Statistical Analysis and Modelling of Spatial Point Patterns

547 Pages·2008·7.599 MB·English

by  Janine Illian, Antti Penttinen, Helga Stoyan, Dietrich Stoyan(auth.)

Statistical Analysis and Modelling
of Spatial Point Patterns
StatisticalAnalysisandModellingofSpatialPointPatterns J.Illian,A.Penttinen,H.StoyanandD.Stoyan
©2008JohnWiley&Sons,Ltd. ISBN: 978-0-470-01491-2
STATISTICS IN PRACTICE
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Statistical Analysis and Modelling
of Spatial Point Patterns
Janine Illian
School of Mathematics and Statistics, University of St Andrews,
Scotland, UK
Antti Penttinen
Department of Mathematics and Statistics,
University of Jyväskylä, Finland
Helga Stoyan
Institut für Stochastik, TU Bergakademie Freiberg,
Germany
Dietrich Stoyan
Institut für Stochastik, TU Bergakademie Freiberg,
Germany
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ISBN978-0-470-01491-2(cloth:acid-freepaper)
1. Spatialanalysis(Statistics) I.Illian,Janine.
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2007045547
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Contents
Preface xi
List of examples xvii
1 Introduction 1
1.1 Point process statistics 2
1.2 Examples of point process data 5
1.2.1 A pattern of amacrine cells 5
1.2.2 Gold particles 6
1.2.3 A pattern of Western Australian plants 7
1.2.4 Waterstriders 8
1.2.5 A sample of concrete 10
1.3 Historical notes 10
1.3.1 Determination of number of trees in a forest 10
1.3.2 Number of blood particles in a sample 12
1.3.3 Patterns of points in plant communities 13
1.3.4 Formulating the power law for the pair correlation
function for galaxies 15
1.4 Sampling and data collection 17
1.4.1 General remarks 17
1.4.2 Choosing an appropriate study area 19
1.4.3 Data collection 20
1.5 Fundamentals of the theory of point processes 23
1.6 Stationarity and isotropy 35
1.6.1 Model approach and design approach 35
1.6.2 Finite and infinite point processes 36
1.6.3 Stationarity and isotropy 37
1.6.4 Ergodicity 39
1.7 Summary characteristics for point processes 40
1.7.1 Numerical summary characteristics 41
1.7.2 Functional summary characteristics 42
vi Contents
1.8 Secondary structures of point processes 42
1.8.1 Introduction 42
1.8.2 Random sets 43
1.8.3 Random fields 44
1.8.4 Tessellations 46
1.8.5 Neighbour networks or graphs 49
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
2.2.1 Introduction 59
2.2.2 Basic properties 60
2.2.3 The periodic binomial process 62
2.2.4 Simulation of the binomial process 63
2.3 The homogeneous Poisson point process 66
2.3.1 Introduction 66
2.3.2 Basic properties 67
2.3.3 Characterisations of the homogeneous Poisson process 69
2.4 Simulation of a homogeneous Poisson process 70
2.5 Model characteristics 71
2.5.1 Moments and moment measures 71
2.5.2 The Palm distribution of a homogeneous Poisson process 74
2.5.3 Summary characteristics of the homogeneous
Poisson process 78
2.6 Estimating the intensity 79
2.7 Testing complete spatial randomness 83
2.7.1 Introduction 83
2.7.2 Quadrat counts 86
2.7.3 Distance methods 89
2.7.4 The J-test 91
2.7.5 Two index-based tests 92
2.7.6 Discrepancy tests 93
2.7.7 The L-test 95
2.7.8 Other tests and recommendations 97
3 Finite point processes 99
3.1 Introduction 100
3.2 Distributions of numbers of points 104
3.2.1 The binomial distribution 104
3.2.2 The Poisson distribution 106
3.2.3 Compound distributions 107
3.2.4 Generalised distributions 109
Contents vii
3.3 Intensity functions and their estimation 110
3.3.1 Parametric statistics for the intensity function 111
3.3.2 Non-parametric estimation of the intensity function 114
3.3.3 Estimating the point density distribution function 117
3.4 Inhomogeneous Poisson process and finite Cox process 118
3.4.1 The inhomogeneous Poisson process 118
3.4.2 The finite Cox process 123
3.5 Summary characteristics for finite point processes 125
3.5.1 Nearest-neighbour distances 126
3.5.2 Dilation function 127
3.5.3 Graph-theoretic statistics 129
3.5.4 Second-order characteristics 129
3.6 Finite Gibbs processes 137
3.6.1 Introduction 137
3.6.2 Gibbs processes with fixed number of points 139
3.6.3 Gibbs processes with a random number of points 147
3.6.4 Second-order summary characteristics of finite
Gibbs processes 154
3.6.5 Further discussion 156
3.6.6 Statistical inference for finite Gibbs processes 160
4 Stationary point processes 173
4.1 Basic definitions and notation 174
4.2 Summary characteristics for stationary point processes 179
4.2.1 Introduction 179
4.2.2 Edge-correction methods 180
4.2.3 The intensity (cid:2) 189
4.2.4 Indices as summary characteristics 195
4.2.5 Empty-space statistics and other morphological
summaries 199
4.2.6 The nearest-neighbour distance distribution
function 206
4.2.7 The J-function 213
4.3 Second-order characteristics 214
4.3.1 The three functions: K, L and g 214
4.3.2 Theoretical foundations of second-order
characteristics 223
4.3.3 Estimators of the second-order characteristics 228
4.3.4 Interpretation of pair correlation functions 239
4.4 Higher-order and topological characteristics 244
4.4.1 Introduction 244
4.4.2 Third-order characteristics 244
4.4.3 Delaunay tessellation characteristics 247
4.4.4 The connectivity function 248
viii Contents
4.5 Orientation analysis for stationary point processes 250
4.5.1 Introduction 250
4.5.2 Nearest-neighbour orientation distribution 252
4.5.3 Second-order orientation analysis 254
4.6 Outliers, gaps and residuals 256
4.6.1 Introduction 256
4.6.2 Simple outlier detection 256
4.6.3 Simple gap detection 257
4.6.4 Model-based outliers 257
4.6.5 Residuals 259
4.7 Replicated patterns 260
4.7.1 Introduction 260
4.7.2 Aggregation recipes 261
4.8 Choosing appropriate observation windows 264
4.8.1 General ideas 264
4.8.2 Representative windows 265
4.9 Multivariate analysis of series of point patterns 270
4.10 Summary characteristics for the non-stationary case 279
4.10.1 Formal application of stationary characteristics
and estimators 280
4.10.2 Intensity reweighting 281
4.10.3 Local rescaling 282
5 Stationary marked point processes 293
5.1 Basic definitions and notation 294
5.1.1 Introduction 294
5.1.2 Marks and their properties 295
5.1.3 Marking models 296
5.1.4 Stationarity 299
5.1.5 First-order characteristics 300
5.1.6 Mark-sum measure 304
5.1.7 Palm distribution 304
5.2 Summary characteristics 306
5.2.1 Introduction 306
5.2.2 Intensity and mark-sum intensity 306
5.2.3 Mean mark, mark d.f. and mark probabilities 309
5.2.4 Indices for stationary marked point
processes 311
5.2.5 Nearest-neighbour distributions 320
5.3 Second-order characteristics for marked point processes 323
5.3.1 Introduction 323
5.3.2 Definitions for qualitative marks 323
5.3.3 Definitions for quantitative marks 341
5.3.4 Estimation of second-order characteristics 352
Contents ix
5.4 Orientation analysis for marked point processes 355
5.4.1 Introduction 355
5.4.2 Orientation analysis for anisotropic processes
with angular marks 357
5.4.3 Orientation analysis for isotropic processes
with angular marks 357
5.4.4 Orientation analysis with constructed marks 359
6 Modelling and simulation of stationary point processes 363
6.1 Introduction 364
6.2 Operations with point processes 364
6.2.1 Thinning 365
6.2.2 Clustering 368
6.2.3 Superposition 370
6.3 Cluster processes 371
6.3.1 General cluster processes 371
6.3.2 Neyman–Scott processes 374
6.4 Stationary Cox processes 379
6.4.1 Introduction 379
6.4.2 Properties of stationary Cox processes 383
6.4.3 Statistics for Cox processes 386
6.5 Hard-core point processes 387
6.5.1 Introduction 387
6.5.2 Matérn hard-core processes 388
6.5.3 The dead leaves model 391
6.5.4 The RSA process 393
6.5.5 Random dense packings of hard spheres 394
6.6 Stationary Gibbs processes 398
6.6.1 Basic ideas and equations 398
6.6.2 Simulation of stationary Gibbs processes 402
6.6.3 Statistics for stationary Gibbs processes 402
6.7 Reconstruction of point patterns 407
6.7.1 Reconstructing point patterns without a specified
model 407
6.7.2 An example: reconstruction of Neyman–Scott
processes 410
6.7.3 Practical application of the reconstruction algorithm 415
6.8 Formulas for marked point process models 417
6.8.1 Introduction 417
6.8.2 Independent marks 418
6.8.3 Random field model 420
6.8.4 Intensity-weighted marks 421
6.9 Moment formulas for stationary
shot-noise fields 423
x Contents
6.10 Space–time point processes 425
6.10.1 Introduction 425
6.10.2 Space–time Poisson processes 428
6.10.3 Second-order statistics for completely stationary
event processes 430
6.10.4 Two examples of space–time processes 434
6.11 Correlations between point processes and other random
structures 437
6.11.1 Introduction 437
6.11.2 Correlations between point processes and
random fields 438
6.11.3 Correlations between point processes and fibre
processes 442
7 Fitting and testing point process models 445
7.1 Choice of model 445
7.2 Parameter estimation 448
7.2.1 Maximum likelihood method 448
7.2.2 Method of moments 450
7.2.3 Trial-and-error estimation 452
7.3 Variance estimation by bootstrap 453
7.4 Goodness-of-fit tests 455
7.4.1 Envelope test 455
7.4.2 Deviation test 457
7.5 Testing mark hypotheses 460
7.5.1 Introduction 460
7.5.2 Testing independent marking, test of association 460
7.5.3 Testing geostatistical marking 468
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