Statistical.Analysis.and.Modelling.of.Spatial.Point.Patterns.pdf

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Statistical Analysis and Modelling of Spatial Point Patterns

Contents

Preface

List of examples

1 Introduction

1.1 Point process statistics

1.2 Examples of point process data

1.2.1 A pattern of amacrine cells

1.2.2 Gold particles

1.2.3 A pattern of Western Australian plants

1.2.4 Waterstriders

1.2.5 A sample of concrete

1.3 Historical notes

1.3.1 Determination of number of trees in a forest

1.3.2 Number of blood particles in a sample

1.3.3 Patterns of points in plant communities

1.3.4 Formulating the power law for the pair correlation function for galaxies

1.4 Sampling and data collection

1.4.1 General remarks

1.4.2 Choosing an appropriate study area

1.4.3 Data collection

1.5 Fundamentals of the theory of point processes

1.6 Stationarity and isotropy

1.6.1 Model approach and design approach

1.6.2 Finite and infinite point processes

1.6.3 Stationarity and isotropy

1.6.4 Ergodicity

1.7 Summary characteristics for point processes

1.7.1 Numerical summary characteristics

1.7.2 Functional summary characteristics

1.8 Secondary structures of point processes

1.8.1 Introduction

1.8.2 Random sets

1.8.3 Random fields

1.8.4 Tessellations

1.8.5 Neighbour networks or graphs

1.9 Simulation of point processes

2 The homogeneous Poisson point process

2.1 Introduction

2.2 The binomial point process

2.2.1 Introduction

2.2.2 Basic properties

2.2.3 The periodic binomial process

2.2.4 Simulation of the binomial process

2.3 The homogeneous Poisson point process

2.3.1 Introduction

2.3.2 Basic properties

2.3.3 Characterisations of the homogeneous Poisson process

2.4 Simulation of a homogeneous Poisson process

2.5 Model characteristics

2.5.1 Moments and moment measures

2.5.2 The Palm distribution of a homogeneous Poisson process

2.5.3 Summary characteristics of the homogeneous Poisson process

2.6 Estimating the intensity

2.7 Testing complete spatial randomness

2.7.1 Introduction

2.7.2 Quadrat counts

2.7.3 Distance methods

2.7.4 The J-test

2.7.5 Two index-based tests

2.7.6 Discrepancy tests

2.7.7 The L-test

2.7.8 Other tests and recommendations

3 Finite point processes

3.1 Introduction

3.2 Distributions of numbers of points

3.2.1 The binomial distribution

3.2.2 The Poisson distribution

3.2.3 Compound distributions

3.2.4 Generalised distributions

3.3 Intensity functions and their estimation

3.3.1 Parametric statistics for the intensity function

3.3.2 Non-parametric estimation of the intensity function

3.3.3 Estimating the point density distribution function

3.4 Inhomogeneous Poisson process and finite Cox process

3.4.1 The inhomogeneous Poisson process

3.4.2 The finite Cox process

3.5 Summary characteristics for finite point processes

3.5.1 Nearest-neighbour distances

3.5.2 Dilation function

3.5.3 Graph-theoretic statistics

3.5.4 Second-order characteristics

3.6 Finite Gibbs processes

3.6.1 Introduction

3.6.2 Gibbs processes with fixed number of points

3.6.3 Gibbs processes with a random number of points

3.6.4 Second-order summary characteristics of finite Gibbs processes

3.6.5 Further discussion

3.6.6 Statistical inference for finite Gibbs processes

4 Stationary point processes

4.1 Basic definitions and notation

4.2 Summary characteristics for stationary point processes

4.2.1 Introduction

4.2.2 Edge-correction methods

4.2.3 The intensity λ

4.2.4 Indices as summary characteristics

4.2.5 Empty-space statistics and other morphological summaries

4.2.6 The nearest-neighbour distance distribution function

4.2.7 The J-function

4.3 Second-order characteristics

4.3.1 The three functions: K, L and g

4.3.2 Theoretical foundations of second-order characteristics

4.3.3 Estimators of the second-order characteristics

4.3.4 Interpretation of pair correlation functions

4.4 Higher-order and topological characteristics

4.4.1 Introduction

4.4.2 Third-order characteristics

4.4.3 Delaunay tessellation characteristics

4.4.4 The connectivity function

4.5 Orientation analysis for stationary point processes

4.5.1 Introduction

4.5.2 Nearest-neighbour orientation distribution

4.5.3 Second-order orientation analysis

4.6 Outliers, gaps and residuals

4.6.1 Introduction

4.6.2 Simple outlier detection

4.6.3 Simple gap detection

4.6.4 Model-based outliers

4.6.5 Residuals

4.7 Replicated patterns

4.7.1 Introduction

4.7.2 Aggregation recipes

4.8 Choosing appropriate observation windows

4.8.1 General ideas

4.8.2 Representative windows

4.9 Multivariate analysis of series of point patterns

4.10 Summary characteristics for the non-stationary case

4.10.1 Formal application of stationary characteristics and estimators

4.10.2 Intensity reweighting

4.10.3 Local rescaling

5 Stationary marked point processes

5.1 Basic definitions and notation

5.1.1 Introduction

5.1.2 Marks and their properties

5.1.3 Marking models

5.1.4 Stationarity

5.1.5 First-order characteristics

5.1.6 Mark-sum measure

5.1.7 Palm distribution

5.2 Summary characteristics

5.2.1 Introduction

5.2.2 Intensity and mark-sum intensity

5.2.3 Mean mark, mark d.f. and mark probabilities

5.2.4 Indices for stationary marked point processes

5.2.5 Nearest-neighbour distributions

5.3 Second-order characteristics for marked point processes

5.3.1 Introduction

5.3.2 Definitions for qualitative marks

5.3.3 Definitions for quantitative marks

5.3.4 Estimation of second-order characteristics

5.4 Orientation analysis for marked point processes

5.4.1 Introduction

5.4.2 Orientation analysis for anisotropic processes with angular marks

5.4.3 Orientation analysis for isotropic processes with angular marks

5.4.4 Orientation analysis with constructed marks

6 Modelling and simulation of stationary point processes

6.1 Introduction

6.2 Operations with point processes

6.2.1 Thinning

6.2.2 Clustering

6.2.3 Superposition

6.3 Cluster processes

6.3.1 General cluster processes

6.3.2 Neyman–Scott processes

6.4 Stationary Cox processes

6.4.1 Introduction

6.4.2 Properties of stationary Cox processes

6.4.3 Statistics for Cox processes

6.5 Hard-core point processes

6.5.1 Introduction

6.5.2 Matérn hard-core processes

6.5.3 The dead leaves model

6.5.4 The RSA process

6.5.5 Random dense packings of hard spheres

6.6 Stationary Gibbs processes

6.6.1 Basic ideas and equations

6.6.2 Simulation of stationary Gibbs processes

6.6.3 Statistics for stationary Gibbs processes

6.7 Reconstruction of point patterns

6.7.1 Reconstructing point patterns without a specified model

6.7.2 An example: reconstruction of Neyman–Scott processes

6.7.3 Practical application of the reconstruction algorithm

6.8 Formulas for marked point process models

6.8.1 Introduction

6.8.2 Independent marks

6.8.3 Random field model

6.8.4 Intensity-weighted marks

6.9 Moment formulas for stationary shot-noise fields

6.10 Space–time point processes

6.10.1 Introduction

6.10.2 Space–time Poisson processes

6.10.3 Second-order statistics for completely stationary event processes

6.10.4 Two examples of space–time processes

6.11 Correlations between point processes and other random structures

6.11.1 Introduction

6.11.2 Correlations between point processes and random fields

6.11.3 Correlations between point processes and fibre processes

7 Fitting and testing point process models

7.1 Choice of model

7.2 Parameter estimation

7.2.1 Maximum likelihood method

7.2.2 Method of moments

7.2.3 Trial-and-error estimation

7.3 Variance estimation by bootstrap

7.4 Goodness-of-fit tests

7.4.1 Envelope test

7.4.2 Deviation test

7.5 Testing mark hypotheses

7.5.1 Introduction

7.5.2 Testing independent marking, test of association

7.5.3 Testing geostatistical marking

7.6 Bayesian methods for point pattern analysis

Appendix A Fundamentals of statistics

Appendix B Geometrical characteristics of sets

Appendix C Fundamentals of geostatistics

References

Notation index

Author index

Subject index

Statistical Analysis and Modelling of Spatial Point Patterns School of Mathematics and Statistics, University of St Andrews, Janine Illian Scotland, UK Antti Penttinen Department of Mathematics and Statistics, University of Jyväskylä, Finland Institut für Stochastik, TU Bergakademie Freiberg, Helga Stoyan Institut für Stochastik, TU Bergakademie Freiberg, Germany Dietrich Stoyan Germany

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Statistical Analysis and Modelling of Spatial Point Patterns

STATISTICS IN PRACTICE Advisory Editors Stephen Senn University of Glasgow, UK Marion Scott University of Glasgow, UK Founding Editor Vic Barnett Nottingham Trent University, UK Statistics in Practice is an important international series of texts which provide detailed coverage of statistical concepts, methods and worked case studies in specific fields of investigation and study. With sound motivation and many worked practical examples, the books show in down-to-earth terms how to select and use an appropriate range of statistical techniques in a particular practical field within each title’s special topic area. The books provide statistical support for professionals and research workers across a range of employment fields and research environments. Subject areas covered include medicine and pharmaceutics; industry, finance and commerce; public services; the earth and environmental sciences, and so on. The books also provide support to students studying statistical courses applied to the above areas. The demand for graduates to be equipped for the work environment has led to such courses becoming increasingly prevalent at universities and colleges. It is our aim to present judiciously chosen and well-written workbooks to meet everyday practical needs. Feedback of views from readers will be most valuable to monitor the success of this aim. A complete list of titles in this series appears at the end of the volume.

Copyright © 2008 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England +44 1243 779777 Telephone Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to permreq@wiley.co.uk, or faxed to (+44) 1243 770620. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 6045 Freemont Blvd, Mississauga, ONT, L5R 4J3 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. 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) QA278.2.S72 2008 519.5—dc22 I. Illian, Janine. 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, India Printed and bound in Great Britain by TJ International, Padstow, Cornwall This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.

Contents Preface List of examples 1 Introduction 1.1 Point process statistics 1.2 Examples of point process data 1.2.1 A pattern of amacrine cells 1.2.2 Gold particles 1.2.3 A pattern of Western Australian plants 1.2.4 Waterstriders 1.2.5 A sample of concrete 1.3 Historical notes 1.3.1 Determination of number of trees in a forest 1.3.2 Number of blood particles in a sample 1.3.3 Patterns of points in plant communities 1.3.4 Formulating the power law for the pair correlation function for galaxies 1.4 Sampling and data collection 1.4.1 General remarks 1.4.2 Choosing an appropriate study area 1.4.3 Data collection 1.5 Fundamentals of the theory of point processes 1.6 Stationarity and isotropy 1.6.1 Model approach and design approach 1.6.2 Finite and infinite point processes 1.6.3 Stationarity and isotropy 1.6.4 Ergodicity 1.7 Summary characteristics for point processes 1.7.1 Numerical summary characteristics 1.7.2 Functional summary characteristics xi xvii 1 2 5 5 6 7 8 10 10 10 12 13 15 17 17 19 20 23 35 35 36 37 39 40 41 42

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