Spatial Statistics and Modeling (eBook)

Carlo Gaetan, Xavier Guyon (Autoren)

eBook Download: PDF

2009 | 2010
XIV, 302 Seiten
Springer New York (Verlag)
978-0-387-92257-7 (ISBN)

Infos

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.

This book is the English translation of Modélisation et Statistique Spatiales published by Springer in the series Mathématiques & Applications, a series established by Société de Mathématiques Appliquées et Industrielles (SMAI).

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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.This book is the English translation of Modelisation et Statistique Spatiales published by Springer in the series Mathematiques & Applications, a series established by Societe de Mathematiques Appliquees et Industrielles (SMAI).

Inhalt

Preface 5
Contents 9
Abbreviations and notation 13
1 Second-order spatial models and geostatistics 16
1.1 Some background in stochastic processes 17
1.2 Stationary processes 18
1.2.1 Definitions and examples 18
1.2.2 Spectral representation of covariances 20
1.3 Intrinsic processes and variograms 23
1.3.1 Definitions, examples and properties 23
1.3.2 Variograms for stationary processes 25
1.3.3 Examples of covariances and variograms 26
1.3.4 Anisotropy 29
1.4 Geometric properties: continuity, differentiability 30
1.4.1 Continuity and differentiability: the stationary case 32
1.5 Spatial modeling using convolutions 34
1.5.1 Continuous model 34
1.5.2 Discrete convolution 36
1.6 Spatio-temporal models 37
1.7 Spatial autoregressive models 40
1.7.1 Stationary MA and ARMA models 41
1.7.2 Stationary simultaneous autoregression 43
1.7.3 Stationary conditional autoregression 45
1.7.4 Non-stationary autoregressive models on finite networks S 49
1.7.5 Autoregressive models with covariates 52
1.8 Spatial regression models 53
1.9 Prediction when the covariance is known 57
1.9.1 Simple kriging 58
1.9.2 Universal kriging 59
1.9.3 Simulated experiments 60
Exercises 62
2 Gibbs-Markov random fields on networks 68
2.1 Compatibility of conditional distributions 69
2.2 Gibbs random fields on S 70
2.2.1 Interaction potential and Gibbs specification 70
2.2.2 Examples of Gibbs specifications 72
2.3 Markov random fields and Gibbs random fields 79
2.3.1 Definitions: cliques, Markov random field 79
2.3.2 The Hammersley-Clifford theorem 80
2.4 Besag auto-models 82
2.4.1 Compatible conditional distributions and auto-models 82
2.4.2 Examples of auto-models 83
2.5 Markov random field dynamics 88
2.5.1 Markov chain Markov random field dynamics 89
2.5.2 Examples of dynamics 89
Exercises 91
3 Spatial point processes 96
3.1 Definitions and notation 97
3.1.1 Exponential spaces 98
3.1.2 Moments of a point process 100
3.1.3 Examples of point processes 102
3.2 Poisson point process 104
3.3 Cox point process 106
3.3.1 log-Gaussian Cox process 106
3.3.2 Doubly stochastic Poisson point process 107
3.4 Point process density 107
3.4.1 Definition 108
3.4.2 Gibbs point process 109
3.5 Nearest neighbor distances for point processes 113
3.5.1 Palm measure 113
3.5.2 Two nearest neighbor distances for X 114
3.5.3 Second-order reduced moments 115
3.6 Markov point process 117
3.6.1 The Ripley-Kelly Markov property 117
3.6.2 Markov nearest neighbor property 119
3.6.3 Gibbs point process on Rd 122
Exercises 123
4 Simulation of spatial models 125
4.1 Convergence of Markov chains 126
4.1.1 Strong law of large numbers and central limit theorem for a homogeneous Markov chain 131
4.2 Two Markov chain simulation algorithms 132
4.2.1 Gibbs sampling on product spaces 132
4.2.2 The Metropolis-Hastings algorithm 134
4.3 Simulating a Markov random field on a network 138
4.3.1 The two standard algorithms 138
4.3.2 Examples 139
4.3.3 Constrained simulation 142
4.3.4 Simulating Markov chain dynamics 143
4.4 Simulation of a point process 143
4.4.1 Simulation conditional on a fixed number of points 144
4.4.2 Unconditional simulation 144
4.4.3 Simulation of a Cox point process 145
4.5 Performance and convergence of MCMC methods 146
4.5.1 Performance of MCMC methods 146
4.5.2 Two methods for quantifying rates of convergence 147
4.6 Exact simulation using coupling from the past 150
4.6.1 The Propp-Wilson algorithm 150
4.6.2 Two improvements to the algorithm 152
4.7 Simulating Gaussian random fields on SRd 154
4.7.1 Simulating stationary Gaussian random fields 154
4.7.2 Conditional Gaussian simulation 158
Exercises 158
5 Statistics for spatial models 163
5.1 Estimation in geostatistics 164
5.1.1 Analyzing the variogram cloud 164
5.1.2 Empirically estimating the variogram 165
5.1.3 Parametric estimation for variogram models 168
5.1.4 Estimating variograms when there is a trend 170
5.1.5 Validating variogram models 172
5.2 Autocorrelation on spatial networks 179
5.2.1 Moran's index 180
5.2.2 Asymptotic test of spatial independence 181
5.2.3 Geary's index 183
5.2.4 Permutation test for spatial independence 184
5.3 Statistics for second-order random fields 187
5.3.1 Estimating stationary models on bold0mu mumu ZZunitsZZZZd 187
5.3.2 Estimating autoregressive models 191
5.3.3 Maximum likelihood estimation 192
5.3.4 Spatial regression estimation 193
5.4 Markov random field estimation 202
5.4.1 Maximum likelihood 203
5.4.2 Besag's conditional pseudo-likelihood 205
5.4.3 The coding method 212
5.4.4 Comparing asymptotic variance of estimators 215
5.4.5 Identification of the neighborhood structure of a Markov random field 217
5.5 Statistics for spatial point processes 221
5.5.1 Testing spatial homogeneity using quadrat counts 221
5.5.2 Estimating point process intensity 222
5.5.3 Estimation of second-order characteristics 224
5.5.4 Estimation of a parametric model for a point process 232
5.5.5 Conditional pseudo-likelihood of a point process 233
5.5.6 Monte Carlo approximation of Gibbs likelihood 237
5.5.7 Point process residuals 240
5.6 Hierarchical spatial models and Bayesian statistics 244
5.6.1 Spatial regression and Bayesian kriging 245
5.6.2 Hierarchical spatial generalized linear models 246
Exercises 254
A Simulation of random variables 263
A.1 The inversion method 263
A.2 Simulation of a Markov chain with a finite number of states 265
A.3 The acceptance-rejection method 265
A.4 Simulating normal distributions 266
B Limit theorems for random fields 268
B.1 Ergodicity and laws of large numbers 268
B.1.1 Ergodicity and the ergodic theorem 268
B.1.2 Examples of ergodic processes 269
B.1.3 Ergodicity and the weak law of large numbers in L2 270
B.1.4 Strong law of large numbers under L2 conditions 271
B.2 Strong mixing coefficients 271
B.3 Central limit theorem for mixing random fields 273
B.4 Central limit theorem for a functional of a Markov random field 274
C Minimum contrast estimation 276
C.1 Definitions and examples 277
C.2 Asymptotic properties 282
C.2.1 Convergence of the estimator 282
C.2.2 Asymptotic normality 284
C.3 Model selection by penalized contrast 287
C.4 Proof of two results in Chapter 5 288
C.4.1 Variance of the maximum likelihood estimator for Gaussian regression 288
C.4.2 Consistency of maximum likelihood for stationary Markov random fields 289
D Software 292
References 295
Index 304

Produkt-Details

• Erscheint lt. Verlag: 10.11.2009
• Reihe/Serie: Springer Series in Statistics
• Reihe/Serie: Springer Series in Statistics
• Zusatzinfo: XIV, 302 p.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Themenwelt: Wirtschaft ► Volkswirtschaftslehre ► Ökonometrie
• Schlagworte: algorithms • Applied Statistics • Geostatistics • markov random field • Mathematical Statistics • Point Process • Spatial statistics and modeling
• ISBN-10: 0-387-92257-1 / 0387922571
• ISBN-13: 978-0-387-92257-7 / 9780387922577