Statistical inference and simulation for spatial point processes

Table of Contents......Page 0
Statistical Inference and Simulation for Spatial Point Processes......Page 1
Contents......Page 3
Preface......Page 9
Acknowledgments......Page 11
CHAPTER 1: Examples of spatial point patterns......Page 12
2.1 Point processes on Rd......Page 18
2.3 Unified framework for point processes and marked point processes......Page 19
2.3.2 Characterisation using the generating functional......Page 20
2.4 Space-time processes......Page 21
3.1.1 Definitions......Page 23
3.1.2 Existence and independent scattering property......Page 25
3.1.3 Constructions of stationary Poisson processes......Page 27
3.2.1 Slivnyak-Mecke’s theorem......Page 30
3.2.2 Superpositioning and thinning......Page 32
3.2.4 Densities for Poisson processes......Page 34
3.3 Marked Poisson processes......Page 35
3.3.2 Multivariate Poisson processes and random labelling......Page 37
4.1 First and second order properties of a point process......Page 39
4.1.1 Basic definitions and results......Page 40
4.1.2 The second order reduced moment measure......Page 42
4.2.1 Second order summary statistics......Page 43
4.2.2 Directional K-functions......Page 44
4.2.3 Summary statistics based on interpoint distances......Page 45
4.3.1 Nonparametric estimation of intensity functions......Page 46
4.3.2 Nonparametric estimation of K and L......Page 47
4.3.3 Edge correction......Page 49
4.3.4 Envelopes for summary statistics......Page 50
4.3.5 Nonparametric estimation of g......Page 54
4.3.6 Nonparametric estimation of F, G, and J-functions......Page 56
4.4 Summary statistics for multivariate point processes......Page 57
4.4.1 Definitions and properties......Page 58
4.4.2 The stationary case......Page 60
4.4.3 Nonparametric estimation......Page 61
4.5 Summary statistics for marked point processes......Page 63
5.1 Definition and simple examples......Page 66
5.2 Basic properties......Page 69
5.3 Neyman-Scott processes as Cox processes......Page 70
5.4 Shot noise Cox processes......Page 71
5.4.1 Shot noise Cox processes as cluster processes......Page 72
5.4.3 Examples......Page 73
5.4.4 Summary statistics......Page 75
5.5 Approximate simulation of SNCPs......Page 77
5.6 Log Gaussian Cox processes......Page 81
5.6.1 Conditions on the covariance function......Page 82
5.6.2 Summary statistics......Page 84
5.7 Simulation of Gaussian fields and LGCPs......Page 85
5.8.1 Summary statistics......Page 87
5.8.2 Multivariate log Gaussian Cox processes......Page 88
5.8.3 Multivariate shot noise Cox processes......Page 89
6.1 Finite point processes with a density......Page 90
6.1.1 Papangelou conditional intensity and stability conditions......Page 92
6.2.1 Definitions and properties......Page 93
6.2.2 Examples of pairwise interaction point processes......Page 94
6.3.1 Definition and characterisation......Page 97
6.3.2 Examples......Page 100
6.3.3 A spatial Markov property......Page 102
6.4.1 Infinite Gibbs point processes......Page 103
6.4.2 Summary statistics......Page 105
6.5 Inhomogeneous Markov point processes......Page 106
6.5.3 Transformation of homogeneous Markov point processes......Page 107
6.6.1 Finite marked and multivariate point processes with a density......Page 108
6.6.2 Definition and characterisation of marked and multivariate Markov point processes......Page 109
6.6.3 Examples of marked and multivariate Markov point processes......Page 110
6.6.4 Summary statistics for multivariate Markov point processes......Page 113
7.1 Description of algorithms......Page 115
7.1.1 Metropolis-Hastings algorithms for the conditional case of point processes witha density......Page 116
7.1.2 Metropolis-Hastings algorithms for the unconditional case......Page 120
7.1.3 Simulation of marked and multivariate point processes witha density......Page 123
7.2 Background material for Markov chains obtained by MCMC algorithms......Page 126
7.2.1 Irreducibility and Harris recurrence......Page 127
7.2.2 Aperiodicity and ergodicity......Page 129
7.2.3 Geometric and uniform ergodicity......Page 130
7.3.1 The conditional case......Page 133
7.3.2 The unconditional case......Page 136
7.3.3 The case of marked and multivariate point processes......Page 140
8.1 Introduction to Monte Carlo methods and output analysis......Page 142
8.1.2 Assessment of convergence......Page 143
8.1.3 Estimation of correlations and asymptotic variances......Page 144
8.1.4 Subsampling......Page 146
8.2.1 Setting and assumptions......Page 147
8.2.2 Exponential family models......Page 148
8.2.3 Importance sampling......Page 149
8.2.4 Bridge sampling and related methods......Page 151
8.2.5 Path sampling Path sampling is a useful......Page 152
8.3.1 Some basic ingredients in likelihood inference......Page 153
8.3.2 Estimation and maximisation of log likelihood functions......Page 154
8.4 Monte Carlo error for path sampling and Monte Carlo maximum likelihood estimates......Page 156
8.5 Distribution of estimates and hypothesis tests......Page 157
8.6 Approximate missing data likelihoods and maximum likelihood estimates......Page 158
8.6.1 Importance, bridge, and path sampling for missing data likelihoods......Page 159
8.6.2 Derivatives of missing data likelihoods and approximate maximum likelihood estimates......Page 160
8.6.3 Monte Carlo EM algorithm......Page 161
CHAPTER 9: Inference for Markov point processes......Page 163
9.1.1 Likelihood functions for Markov point processes......Page 164
9.1.3 Asymptotic properties of maximum likelihood estimates......Page 167
9.1.4 Monte Carlo maximum likelihood......Page 168
9.1.5 Examples......Page 169
9.2.1 Pseudo likelihood functions......Page 177
9.2.2 Practical implementation of pseudo likelihood estimation......Page 180
9.2.3 Consistency and asymptotic normality of pseudo likelihood estimates......Page 182
9.2.4 Relation to Takacs-Fiksel estimation......Page 183
9.2.5 Time-space processes......Page 184
9.3 Bayesian inference......Page 185
CHAPTER 10 Inference for Cox processes......Page 187
10.1 Minimum contrast estimation......Page 188
10.2 Conditional simulation and prediction......Page 190
10.2.1 Conditional simulation for Neyman-Scott processes......Page 191
10.2.2 Conditional simulation for SNCPs......Page 192
10.2.3 Conditional simulation for LGCPs......Page 196
10.3.1 Likelihood inference for a Thomas process......Page 198
10.3.2 Likelihood inference for a Poisson-gamma process......Page 203
10.3.3 Likelihood inference for LGCPs......Page 205
10.4 Bayesian inference......Page 206
10.4.1 Bayesian inference for cluster processes......Page 210
11.1 Simulation based on spatial birth-death processes......Page 211
11.1.1 General definition and description of spatial birth-death processes......Page 212
11.1.2 General algorithms......Page 213
11.1.3 Simulation of spatial point processes with a density......Page 215
11.1.4 A useful coupling construction in the locally stable and constant death rate case......Page 217
11.1.5 Ergodic averages for spatial birth-death processes......Page 220
11.2 Perfect simulation......Page 222
11.2.1 General CFTP algorithms......Page 223
11.2.2 Propp-Wilson’s CFTP algorithm......Page 226
11.2.3 Propp-Wilson’s monotone CFTP algorithm......Page 227
11.2.4 Perfect simulation of continuum Ising models......Page 229
11.2.5 Wilson’s read-once algorithm......Page 231
11.2.6 Perfect simulation for locally stable point processes using dominated CFTP......Page 233
11.2.7 Perfect simulation for locally stable point processes using clans of ancestors......Page 238
11.2.8 Empirical findings......Page 239
11.2.9 Other perfect simulation algorithms......Page 242
References......Page 244
A.1 Brief history......Page 258
A.3 Software......Page 259
B.2 Formal definition of point processes......Page 260
B.3 Some useful conditions and results......Page 262
C.1.1 Moment measures in a general setting......Page 265
C.2.1 Campbell measures and Palm distributions in a general setting......Page 266
C.2.2 Palm distributions in the stationary case......Page 269
C.2.3 Interpretation of K and G as Palm expectations......Page 270
APPENDIX D: Simulation of SNCPs without edge effects and truncation......Page 271
APPENDIX E: Simulation of Gaussian .elds......Page 275
F.1 Definition and characterisation......Page 278
F.2 Examples......Page 280
F.3 Connected component Markov point processes......Page 282
G.1 Jump processes......Page 286
G.2 Coupling constructions......Page 287
G.3 Detailed balance......Page 289
G.4 Ergodicity properties......Page 291