Spatial Statistics and Spatio-Temporal Data: Covariance Functions and Directional Properties (Wiley Series in Probability and Statistics)

Spatial Statistics and Spatio-Temporal Data......Page 6
Contents......Page 8
Preface......Page 14
1 Introduction......Page 18
1.1 Stationarity......Page 21
1.2.1 Estimation......Page 22
1.2.2 Prediction......Page 29
1.3 Texas tidal data......Page 31
2 Geostatistics......Page 38
2.1 A model for optimal prediction and error assessment......Page 40
2.2 Optimal prediction (kriging)......Page 42
2.2.1 An example: phosphorus prediction......Page 45
2.2.2 An example in the power family of variogram functions......Page 49
2.3 Prediction intervals......Page 51
2.3.1 Predictions and prediction intervals for lognormal observations......Page 52
2.4 Universal kriging......Page 55
2.4.1 Optimal prediction in universal kriging......Page 56
2.5 The intuition behind kriging......Page 57
2.5.1 An example: the kriging weights in the phosphorus data......Page 58
3.1 Empirical estimation of the variogram or covariance function......Page 62
3.1.1 Robust estimation......Page 63
3.2 On the necessity of parametric variogram and covariance models......Page 64
3.3 Covariance and variogram models......Page 65
3.3.1 Spectral methods and the Matérn covariance model......Page 68
3.4 Convolution methods and extensions......Page 72
3.4.2 Jumps at the origin and the nugget effect......Page 73
3.5 Parameter estimation for variogram and covariance models......Page 74
3.5.1 Estimation with a nonconstant mean function......Page 79
3.6 Prediction for the phosphorus data......Page 80
3.7 Nonstationary covariance models......Page 86
4 Spatial models and statistical inference......Page 88
4.1 Estimation in the Gaussian case......Page 91
4.1.1 A data example: model fitting for the wheat yield data......Page 92
4.2 Estimation for binary spatial observations......Page 95
4.2.1 Edge effects......Page 100
4.2.2 Goodness of model fit......Page 101
5 Isotropy......Page 104
5.1 Geometric anisotropy......Page 108
5.2 Other types of anisotropy......Page 109
5.3 Covariance modeling under anisotropy......Page 110
5.4 Detection of anisotropy: the rose plot......Page 111
5.5 Parametric methods to assess isotropy......Page 113
5.6.1 Regularly spaced data case......Page 114
5.6.2 Irregularly spaced data case......Page 118
5.6.3 Choice of spatial lags for assessment of isotropy......Page 121
5.6.4 Test statistics......Page 122
5.6.5 Numerical results......Page 124
5.7.1 A stochastic sampling design......Page 128
5.7.2 Covariogram estimation and asymptotic properties......Page 129
5.7.3 Testing for spatial isotropy......Page 130
5.7.4 Numerical results for general spatial designs......Page 132
5.7.5 Effect of bandwidth and block size choice......Page 134
5.8 An assessment of isotropy for the longleaf pine sizes......Page 137
6.1 Space–time observations......Page 140
6.2 Spatio-temporal stationarity and spatio-temporal prediction......Page 141
6.3 Empirical estimation of the variogram, covariance models, and estimation......Page 142
6.3.1 Space–time symmetry and separability......Page 143
6.4 Spatio-temporal covariance models......Page 144
6.4.1 Nonseparable space–time covariance models......Page 145
6.5 Space–time models......Page 147
6.6 Parametric methods of assessing full symmetry and space–time separability......Page 149
6.7 Nonparametric methods of assessing full symmetry and space–time separability......Page 150
6.7.1 Irish wind data......Page 156
6.7.2 Pacific Ocean wind data......Page 158
6.7.3 Numerical experiments based on the Irish wind data......Page 159
6.7.4 Numerical experiments on the test for separability for data on a grid......Page 161
6.7.5 Taylor’s hypothesis......Page 162
6.8 Nonstationary space–time covariance models......Page 164
7 Spatial point patterns......Page 166
7.1 The Poisson process and spatial randomness......Page 167
7.2 Inhibition models......Page 173
7.3 Clustered models......Page 175
8 Isotropy for spatial point patterns......Page 184
8.1 Some large sample results......Page 186
8.2 A test for isotropy......Page 187
8.3 Practical issues......Page 188
8.4.1 Poisson cluster processes......Page 190
8.4.2 Simple inhibition processes......Page 193
8.5 An application to leukemia data......Page 194
9 Multivariate spatial and spatio-temporal models......Page 198
9.1 Cokriging......Page 200
9.2 An alternative to cokriging......Page 203
9.2.1 Statistical model......Page 204
9.2.2 Model fitting......Page 205
9.2.3 Prediction......Page 208
9.2.4 Validation......Page 209
9.3 Multivariate covariance functions......Page 211
9.3.1 Variogram function or covariance function?......Page 212
9.3.2 Intrinsic correlation, separable models......Page 213
9.3.3 Coregionalization and kernel convolution models......Page 214
9.4 Testing and assessing intrinsic correlation......Page 215
9.4.1 Testing procedures for intrinsic correlation and symmetry......Page 218
9.4.2 Determining the order of a linear model of coregionalization......Page 219
9.4.3 Covariance estimation......Page 221
9.5.1 Symmetry......Page 222
9.5.2 Intrinsic correlation......Page 224
9.6 A data application to pollutants......Page 226
9.7 Discussion......Page 230
10 Resampling for correlated observations......Page 232
10.1.1 U-statistics......Page 235
10.1.2 The jackknife......Page 237
10.1.3 The bootstrap......Page 238
10.2 Other data structures......Page 241
10.3.1 Regression......Page 242
10.3.2 Time series: autoregressive models......Page 244
10.4 Model-free resampling methods......Page 245
10.4.1 Resampling for stationary dependent observations......Page 247
10.4.2 Block bootstrap......Page 249
10.4.4 A numerical experiment......Page 250
10.5 Spatial resampling......Page 253
10.5.1 Model-based resampling......Page 254
10.5.2 Monte Carlo maximum likelihood......Page 255
10.6 Model-free spatial resampling......Page 257
10.6.1 A spatial numerical experiment......Page 261
10.7 Unequally spaced observations......Page 263
Bibliography......Page 268
Index......Page 280