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Spatial Statistics and Modeling

:	Carlo Gaetan, Xavier Guyon
:	Spatial Statistics and Modeling
:	Springer-Verlag
:	9780387922577
:	1
:	CHF 150.60
:

:	Wahrscheinlichkeitstheorie, Stochastik, Mathematische Statistik
:	English

:	308
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

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).

	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