Statistical Inference and Simulation for Spatial Point Processes

Front Cover
CRC Press, Sep 25, 2003 - Mathematics - 320 pages
Spatial point processes play a fundamental role in spatial statistics and today they are an active area of research with many new applications. Although other published works address different aspects of spatial point processes, most of the classical literature deals only with nonparametric methods, and a thorough treatment of the theory and applications of simulation-based inference is difficult to find. Written by researchers at the top of the field, this book collects and unifies recent theoretical advances and examples of applications. The authors examine Markov chain Monte Carlo algorithms and explore one of the most important recent developments in MCMC: perfect simulation procedures.
 

Selected pages

Contents

Examples of spatial point patterns
1
Introduction to point processes
7
22 Marked point processes and multivariate point processes
8
231 Characterisation using void events
9
233 The standard proof
10
Poisson point processes
13
312 Existence and independent scattering property
15
3 13 Constructions of stationary Poisson processes
17
814 Subsampling
139
82 Estimation of ratios of normalising constants
140
822 Exponential family models
141
823 Importance sampling
142
824 Bridge sampling and related methods
144
825 Path sampling
145
83 Approximate likelihood inference using MCMC
146
832 Estimation and maximisation of log likelihood functions
147

32 Further results
20
322 Superpositioning and thinning
22
323 Simulation of Poisson processes
24
33 Marked Poisson processes
25
331 Random independent displacements of the points in a Poisson process
27
Summary statistics
29
411 Basic definitions and results
30
412 The second order reduced moment measure
32
42 Summary statistics
33
422 Directional Kfunctions
34
423 Summary statistics based on interpoint distances
35
43 Nonparametric estimation
36
432 Nonparametric estimation of K and L
37
433 Edge correction
39
434 Envelopes for summary statistics
40
435 Nonparametric estimation of g
44
436 Nonparametric estimation of F G and Jfunctions
46
44 Summary statistics for multivariate point processes
47
441 Definitions and properties
48
442 The stationary case
50
443 Nonparametric estimation
51
45 Summary statistics for marked point processes
53
Cox processes
57
52 Basic properties
60
53 NeymanScott processes as Cox processes
61
54 Shot noise Cox processes
62
541 Shot noise Cox processes as cluster processes
63
542 Relation to marked point processes
64
544 Summary statistics
66
55 Approximate simulation of SNCPs
68
56 Log Gaussian Cox processes
72
561 Conditions on the covariance function
73
562 Summary statistics
75
57 Simulation of Gaussian fields and LGCPs
76
58 Multivariate Cox processes
78
582 Multivariate log Gaussian Cox processes
79
583 Multivariate shot noise Cox processes
80
Markov point processes
81
611 Papangelou conditional intensity and stability conditions
83
62 Pairwise interaction point processes
84
622 Examples of pairwise interaction point processes
85
63 Markov point processes
88
632 Examples
91
633 A spatial Markov property
93
64 Extensions of Markov point processes to Rd
94
642 Summary statistics
96
65 Inhomogeneous Markov point processes
97
651 First order inhomogeneity
98
66 Marked and multivariate Markov point processes
99
66S Definition and characterisation of marked and multivariate Markov point processes
100
663 Examples of marked and multivariate Markov point processes
101
664 Summary statistics for multivariate Markov point processes
104
MetropolisHastings algorithms for point processes with an unnormalised density
107
711 MetropolisHastings algorithms for the conditional case of point processes with a density
108
712 MetropolisHastings algorithms for the unconditional case
112
713 Simulation of marked and multivariate point processes with a density
115
72 Background material for Markov chains obtained by MCMC algorithms
118
721 Irreducibility and Harris recurrence
119
722 Aperiodicity and ergodicity
121
723 Geometric and uniform ergodicity
122
73 Convergence properties of algorithms
125
732 The unconditional case
128
733 The case of marked and multivariate point processes
132
Simulationbased inference
135
811 Ergodic averages
136
813 Estimation of correlations and asymptotic variances
137
84 Monte Carlo error for path sampling and Monte Carlo maximum likelihood estimates
149
85 Distribution of estimates and hypothesis tests
150
86 Approximate missing data likelihoods and maximum likelihood estimates
151
861 Importance bridge and path sampling for missing data likelihoods
152
862 Derivatives of missing data likelihoods and approximate maximum likelihood estimates
153
863 Monte Carlo EM algorithm
154
Inference for Markov point processes
157
91 Maximum likelihood inference
158
912 Conditioning on the number of points
161
914 Monte Carlo maximum likelihood
162
915 Examples
163
92 Pseudo likelihood
171
922 Practical implementation of pseudo likelihood estimation
174
923 Consistency and asymptotic normality of pseudo likelihood estimates
176
924 Relation to TakacsFiksel estimation
177
925 Timespace processes
178
93 Bayesian inference
179
Inference for Cox processes
181
101 Minimum contrast estimation
182
102 Conditional simulation and prediction
184
1021 Conditional simulation for NeymanScott processes
185
1022 Conditional simulation for SNCPs
186
1023 Conditional simulation for LGCPs
190
103 Maximum likelihood inference
192
1032 Likelihood inference for a Poissongamma process
197
1033 Likelihood inference for LGCPs
199
104 Bayesian inference
200
1041 Bayesian inference for cluster processes
204
Spatial birthdeath processes and perfect simulation
205
1111 General definition and description of spatial birthdeath processes
206
1112 General algorithms
207
1113 Simulation of spatial point processes with a density
209
1114 A useful coupling construction in the locally stable and constant death rate case
211
1115 Ergodic averages for spatial birthdeath processes
214
112 Perfect simulation
216
1121 General CFTP algorithms
217
1122 ProppWilsons CFTP algorithm
220
1123 Propp Wilsons monotone CFTP algorithm
221
1124 Perfect simulation of continuum Ising models
223
1125 Wilsons readonce algorithm
225
1126 Perfect simulation for locally stable point processes using dominated CFTP
227
1127 Perfect simulation for locally stable point processes using clans of ancestors
232
1128 Empirical findings
233
1129 Other perfect simulation algorithms
236
History bibliography and software
239
A2 Brief bibliography
240
Measure theoretical details
241
B3 Some useful conditions and results
243
Moment measures and Palm distributions
247
C2 Campbell measures and Palm distributions
248
C22 Palm distributions in the stationary case
251
C23 Interpretation of K and G as Palm expectations
252
Simulation of SNCPs without edge effects and truncation
253
Simulation of Gaussian fields
257
Nearestneighbour Markov point processes
261
F2 Examples
263
F3 Connected component Markov point processes
265
Ergodicity properties and other results for spatial birthdeath processes
269
G2 Coupling constructions
270
G3 Detailed balance
272
G4 Ergodicity properties
274
References
279
Subject index
293
Notation index
299
Copyright

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Statistical Inference and Simulation for Spatial Point Processes
Jesper Moller,Rasmus Plenge Waagepetersen
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Common terms and phrases

Algorithm 7.4 approximate asymptotic Baddeley Bayesian Bext bounded Brix cells CFTP algorithm cluster compute conditional density consider convergence correlation function Cox processes defined DEFINITION denote Diggle ergodic Example exponential family finite point follows g-function Gaussian Geyer Gibbs point process Gibbs sampler given homogeneous Poisson process importance sampling inhomogeneous intensity function interaction function Ising model kernel Lemma LGCP Lieshout likelihood function locally stable log likelihood marked Markov chain Markov point processes MCMC measure Metropolis-Hastings algorithms Monte Carlo methods Nonparametric estimation normalising constant obtained parameter path sampling perfect simulation point patterns Poisson process probability Proof Proposition pseudo likelihood rejection sampling Remark respect second order Section shot noise SNCP space spatial birth-death processes spatial point processes stationary stochastic Stoyan Strauss process summary statistics Suppose Theorem Thomas process tion uniformly ergodic unnormalised density values Ym+1

References to this book

Bibliographic information

TitleStatistical Inference and Simulation for Spatial Point Processes
Chapman & Hall/CRC Monographs on Statistics & Applied Probability
AuthorsJesper Moller, Rasmus Plenge Waagepetersen
Editionillustrated
PublisherCRC Press, 2003
ISBN0203496930, 9780203496930
Length320 pages
Subjects ›  › 

Mathematics / Probability & Statistics / Bayesian Analysis
Mathematics / Probability & Statistics / General
Science / Earth Sciences / Geology
  
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