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