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 Matern 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
