ozone

12

3

4

5

6

789

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

444546

47

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

17

18

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

Posterior samples

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 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

44

4546

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

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

444546

47

48

49

50

5152

53

54

55

56

57

58

596061

62

63

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

Other applications

Point process deformation (Jensen amp Nielsen Bernoulli 2000)

Deformation of brain images (Worseley et al 1999)

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions pq the angle between them and Pi the Legendre polynomials

Example ai=(2i+1)ρi

euro

C(pq) = aii= 0

infin

sum Pi (cosγpq )

euro

C(pq) =1minus ρ2

1minus 2ρcos γpq + ρ2 minus 1

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

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) γ(u) is differentiable for ugt0then (fγ) is unique up to a scaling for γand a homothetic transformation for f(rotation scaling reflection)

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

Thin-plate splines

euro

f(s) = c + As + W ˜ σ (s)

Linear part

euro

˜ σ (s) = σ(s minus x1)σ(s minus xn)( )

euro

σ(h) = h 2 log( h )

euro

1 W = 0 X W = 0

euro

J(f) = tr(W ˜ S W)

A Bayesian implementation

Likelihood

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

euro

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

trΣminus1S ⎧ ⎨ ⎩

⎫ ⎬ ⎭

euro

p(W) prop exp minus1

2τWi

˜ S Wii=1

2

sum ⎛

⎝ ⎜

⎞

⎠ ⎟

California ozone

12

3

4

5

6

789

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

444546

47

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

17

18

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

Posterior samples

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 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

44

4546

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

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

444546

47

48

49

50

5152

53

54

55

56

57

58

596061

62

63

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

Other applications

Point process deformation (Jensen amp Nielsen Bernoulli 2000)

Deformation of brain images (Worseley et al 1999)

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions pq the angle between them and Pi the Legendre polynomials

Example ai=(2i+1)ρi

euro

C(pq) = aii= 0

infin

sum Pi (cosγpq )

euro

C(pq) =1minus ρ2

1minus 2ρcos γpq + ρ2 minus 1

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

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) γ(u) is differentiable for ugt0then (fγ) is unique up to a scaling for γand a homothetic transformation for f(rotation scaling reflection)

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

Thin-plate splines

euro

f(s) = c + As + W ˜ σ (s)

Linear part

euro

˜ σ (s) = σ(s minus x1)σ(s minus xn)( )

euro

σ(h) = h 2 log( h )

euro

1 W = 0 X W = 0

euro

J(f) = tr(W ˜ S W)

A Bayesian implementation

Likelihood

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

euro

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

trΣminus1S ⎧ ⎨ ⎩

⎫ ⎬ ⎭

euro

p(W) prop exp minus1

2τWi

˜ S Wii=1

2

sum ⎛

⎝ ⎜

⎞

⎠ ⎟

California ozone

12

3

4

5

6

789

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

444546

47

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

17

18

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

Posterior samples

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 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

44

4546

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

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

444546

47

48

49

50

5152

53

54

55

56

57

58

596061

62

63

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

Other applications

Point process deformation (Jensen amp Nielsen Bernoulli 2000)

Deformation of brain images (Worseley et al 1999)

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions pq the angle between them and Pi the Legendre polynomials

Example ai=(2i+1)ρi

euro

C(pq) = aii= 0

infin

sum Pi (cosγpq )

euro

C(pq) =1minus ρ2

1minus 2ρcos γpq + ρ2 minus 1

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

D-plane Equicorrelation Contours

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn(2) γ(u) is differentiable for ugt0then (fγ) is unique up to a scaling for γand a homothetic transformation for f(rotation scaling reflection)

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

Thin-plate splines

euro

f(s) = c + As + W ˜ σ (s)

Linear part

euro

˜ σ (s) = σ(s minus x1)σ(s minus xn)( )

euro

σ(h) = h 2 log( h )

euro

1 W = 0 X W = 0

euro

J(f) = tr(W ˜ S W)

A Bayesian implementation

Likelihood

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

euro

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

trΣminus1S ⎧ ⎨ ⎩

⎫ ⎬ ⎭

euro

p(W) prop exp minus1

2τWi

˜ S Wii=1

2

sum ⎛

⎝ ⎜

⎞

⎠ ⎟

California ozone

12

3

4

5

6

789

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

444546

47

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

17

18

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

Posterior samples

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 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

44

4546

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

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

444546

47

48

49

50

5152

53

54

55

56

57

58

596061

62

63

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

Other applications

Point process deformation (Jensen amp Nielsen Bernoulli 2000)

Deformation of brain images (Worseley et al 1999)

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions pq the angle between them and Pi the Legendre polynomials

Example ai=(2i+1)ρi

euro

C(pq) = aii= 0

infin

sum Pi (cosγpq )

euro

C(pq) =1minus ρ2

1minus 2ρcos γpq + ρ2 minus 1

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

Theoretical properties of the deformation model

IdentifiabilityPerrin and Meiring (1999) Let

If (1) and are differentiable in Rn(2) γ(u) is differentiable for ugt0then (fγ) is unique up to a scaling for γand a homothetic transformation for f(rotation scaling reflection)

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

Thin-plate splines

euro

f(s) = c + As + W ˜ σ (s)

Linear part

euro

˜ σ (s) = σ(s minus x1)σ(s minus xn)( )

euro

σ(h) = h 2 log( h )

euro

1 W = 0 X W = 0

euro

J(f) = tr(W ˜ S W)

A Bayesian implementation

Likelihood

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

euro

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

trΣminus1S ⎧ ⎨ ⎩

⎫ ⎬ ⎭

euro

p(W) prop exp minus1

2τWi

˜ S Wii=1

2

sum ⎛

⎝ ⎜

⎞

⎠ ⎟

California ozone

12

3

4

5

6

789

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

444546

47

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

17

18

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

Posterior samples

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 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

44

4546

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

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

444546

47

48

49

50

5152

53

54

55

56

57

58

596061

62

63

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

Other applications

Point process deformation (Jensen amp Nielsen Bernoulli 2000)

Deformation of brain images (Worseley et al 1999)

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions pq the angle between them and Pi the Legendre polynomials

Example ai=(2i+1)ρi

euro

C(pq) = aii= 0

infin

sum Pi (cosγpq )

euro

C(pq) =1minus ρ2

1minus 2ρcos γpq + ρ2 minus 1

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

Thin-plate splines

euro

f(s) = c + As + W ˜ σ (s)

Linear part

euro

˜ σ (s) = σ(s minus x1)σ(s minus xn)( )

euro

σ(h) = h 2 log( h )

euro

1 W = 0 X W = 0

euro

J(f) = tr(W ˜ S W)

A Bayesian implementation

Likelihood

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

euro

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

trΣminus1S ⎧ ⎨ ⎩

⎫ ⎬ ⎭

euro

p(W) prop exp minus1

2τWi

˜ S Wii=1

2

sum ⎛

⎝ ⎜

⎞

⎠ ⎟

California ozone

12

3

4

5

6

789

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

444546

47

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

17

18

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

Posterior samples

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 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

44

4546

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

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

444546

47

48

49

50

5152

53

54

55

56

57

58

596061

62

63

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

Other applications

Point process deformation (Jensen amp Nielsen Bernoulli 2000)

Deformation of brain images (Worseley et al 1999)

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions pq the angle between them and Pi the Legendre polynomials

Example ai=(2i+1)ρi

euro

C(pq) = aii= 0

infin

sum Pi (cosγpq )

euro

C(pq) =1minus ρ2

1minus 2ρcos γpq + ρ2 minus 1

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

A Bayesian implementation

Likelihood

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

euro

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

trΣminus1S ⎧ ⎨ ⎩

⎫ ⎬ ⎭

euro

p(W) prop exp minus1

2τWi

˜ S Wii=1

2

sum ⎛

⎝ ⎜

⎞

⎠ ⎟

California ozone

12

3

4

5

6

789

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

444546

47

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

17

18

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

Posterior samples

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 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

44

4546

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

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

444546

47

48

49

50

5152

53

54

55

56

57

58

596061

62

63

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

Other applications

Point process deformation (Jensen amp Nielsen Bernoulli 2000)

Deformation of brain images (Worseley et al 1999)

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions pq the angle between them and Pi the Legendre polynomials

Example ai=(2i+1)ρi

euro

C(pq) = aii= 0

infin

sum Pi (cosγpq )

euro

C(pq) =1minus ρ2

1minus 2ρcos γpq + ρ2 minus 1

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

California ozone

12

3

4

5

6

789

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

444546

47

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

17

18

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

Posterior samples

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 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

44

4546

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

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

444546

47

48

49

50

5152

53

54

55

56

57

58

596061

62

63

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

Other applications

Point process deformation (Jensen amp Nielsen Bernoulli 2000)

Deformation of brain images (Worseley et al 1999)

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions pq the angle between them and Pi the Legendre polynomials

Example ai=(2i+1)ρi

euro

C(pq) = aii= 0

infin

sum Pi (cosγpq )

euro

C(pq) =1minus ρ2

1minus 2ρcos γpq + ρ2 minus 1

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

Posterior samples

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 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

44

4546

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

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

444546

47

48

49

50

5152

53

54

55

56

57

58

596061

62

63

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

Other applications

Point process deformation (Jensen amp Nielsen Bernoulli 2000)

Deformation of brain images (Worseley et al 1999)

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions pq the angle between them and Pi the Legendre polynomials

Example ai=(2i+1)ρi

euro

C(pq) = aii= 0

infin

sum Pi (cosγpq )

euro

C(pq) =1minus ρ2

1minus 2ρcos γpq + ρ2 minus 1

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

Other applications

Point process deformation (Jensen amp Nielsen Bernoulli 2000)

Deformation of brain images (Worseley et al 1999)

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions pq the angle between them and Pi the Legendre polynomials

Example ai=(2i+1)ρi

euro

C(pq) = aii= 0

infin

sum Pi (cosγpq )

euro

C(pq) =1minus ρ2

1minus 2ρcos γpq + ρ2 minus 1

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

Isotropic covariances on the sphere

Isotropic covariances on a sphere are of the form

where p and q are directions pq the angle between them and Pi the Legendre polynomials

Example ai=(2i+1)ρi

euro

C(pq) = aii= 0

infin

sum Pi (cosγpq )

euro

C(pq) =1minus ρ2

1minus 2ρcos γpq + ρ2 minus 1

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

A class of global transformations

Iteration between simple parametric deformation of latitude (with parameters changing with longitude) and similar deformations of longitude (changing smoothly with latitude) (Das 2000)

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

Three iterations

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

Global temperature

Global Historical Climatology Network 7280 stations with at least 10 years of data Subset with 839 stations with data 1950-1991 selected

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

Isotropic correlations

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

Deformation

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

Assessing uncertainty

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

Other approaches Haas 1990 Moving window kriging Nott amp Dunsmuir 2002 Biometrikamdashcomputationally convenient but hellip Higdon 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 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

Gaussian moving averages

Higdon (1998) Swall (2000)Let 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

euro

X(s) = b(s minus u)dξ (u)R2int

euro

ρ(d) = b(u)R2int b(u minus d)du

euro

ρ(s1s2 ) = bs 1R2int (u)bs 2(u)du

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds

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

euro

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

K

sum

euro

ρ(s1s2 ) = wk(s1)wk(s2 )ρkk=1

K

sum (s1 minus s2 )

euro

Z(s) = w(x minus s)Zθ (s )int (x)ds