Spatial Statistics and Spatio-Temporal Data: Covariance Functions and Directional Properties

by Michael Sherman

Overview

In the spatial or spatio-temporal context, specifying the correct covariance function is fundamental to obtain efficient predictions, and to understand the underlying physical process of interest. This book focuses on covariance and variogram functions, their role in prediction, and appropriate choice of these functions in applications. Both recent and more established methods are illustrated to assess many common assumptions on these functions, such as, isotropy, separability, symmetry, and intrinsic correlation.

After an extensive introduction to spatial methodology, the book details the effects of common covariance assumptions and addresses methods to assess the appropriateness of such assumptions for various data structures.

Key features:

• An extensive introduction to spatial methodology including a survey of spatial covariance functions and their use in spatial prediction (kriging) is given.
• Explores methodology for assessing the appropriateness of assumptions on covariance functions in the spatial, spatio-temporal, multivariate spatial, and point pattern settings.
• Provides illustrations of all methods based on data and simulation experiments to demonstrate all methodology and guide to proper usage of all methods.
• Presents a brief survey of spatial and spatio-temporal models, highlighting the Gaussian case and the binary data setting, along with the different methodologies for estimation and model fitting for these two data structures.
• Discusses models that allow for anisotropic and nonseparable behaviour in covariance functions in the spatial, spatio-temporal and multivariate settings.
• Gives an introduction to point pattern models, including testing for randomness, and fitting regular and clustered point patterns. The importance and assessment of isotropy of point patterns is detailed.

Statisticians, researchers, and data analysts working with spatial and space-time data will benefit from this book as well as will graduate students with a background in basic statistics following courses in engineering, quantitative ecology or atmospheric science.

Product Details

ISBN-13:	9780470974926
Publisher:	Wiley
Publication date:	01/06/2011
Series:	Wiley Series in Probability and Statistics
Sold by:	JOHN WILEY & SONS
Format:	eBook
Pages:	304
File size:	10 MB

About the Author

Michael Sherman, Professor of Statistics, Texas A&M University
Michael Sherman has done extensive research on re-sampling methods for temporally or spatially dependent data and spatial statistics. He has published various papers in JASA, Biometrics and JRSS-B.  In 2000 he created a course in Spatial Statistics at Texas A&M University and has given over 35 invited presentations at University seminars, ASA meetings and special topic meetings.

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Table of Contents

Preface xi

1 Introduction 1

1.1 Stationarity 4

1.2 The effect of correlation in estimation and prediction 5

1.2.1 Estimation 5

1.2.2 Prediction 12

1.3 Texas tidal data 14

2 Geostatistics 21

2.1 A model for optimal prediction and error assessment 23

2.2 Optimal prediction (kriging) 25

2.2.1 An example: phosphorus prediction 28

2.2.2 An example in the power family of variogram functions 32

2.3 Prediction intervals 34

2.3.1 Predictions and prediction intervals for lognormal observations 35

2.4 Universal kriging 38

2.4.1 Optimal prediction in universal kriging 39

2.5 The intuition behind kriging 40

2.5.1 An example: the kriging weights in the phosphorus data 41

3 Variogram and covariance models and estimation 45

3.1 Empirical estimation of the variogram or covariance function 45

3.1.1 Robust estimation 46

3.1.2 Kernel smoothing 47

3.2 On the necessity of parametric variogram and covariance models 47

3.3 Covariance and variogram models 48

3.3.1 Spectral methods and the Matérn covariance model 51

3.4 Convolution methods and extensions 55

3.4.1 Variogram models where no covariance function exists 56

3.4.2 Jumps at the origin and the nugget effect 56

3.5 Parameter estimation for variogram and covariance models 57

3.5.1 Estimation with a nonconstant mean function 62

3.6 Prediction for the phosphorus data 63

3.7 Nonstationary covariance models 69

4 Spatial models and statistical inference 71

4.1 Estimation in the Gaussian case 74

4.1.1 A data example: model fitting for the wheat yield data 75

4.2 Estimation for binary spatial observations 78

4.2.1 Edge effects 83

4.2.2 Goodness of model fit 84

5 Isotropy 87

5.1 Geometric anisotropy 91

5.2 Other types of anisotropy 92

5.3 Covariance modeling under anisotropy 93

5.4 Detection of anisotropy: the rose plot 94

5.5 Parametric methods to assess isotropy 96

5.6 Nonparametric methods of assessing anisotropy 97

5.6.1 Regularly spaced data case 97

5.6.2 Irregularly spaced data case 101

5.6.3 Choice of spatial lags for assessment of isotropy 104

5.6.4 Test statistics 105

5.6.5 Numerical results 107

5.7 Assessment of isotropy for general sampling designs 111

5.7.1 A stochastic sampling design 111

5.7.2 Covariogram estimation and asymptotic properties 112

5.7.3 Testing for spatial isotropy 113

5.7.4 Numerical results for general spatial designs 115

5.7.5 Effect of bandwidth and block size choice 117

5.8 An assessment of isotropy for the longleaf pine sizes 120

6 Space–time data 123

6.1 Space–time observations 123

6.2 Spatio-temporal stationarity and spatio-temporal prediction 124

6.3 Empirical estimation of the variogram, covariance models, and estimation 125

6.3.1 Space–time symmetry and separability 126

6.4 Spatio-temporal covariance models 127

6.4.1 Nonseparable space–time covariance models 128

6.5 Space–time models 130

6.6 Parametric methods of assessing full symmetry and space–time separability 132

6.7 Nonparametric methods of assessing full symmetry and space–time separability 133

6.7.1 Irish wind data 139

6.7.2 Pacific Ocean wind data 141

6.7.3 Numerical experiments based on the Irish wind data 142

6.7.4 Numerical experiments on the test for separability for data on a grid 144

6.7.5 Taylor’s hypothesis 145

6.8 Nonstationary space–time covariance models 147

7 Spatial point patterns 149

7.1 The Poisson process and spatial randomness 150

7.2 Inhibition models 156

7.3 Clustered models 158

8 Isotropy for spatial point patterns 167

8.1 Some large sample results 169

8.2 A test for isotropy 170

8.3 Practical issues 171

8.4 Numerical results 173

8.4.1 Poisson cluster processes 173

8.4.2 Simple inhibition processes 176

8.5 An application to leukemia data 177

9 Multivariate spatial and spatio-temporal models 181

9.1 Cokriging 183

9.2 An alternative to cokriging 186

9.2.1 Statistical model 187

9.2.2 Model fitting 188

9.2.3 Prediction 191

9.2.4 Validation 192

9.3 Multivariate covariance functions 194

9.3.1 Variogram function or covariance function? 195

9.3.2 Intrinsic correlation, separable models 196

9.3.3 Coregionalization and kernel convolution models 197

9.4 Testing and assessing intrinsic correlation 198

9.4.1 Testing procedures for intrinsic correlation and symmetry 201

9.4.2 Determining the order of a linear model of coregionalization 202

9.4.3 Covariance estimation 204

9.5 Numerical experiments 205

9.5.1 Symmetry 205

9.5.2 Intrinsic correlation 207

9.5.3 Linear model of coregionalization 209

9.6 A data application to pollutants 209

9.7 Discussion 213

10 Resampling for correlated observations 215

10.1 Independent observations 218

10.1.1 U-statistics 218

10.1.2 The jackknife 220

10.1.3 The bootstrap 221

10.2 Other data structures 224

10.3 Model-based bootstrap 225

10.3.1 Regression 225

10.3.2 Time series: autoregressive models 227

10.4 Model-free resampling methods 228

10.4.1 Resampling for stationary dependent observations 230

10.4.2 Block bootstrap 232

10.4.3 Block jackknife 233

10.4.4 A numerical experiment 233

10.5 Spatial resampling 236

10.5.1 Model-based resampling 237

10.5.2 Monte Carlo maximum likelihood 238

10.6 Model-free spatial resampling 240

10.6.1 A spatial numerical experiment 244

10.6.2 Spatial bootstrap 246

10.7 Unequally spaced observations 246

Bibliography 251

Index 263