## Statistical Inference and Simulation for Spatial Point ProcessesSpatial 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. |

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### Contents

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 |

279 | |

293 | |

299 | |

### Other editions - View all

Statistical Inference and Simulation for Spatial Point Processes Jesper Moller,Rasmus Plenge Waagepetersen No preview available - 2003 |

### Common terms and phrases

Algorithm 7.4 approximate asymptotic B C Rd Baddeley Bayesian Bext bounded Brix cells CFTP algorithm compute conditional density consider convergence correlation function Cox process defined DEFINITION denote ergodic Example exponential family follows Gaussian Gaussian field Geyer Gibbs point process Gibbs sampler given homogeneous Poisson process importance sampling inhomogeneous interaction function irreducible kernel L-function Lemma LGCP likelihood function locally stable log likelihood M0ller Markov chain Markov point processes MCMC measure Metropolis-Hastings algorithms Monte Carlo estimates Monte Carlo methods Nonparametric estimation normalising constant number of points obtained parameter path sampling perfect simulation plot in Figure point pattern Poisson process process with intensity Proof Proposition pseudo likelihood rejection sampling Remark respect right plot second order Section shot noise SNCPs space spatial birth-death processes spatial point processes stationary Stoyan Strauss process summary statistics Suppose Theorem Thomas process tion uniformly ergodic unnormalised density values Waagepetersen

### Popular passages

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Page xv - by the Centre for Mathematical Physics and Stochastics (MaPhySto), funded by a grant from the Danish National Research Foundation, and by the Danish Natural Science Research Council.

### References to this book

Applied Spatial Data Analysis with R Roger S. Bivand,Edzer J. Pebesma,Virgilio Gómez-Rubio Limited preview - 2008 |