Statistical Analysis and Modellingof Spatial Point Patterns

Janine IllianSchool of Mathematics and Statistics, University of St Andrews,

Scotland, UK

Antti PenttinenDepartment of Mathematics and Statistics,

University of Jyväskylä, Finland

Helga StoyanInstitut für Stochastik, TU Bergakademie Freiberg,

Germany

Dietrich StoyanInstitut für Stochastik, TU Bergakademie Freiberg,

Germany

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Statistical Analysis and Modellingof Spatial Point Patterns

STATISTICS IN PRACTICE

Advisory Editors

Stephen SennUniversity of Glasgow, UK

Marion ScottUniversity of Glasgow, UK

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Vic BarnettNottingham Trent University, UK

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A complete list of titles in this series appears at the end of the volume.

Statistical Analysis and Modellingof Spatial Point Patterns

Janine IllianSchool of Mathematics and Statistics, University of St Andrews,

Scotland, UK

Antti PenttinenDepartment of Mathematics and Statistics,

University of Jyväskylä, Finland

Helga StoyanInstitut für Stochastik, TU Bergakademie Freiberg,

Germany

Dietrich StoyanInstitut für Stochastik, TU Bergakademie Freiberg,

Germany

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Library of Congress Cataloging in Publication Data

Statistical analysis and modelling of spatial point patterns / Janine Illian # # # [et al].p. cm. — (Statistics in practice)

Includes bibliographical references and index.ISBN 978-0-470-01491-2 (cloth : acid-free paper)1. Spatial analysis (Statistics) I. Illian, Janine.QA278.2.S72 2008519.5—dc22

2007045547

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 978-0-470-01491-2 (HB)

Typeset in 10/12pt Times by Integra Software Services Pvt. Ltd, Pondicherry, IndiaPrinted and bound in Great Britain by TJ International, Padstow, CornwallThis book is printed on acid-free paper responsibly manufactured from sustainable forestryin which at least two trees are planted for each one used for paper production.

Contents

Preface xi

List of examples xvii

1 Introduction 11.1 Point process statistics 21.2 Examples of point process data 5

1.2.1 A pattern of amacrine cells 51.2.2 Gold particles 61.2.3 A pattern of Western Australian plants 71.2.4 Waterstriders 81.2.5 A sample of concrete 10

1.3 Historical notes 101.3.1 Determination of number of trees in a forest 101.3.2 Number of blood particles in a sample 121.3.3 Patterns of points in plant communities 131.3.4 Formulating the power law for the pair correlation

function for galaxies 151.4 Sampling and data collection 17

1.4.1 General remarks 171.4.2 Choosing an appropriate study area 191.4.3 Data collection 20

1.5 Fundamentals of the theory of point processes 231.6 Stationarity and isotropy 35

1.6.1 Model approach and design approach 351.6.2 Finite and infinite point processes 361.6.3 Stationarity and isotropy 371.6.4 Ergodicity 39

1.7 Summary characteristics for point processes 401.7.1 Numerical summary characteristics 411.7.2 Functional summary characteristics 42

vi Contents

1.8 Secondary structures of point processes 421.8.1 Introduction 421.8.2 Random sets 431.8.3 Random fields 441.8.4 Tessellations 461.8.5 Neighbour networks or graphs 49

1.9 Simulation of point processes 52

2 The homogeneous Poisson point process 572.1 Introduction 582.2 The binomial point process 59

2.2.1 Introduction 592.2.2 Basic properties 602.2.3 The periodic binomial process 622.2.4 Simulation of the binomial process 63

2.3 The homogeneous Poisson point process 662.3.1 Introduction 662.3.2 Basic properties 672.3.3 Characterisations of the homogeneous Poisson process 69

2.4 Simulation of a homogeneous Poisson process 702.5 Model characteristics 71

2.5.1 Moments and moment measures 712.5.2 The Palm distribution of a homogeneous Poisson process 742.5.3 Summary characteristics of the homogeneous

Poisson process 782.6 Estimating the intensity 792.7 Testing complete spatial randomness 83

2.7.1 Introduction 832.7.2 Quadrat counts 862.7.3 Distance methods 892.7.4 The J -test 912.7.5 Two index-based tests 922.7.6 Discrepancy tests 932.7.7 The L-test 952.7.8 Other tests and recommendations 97

3 Finite point processes 99

3.1 Introduction 1003.2 Distributions of numbers of points 104

3.2.1 The binomial distribution 1043.2.2 The Poisson distribution 1063.2.3 Compound distributions 1073.2.4 Generalised distributions 109

Contents vii

3.3 Intensity functions and their estimation 1103.3.1 Parametric statistics for the intensity function 1113.3.2 Non-parametric estimation of the intensity function 1143.3.3 Estimating the point density distribution function 117

3.4 Inhomogeneous Poisson process and finite Cox process 1183.4.1 The inhomogeneous Poisson process 1183.4.2 The finite Cox process 123

3.5 Summary characteristics for finite point processes 1253.5.1 Nearest-neighbour distances 1263.5.2 Dilation function 1273.5.3 Graph-theoretic statistics 1293.5.4 Second-order characteristics 129

3.6 Finite Gibbs processes 1373.6.1 Introduction 1373.6.2 Gibbs processes with fixed number of points 1393.6.3 Gibbs processes with a random number of points 1473.6.4 Second-order summary characteristics of finite

Gibbs processes 1543.6.5 Further discussion 1563.6.6 Statistical inference for finite Gibbs processes 160

4 Stationary point processes 1734.1 Basic definitions and notation 1744.2 Summary characteristics for stationary point processes 179

4.2.1 Introduction 1794.2.2 Edge-correction methods 1804.2.3 The intensity ! 1894.2.4 Indices as summary characteristics 1954.2.5 Empty-space statistics and other morphological

summaries 1994.2.6 The nearest-neighbour distance distribution

function 2064.2.7 The J -function 213

4.3 Second-order characteristics 2144.3.1 The three functions: K, L and g 2144.3.2 Theoretical foundations of second-order

characteristics 2234.3.3 Estimators of the second-order characteristics 2284.3.4 Interpretation of pair correlation functions 239

4.4 Higher-order and topological characteristics 2444.4.1 Introduction 2444.4.2 Third-order characteristics 2444.4.3 Delaunay tessellation characteristics 2474.4.4 The connectivity function 248

viii Contents

4.5 Orientation analysis for stationary point processes 2504.5.1 Introduction 2504.5.2 Nearest-neighbour orientation distribution 2524.5.3 Second-order orientation analysis 254

4.6 Outliers, gaps and residuals 2564.6.1 Introduction 2564.6.2 Simple outlier detection 2564.6.3 Simple gap detection 2574.6.4 Model-based outliers 2574.6.5 Residuals 259

4.7 Replicated patterns 2604.7.1 Introduction 2604.7.2 Aggregation recipes 261

4.8 Choosing appropriate observation windows 2644.8.1 General ideas 2644.8.2 Representative windows 265

4.9 Multivariate analysis of series of point patterns 2704.10 Summary characteristics for the non-stationary case 279

4.10.1 Formal application of stationary characteristicsand estimators 280

4.10.2 Intensity reweighting 2814.10.3 Local rescaling 282

5 Stationary marked point processes 2935.1 Basic definitions and notation 294

5.1.1 Introduction 2945.1.2 Marks and their properties 2955.1.3 Marking models 2965.1.4 Stationarity 2995.1.5 First-order characteristics 3005.1.6 Mark-sum measure 3045.1.7 Palm distribution 304

5.2 Summary characteristics 3065.2.1 Introduction 3065.2.2 Intensity and mark-sum intensity 3065.2.3 Mean mark, mark d.f. and mark probabilities 3095.2.4 Indices for stationary marked point

processes 3115.2.5 Nearest-neighbour distributions 320

5.3 Second-order characteristics for marked point processes 3235.3.1 Introduction 3235.3.2 Definitions for qualitative marks 3235.3.3 Definitions for quantitative marks 3415.3.4 Estimation of second-order characteristics 352

Contents ix

5.4 Orientation analysis for marked point processes 3555.4.1 Introduction 3555.4.2 Orientation analysis for anisotropic processes

with angular marks 3575.4.3 Orientation analysis for isotropic processes

with angular marks 3575.4.4 Orientation analysis with constructed marks 359

6 Modelling and simulation of stationary point processes 3636.1 Introduction 3646.2 Operations with point processes 364

6.2.1 Thinning 3656.2.2 Clustering 3686.2.3 Superposition 370

6.3 Cluster processes 3716.3.1 General cluster processes 3716.3.2 Neyman–Scott processes 374

6.4 Stationary Cox processes 3796.4.1 Introduction 3796.4.2 Properties of stationary Cox processes 3836.4.3 Statistics for Cox processes 386

6.5 Hard-core point processes 3876.5.1 Introduction 3876.5.2 Matérn hard-core processes 3886.5.3 The dead leaves model 3916.5.4 The RSA process 3936.5.5 Random dense packings of hard spheres 394

6.6 Stationary Gibbs processes 3986.6.1 Basic ideas and equations 3986.6.2 Simulation of stationary Gibbs processes 4026.6.3 Statistics for stationary Gibbs processes 402

6.7 Reconstruction of point patterns 4076.7.1 Reconstructing point patterns without a specified

model 4076.7.2 An example: reconstruction of Neyman–Scott

processes 4106.7.3 Practical application of the reconstruction algorithm 415

6.8 Formulas for marked point process models 4176.8.1 Introduction 4176.8.2 Independent marks 4186.8.3 Random field model 4206.8.4 Intensity-weighted marks 421

6.9 Moment formulas for stationaryshot-noise fields 423

x Contents

6.10 Space–time point processes 4256.10.1 Introduction 4256.10.2 Space–time Poisson processes 4286.10.3 Second-order statistics for completely stationary

event processes 4306.10.4 Two examples of space–time processes 434

6.11 Correlations between point processes and other randomstructures 4376.11.1 Introduction 4376.11.2 Correlations between point processes and

random fields 4386.11.3 Correlations between point processes and fibre

processes 442

7 Fitting and testing point process models 4457.1 Choice of model 4457.2 Parameter estimation 448

7.2.1 Maximum likelihood method 4487.2.2 Method of moments 4507.2.3 Trial-and-error estimation 452

7.3 Variance estimation by bootstrap 4537.4 Goodness-of-fit tests 455

7.4.1 Envelope test 4557.4.2 Deviation test 457

7.5 Testing mark hypotheses 4607.5.1 Introduction 4607.5.2 Testing independent marking, test of association 4607.5.3 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