Statistical Analysis and Modelling

Preface xi
List of examples xvii

1 Introduction 1
1.1 Point process statistics 2
1.2 Examples of point process data 5
1.3 Historical notes 10
1.4 Sampling and data collection 17
1.5 Fundamentals of the theory of point processes 23
1.6 Stationarity and isotropy 35
1.7 Summary characteristics for point processes 40
1.8 Secondary structures of point processes 42
1.9 Simulation of point processes 52

2 The homogeneous Poisson point process 57
2.1 Introduction 58
2.2 The binomial point process 59
2.3 The homogeneous Poisson point process 66
2.4 Simulation of a homogeneous Poisson process 70
2.5 Model characteristics 71
2.6 Estimating the intensity 79
2.7 Testing complete spatial randomness 83

3 Finite point processes 99
3.1 Introduction 100
3.2 Distributions of numbers of points 104
3.3 Intensity functions and their estimation 110
3.4 Inhomogeneous Poisson process and finite Cox process 118
3.5 Summary characteristics for finite point processes 125
3.6 Finite Gibbs processes 137

4 Stationary point processes 173
4.1 Basic definitions and notation 174
4.2 Summary characteristics for stationary point processes 179
4.3 Second-order characteristics 214
4.4 Higher-order and topological characteristics 244
4.5 Orientation analysis for stationary point processes 250
4.6 Outliers, gaps and residuals 256
4.7 Replicated patterns 260
4.8 Choosing appropriate observation windows 264
4.9 Multivariate analysis of series of point patterns 270
4.10 Summary characteristics for the non-stationary case 279

5 Stationary marked point processes 293
5.1 Basic definitions and notation 294
5.2 Summary characteristics 306
5.3 Second-order characteristics for marked point processes 323
5.4 Orientation analysis for marked point processes 355

6 Modelling and simulation of stationary point processes 363
6.1 Introduction 364
6.2 Operations with point processes 364
6.3 Cluster processes 371
6.4 Stationary Cox processes 379
6.5 Hard-core point processes 387
6.6 Stationary Gibbs processes 398
6.7 Reconstruction of point patterns 407
6.8 Formulas for marked point process models 417
6.9 Moment formulas for stationary shot-noise fields 423
6.10 Space-time point processes 425
6.11 Correlations between point processes and other random structures 437

7 Fitting and testing point process models 445
7.1 Choice of model 445
7.2 Parameter estimation 448
7.3 Variance estimation by bootstrap 453
7.4 Goodness-of-fit tests 455
7.5 Testing mark hypotheses 460
7.6 Bayesian methods for point pattern analysis 471

Appendix A Fundamentals of statistics 479
Appendix B Geometrical characteristics of sets 483
Appendix C Fundamentals of geostatistics 489

References 493
Notation index 515
Author index 519
Subject index 527