Spatial Statistics and Modeling (Springer Series in Statistics)

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Carlo Gaetan, Xavier Guyon, "Spatial Statistics and Modeling"
Springer | 2009-12-11 | ISBN: 0387922563 | 302 pages | PDF | 8,0 MB

Spatial statistics are useful in subjects as diverse as climatology, ecology, economics, environmental and earth sciences, epidemiology, image analysis and more. This book covers the best-known spatial models for three types of spatial data: geostatistical data (stationarity, intrinsic models, variograms, spatial regression and space-time models), areal data (Gibbs-Markov fields and spatial auto-regression) and point pattern data (Poisson, Cox, Gibbs and Markov point processes). The level is relatively advanced, and the presentation concise but complete.

The most important statistical methods and their asymptotic properties are described, including estimation in geostatistics, autocorrelation and second-order statistics, maximum likelihood methods, approximate inference using the pseudo-likelihood or Monte-Carlo simulations, statistics for point processes and Bayesian hierarchical models. A chapter is devoted to Markov Chain Monte Carlo simulation (Gibbs sampler, Metropolis-Hastings algorithms and exact simulation).
A large number of real examples are studied with R, and each chapter ends with a set of theoretical and applied exercises. While a foundation in probability and mathematical statistics is assumed, three appendices introduce some necessary background. The book is accessible to senior undergraduate students with a solid math background and Ph.D. students in statistics. Furthermore, experienced statisticians and researchers in the above-mentioned fields will find the book valuable as a mathematically sound reference.

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关键词：Statistics statistic Springer Modeling Statist Modeling Statistics Series Springer Spatial

Contents
1 Second-order spatial models and geostatistics . . . . . . . . . . . . . . . . . . . . . 1
1.1 Some background in stochastic processes . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Stationary processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Definitions and examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Spectral representation of covariances . . . . . . . . . . . . . . . . . . . 5
1.3 Intrinsic processes and variograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Definitions, examples and properties . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Variograms for stationary processes . . . . . . . . . . . . . . . . . . . . . 10
1.3.3 Examples of covariances and variograms . . . . . . . . . . . . . . . . 11
1.3.4 Anisotropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Geometric properties: continuity, differentiability . . . . . . . . . . . . . . . . 15
1.4.1 Continuity and differentiability: the stationary case . . . . . . . . 17
1.5 Spatial modeling using convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5.1 Continuous model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5.2 Discrete convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6 Spatio-temporal models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Spatial autoregressive models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.7.1 Stationary MA and ARMA models . . . . . . . . . . . . . . . . . . . . . 26
1.7.2 Stationary simultaneous autoregression . . . . . . . . . . . . . . . . . . 28
1.7.3 Stationary conditional autoregression . . . . . . . . . . . . . . . . . . . 30
1.7.4 Non-stationary autoregressive models on finite networks S . . 34
1.7.5 Autoregressive models with covariates . . . . . . . . . . . . . . . . . . 37
1.8 Spatial regression models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.9 Prediction when the covariance is known . . . . . . . . . . . . . . . . . . . . . . . 42
1.9.1 Simple kriging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
1.9.2 Universal kriging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.9.3 Simulated experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2 Gibbs-Markov random fields on networks . . . . . . . . . . . . . . . . . . . . . . . . 53
2.1 Compatibility of conditional distributions . . . . . . . . . . . . . . . . . . . . . . 54
2.2 Gibbs random fields on S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2.1 Interaction potential and Gibbs specification . . . . . . . . . . . . . . 55
2.2.2 Examples of Gibbs specifications . . . . . . . . . . . . . . . . . . . . . . . 57
2.3 Markov random fields and Gibbs random fields . . . . . . . . . . . . . . . . . 64
2.3.1 Definitions: cliques, Markov random field. . . . . . . . . . . . . . . . 64
2.3.2 The Hammersley-Clifford theorem . . . . . . . . . . . . . . . . . . . . . 65
2.4 Besag auto-models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.4.1 Compatible conditional distributions and auto-models . . . . . 67
2.4.2 Examples of auto-models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.5 Markov random field dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.5.1 Markov chain Markov random field dynamics . . . . . . . . . . . . 74
2.5.2 Examples of dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3 Spatial point processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.1 Definitions and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.1.1 Exponential spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.1.2 Moments of a point process. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.1.3 Examples of point processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.2 Poisson point process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.3 Cox point process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.3.1 log-Gaussian Cox process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.3.2 Doubly stochastic Poisson point process . . . . . . . . . . . . . . . . . 92
3.4 Point process density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.4.2 Gibbs point process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.5 Nearest neighbor distances for point processes . . . . . . . . . . . . . . . . . . 98
3.5.1 Palm measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.5.2 Two nearest neighbor distances for X . . . . . . . . . . . . . . . . . . . 99
3.5.3 Second-order reduced moments . . . . . . . . . . . . . . . . . . . . . . . . 100
3.6 Markov point process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.6.1 The Ripley-Kelly Markov property . . . . . . . . . . . . . . . . . . . . . 102
3.6.2 Markov nearest neighbor property . . . . . . . . . . . . . . . . . . . . . . 104
3.6.3 Gibbs point process on Rd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4 Simulation of spatial models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.1 Convergence of Markov chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.1.1 Strong law of large numbers and central limit theorem
for a homogeneous Markov chain . . . . . . . . . . . . . . . . . . . . . . . 117
4.2 Two Markov chain simulation algorithms . . . . . . . . . . . . . . . . . . . . . . 118
4.2.1 Gibbs sampling on product spaces . . . . . . . . . . . . . . . . . . . . . . 118
4.2.2 TheMetropolis-Hastings algorithm . . . . . . . . . . . . . . . . . . . . . 120
4.3 Simulating a Markov random field on a network . . . . . . . . . . . . . . . . . 124
4.3.1 The two standard algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.3.3 Constrained simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.3.4 Simulating Markov chain dynamics . . . . . . . . . . . . . . . . . . . . . 129
4.4 Simulation of a point process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.4.1 Simulation conditional on a fixed number of points . . . . . . . . 130
4.4.2 Unconditional simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.4.3 Simulation of a Cox point process . . . . . . . . . . . . . . . . . . . . . . 131
4.5 Performance and convergence of MCMC methods . . . . . . . . . . . . . . . 132
4.5.1 Performance of MCMC methods . . . . . . . . . . . . . . . . . . . . . . . 132
4.5.2 Two methods for quantifying rates of convergence . . . . . . . . 133
4.6 Exact simulation using coupling from the past . . . . . . . . . . . . . . . . . . 136
4.6.1 The Propp-Wilson algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.6.2 Two improvements to the algorithm. . . . . . . . . . . . . . . . . . . . . 138
4.7 Simulating Gaussian random fields on S ⊆ Rd . . . . . . . . . . . . . . . . . . 140
4.7.1 Simulating stationary Gaussian random fields . . . . . . . . . . . . 140
4.7.2 Conditional Gaussian simulation . . . . . . . . . . . . . . . . . . . . . . . 144
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5 Statistics for spatial models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.1 Estimation in geostatistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.1.1 Analyzing the variogram cloud . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.1.2 Empirically estimating the variogram . . . . . . . . . . . . . . . . . . . 151
5.1.3 Parametric estimation for variogram models . . . . . . . . . . . . . . 154
5.1.4 Estimating variograms when there is a trend . . . . . . . . . . . . . . 156
5.1.5 Validating variogram models. . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.2 Autocorrelation on spatial networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.2.1 Moran’s index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.2.2 Asymptotic test of spatial independence . . . . . . . . . . . . . . . . . 167
5.2.3 Geary’s index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.2.4 Permutation test for spatial independence . . . . . . . . . . . . . . . . 170
5.3 Statistics for second-order random fields . . . . . . . . . . . . . . . . . . . . . . . 173
5.3.1 Estimating stationary models on Zd . . . . . . . . . . . . . . . . . . . . . 173
5.3.2 Estimating autoregressive models . . . . . . . . . . . . . . . . . . . . . . . 177
5.3.3 Maximum likelihood estimation . . . . . . . . . . . . . . . . . . . . . . . . 178
5.3.4 Spatial regression estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.4 Markov random field estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
5.4.1 Maximum likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
5.4.2 Besag’s conditional pseudo-likelihood . . . . . . . . . . . . . . . . . . . 191
5.4.3 The coding method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
5.4.4 Comparing asymptotic variance of estimators . . . . . . . . . . . . . 201
5.4.5 Identification of the neighborhood structure of a Markov
random field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.5 Statistics for spatial point processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.5.1 Testing spatial homogeneity using quadrat counts . . . . . . . . . 207
5.5.2 Estimating point process intensity . . . . . . . . . . . . . . . . . . . . . . 208
5.5.3 Estimation of second-order characteristics . . . . . . . . . . . . . . . 210
5.5.4 Estimation of a parametric model for a point process . . . . . . 218
5.5.5 Conditional pseudo-likelihood of a point process . . . . . . . . . . 219
5.5.6 Monte Carlo approximation of Gibbs likelihood . . . . . . . . . . 223
5.5.7 Point process residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
5.6 Hierarchical spatial models and Bayesian statistics . . . . . . . . . . . . . . . 230
5.6.1 Spatial regression and Bayesian kriging . . . . . . . . . . . . . . . . . 231
5.6.2 Hierarchical spatial generalized linear models . . . . . . . . . . . . 232
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
A Simulation of random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
A.1 The inversion method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
A.2 Simulation of a Markov chain with a finite number of states . . . . . . . 251
A.3 The acceptance-rejection method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
A.4 Simulating normal distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
B Limit theorems for random fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
B.1 Ergodicity and laws of large numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 255
B.1.1 Ergodicity and the ergodic theorem . . . . . . . . . . . . . . . . . . . . . 255
B.1.2 Examples of ergodic processes . . . . . . . . . . . . . . . . . . . . . . . . . 256
B.1.3 Ergodicity and the weak law of large numbers in L2 . . . . . . . 257
B.1.4 Strong law of large numbers under L2 conditions . . . . . . . . . . 258
B.2 Strong mixing coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
B.3 Central limit theorem for mixing random fields . . . . . . . . . . . . . . . . . . 260
B.4 Central limit theorem for a functional of a Markov random field . . . 261
C Minimum contrast estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
C.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
C.2 Asymptotic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
C.2.1 Convergence of the estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 269
C.2.2 Asymptotic normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
C.3 Model selection by penalized contrast . . . . . . . . . . . . . . . . . . . . . . . . . 274
C.4 Proof of two results in Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
C.4.1 Variance of the maximum likelihood estimator for
Gaussian regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
C.4.2 Consistency of maximum likelihood for stationary Markov
random fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
D Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

好东西，我在找呢。谢谢

good book for spatial statistics!!
Thanks!!

very good, thanks a lot

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不错的。。。。

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