Statistical Analysis and Modelling of Spatial Point Patterns

by Illian, Janine; Penttinen, Antti; Stoyan, Helga; Stoyan, Dietrich

• ISBN13: 9780470014912
• ISBN10: 0470014911
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-02-26
• Publisher: Wiley-Interscience

Summary

Spatial point processes are mathematical models used to describe and analyse the geometrical structure of patterns formed by objects that are irregularly or randomly distributed in one-, two- or three-dimensional space. Examples include locations of trees in a forest, blood particles on a glass plate, galaxies in the universe, and particle centres in samples of material.Numerous aspects of the nature of a specific spatial point pattern may be described using the appropriate statistical methods. Statistical Analysis and Modelling of Spatial Point Patterns provides a practical guide to the use of these specialised methods. The application-oriented approach helps demonstrate the benefits of this increasingly popular branch of statistics to a broad audience.The book:+ Provides an introduction to spatial point patterns for researchers across numerous areas of application.+ Adopts an extremely accessible style, allowing the non-statistician complete understanding.+ Describes the process of extracting knowledge from the data, emphasising the marked point process.+ Demonstrates the analysis of complex datasets, using applied examples from areas including biology, forestry, and materials science.+ Features a supplementary website containing example datasets.Statistical Analysis and Modelling of Spatial Point Patterns is ideally suited for researchers in the many areas of application, including environmental statistics, ecology, physics, materials science, geostatistics, and biology. It is also suitable for students of statistics, mathematics, computer science, biology and geoinformatics.

Author Biography

Janine Illian, SIMBIOS, University of Abertay, Dundee, Scotland

Antti Pentinen, Professor  in the Department of Mathematics and Statistics, University of Jyvaskyla, Finland

Dietrich Stoyan, Professor a the Insitut fur  Stochastik,  University of Freiberg, Germany

Table of Contents

Preface	p. xi
List of examples	p. xvii
Introduction	p. 1
Point process statistics	p. 2
Examples of point process data	p. 5
A pattern of amacrine cells	p. 5
Gold particles	p. 6
A pattern of Western Australian plants	p. 7
Waterstriders	p. 8
A sample of concrete	p. 10
Historical notes	p. 10
Determination of number of trees in a forest	p. 10
Number of blood particles in a sample	p. 12
Patterns of points in plant communities	p. 13
Formulating the power law for the pair correlation function for galaxies	p. 15
Sampling and data collection	p. 17
General remarks	p. 17
Choosing an appropriate study area	p. 19
Data collection	p. 20
Fundamentals of the theory of point processes	p. 23
Stationarity and isotropy	p. 35
Model approach and design approach	p. 35
Finite and infinite point processes	p. 36
Stationarity and isotropy	p. 37
Ergodicity	p. 39
Summary characteristics for point processes	p. 40
Numerical summary characteristics	p. 41
Functional summary characteristics	p. 42
Secondary structures of point processes	p. 42
Introduction	p. 42
Random sets	p. 43
Random fields	p. 44
Tessellations	p. 46
Neighbour networks or graphs	p. 49
Simulation of point processes	p. 52
The homogeneous Poisson point process	p. 57
Introduction	p. 58
The binomial point process	p. 59
Introduction	p. 59
Basic properties	p. 60
The periodic binomial process	p. 62
Simulation of the binomial process	p. 63
The homogeneous Poisson point process	p. 66
Introduction	p. 66
Basic properties	p. 67
Characterisations of the homogeneous Poisson process	p. 69
Simulation of a homogeneous Poisson process	p. 70
Model characteristics	p. 71
Moments and moment measures	p. 71
The Palm distribution of a homogeneous Poisson process	p. 74
Summary characteristics of the homogeneous Poisson process	p. 78
Estimating the intensity	p. 79
Testing complete spatial randomness	p. 83
Introduction	p. 83
Quadrat counts	p. 86
Distance methods	p. 89
The J-test	p. 91
Two index-based tests	p. 92
Discrepancy tests	p. 93
The L-test	p. 95
Other tests and recommendations	p. 97
Finite point processes	p. 99
Introduction	p. 100
Distributions of numbers of points	p. 104
The binomial distribution	p. 104
The Poisson distribution	p. 106
Compound distributions	p. 107
Generalised distributions	p. 109
Intensity functions and their estimation	p. 110
Parametric statistics for the intensity function	p. 111
Non-parametric estimation of the intensity function	p. 114
Estimating the point density distribution function	p. 117
Inhomogeneous Poisson process and finite Cox process	p. 118
The inhomogeneous Poisson process	p. 118
The finite Cox process	p. 123
Summary characteristics for finite point processes	p. 125
Nearest-neighbour distances	p. 126
Dilation function	p. 127
Graph-theoretic statistics	p. 129
Second-order characteristics	p. 129
Finite Gibbs processes	p. 137
Introduction	p. 137
Gibbs processes with fixed number of points	p. 139
Gibbs processes with a random number of points	p. 147
Second-order summary characteristics of finite Gibbs processes	p. 154
Further discussion	p. 156
Statistical inference for finite Gibbs processes	p. 160
Stationary point processes	p. 173
Basic definitions and notation	p. 174
Summary characteristics for stationary point processes	p. 179
Introduction	p. 179
Edge-correction methods	p. 180
The intensity [lambda]	p. 189
Indices as summary characteristics	p. 195
Empty-space statistics and other morphological summaries	p. 199
The nearest-neighbour distance distribution function	p. 206
The J-function	p. 213
Second-order characteristics	p. 214
The three functions: K, L and g	p. 214
Theoretical foundations of second-order characteristics	p. 223
Estimators of the second-order characteristics	p. 228
Interpretation of pair correlation functions	p. 239
Higher-order and topological characteristics	p. 244
Introduction	p. 244
Third-order characteristics	p. 244
Delaunay tessellation characteristics	p. 247
The connectivity function	p. 248
Orientation analysis for stationary point processes	p. 250
Introduction	p. 250
Nearest-neighbour orientation distribution	p. 252
Second-order orientation analysis	p. 254
Outliers, gaps and residuals	p. 256
Introduction	p. 256
Simple outlier detection	p. 256
Simple gap detection	p. 257
Model-based outliers	p. 257
Residuals	p. 259
Replicated patterns	p. 260
Introduction	p. 260
Aggregation recipes	p. 261
Choosing appropriate observation windows	p. 264
General ideas	p. 264
Representative windows	p. 265
Multivariate analysis of series of point patterns	p. 270
Summary characteristics for the non-stationary case	p. 279
Formal application of stationary characteristics and estimators	p. 280
Intensity reweighting	p. 281
Local rescaling	p. 282
Stationary marked point processes	p. 293
Basic definitions and notation	p. 294
Introduction	p. 294
Marks and their properties	p. 295
Marking models	p. 296
Stationarity	p. 299
First-order characteristics	p. 300
Mark-sum measure	p. 304
Palm distribution	p. 304
Summary characteristics	p. 306
Introduction	p. 306
Intensity and mark-sum intensity	p. 306
Mean mark, mark d.f. and mark probabilities	p. 309
Indices for stationary marked point processes	p. 311
Nearest-neighbour distributions	p. 320
Second-order characteristics for marked point processes	p. 323
Introduction	p. 323
Definitions for qualitative marks	p. 323
Definitions for quantitative marks	p. 341
Estimation of second-order characteristics	p. 352
Orientation analysis for marked point processes	p. 355
Introduction	p. 355
Orientation analysis for anisotropic processes with angular marks	p. 357
Orientation analysis for isotropic processes with angular marks	p. 357
Orientation analysis with constructed marks	p. 359
Modelling and simulation of stationary point processes	p. 363
Introduction	p. 364
Operations with point processes	p. 364
Thinning	p. 365
Clustering	p. 368
Superposition	p. 370
Cluster processes	p. 371
General cluster processes	p. 371
Neyman-Scott processes	p. 374
Stationary Cox processes	p. 379
Introduction	p. 379
Properties of stationary Cox processes	p. 383
Statistics for Cox processes	p. 386
Hard-core point processes	p. 387
Introduction	p. 387
Matern hard-core processes	p. 388
The dead leaves model	p. 391
The RSA process	p. 393
Random dense packings of hard spheres	p. 394
Stationary Gibbs processes	p. 398
Basic ideas and equations	p. 398
Simulation of stationary Gibbs processes	p. 402
Statistics for stationary Gibbs processes	p. 402
Reconstruction of point patterns	p. 407
Reconstructing point patterns without a specified model	p. 407
An example: reconstruction of Neyman-Scott processes	p. 410
Practical application of the reconstruction algorithm	p. 415
Formulas for marked point process models	p. 417
Introduction	p. 417
Independent marks	p. 418
Random field model	p. 420
Intensity-weighted marks	p. 421
Moment formulas for stationary shot-noise fields	p. 423
Space-time point processes	p. 425
Introduction	p. 425
Space-time Poisson processes	p. 428
Second-order statistics for completely stationary event processes	p. 430
Two examples of space-time processes	p. 434
Correlations between point processes and other random structures	p. 437
Introduction	p. 437
Correlations between point processes and random fields	p. 438
Correlations between point processes and fibre processes	p. 442
Fitting and testing point process models	p. 445
Choice of model	p. 445
Parameter estimation	p. 448
Maximum likelihood method	p. 448
Method of moments	p. 450
Trial-and-error estimation	p. 452
Variance estimation by bootstrap	p. 453
Goodness-of-fit tests	p. 455
Envelope test	p. 455
Deviation test	p. 457
Testing mark hypotheses	p. 460
Introduction	p. 460
Testing independent marking, test of association	p. 460
Testing geostatistical marking	p. 468
Bayesian methods for point pattern analysis	p. 471
Fundamentals of statistics	p. 479
Geometrical characteristics of sets	p. 483
Fundamentals of geostatistics	p. 489
References	p. 493
Notation index	p. 515
Author index	p. 519
Subject index	p. 527
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