Statistical Analysis and Modelling of Spatial Point Patterns

by Janine Illian (University of St. Andrews, Scotland ; ); Antti Penttinen (University of Jyvaskyla, Finland); Helga Stoyan; Dietrich Stoyan (Technical University, Bergakademie Freiberg Germany)

• ISBN13: 9780470725153
• ISBN10: 047072515X
• Format: eBook
• Copyright: 2008-06-01
• 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.

Table of Contents

Preface

List of Examples

Introduction

Point process statistics

Examples of point process data

A pattern of amacrine cells

Gold particles

A pattern of Western Australian plants

Waterstriders

A sample of concrete

Historical notes

Determination of number of trees in a forest

Number of blood particles in a sample

Patterns of points in plant communities

Formulating the power law for the pair correlation function for galaxies

Sampling and data collection

General remarks

Choosing an appropriate study area

Data collection

Fundamentals of the theory of point processes

Stationarity and isotropy

Model approach and design approach

Finite and infinite point processes

Stationarity and isotropy

Ergodicity

Summary characteristics for point processes

Numerical summary characteristics

Functional summary characteristics

Secondary structures of point processes

Introduction

Random sets

Random fields

Tessellations

Neighbour networks or graphs

Simulation of point processes

The Homogeneous Poisson point process

Introduction

The binomial point process

Introduction

Basic properties

The periodic binomial process

Simulation of the binomial process

The homogeneous Poisson point process

Introduction

Basic properties

Characterisations of the homogeneous Poisson process

Simulation of a homogeneous Poisson process

Model characteristics

Moments and moment measures

The Palm distribution of a homogeneous Poisson process

Summary characteristics of the homogeneous Poisson process

Estimating the intensity

Testing complete spatial randomness

Introduction

Quadrat counts

Distance methods

The J-test

Two index-based tests

Discrepancy tests

The L-test

Other tests and recommendations

Finite point processes

Introduction

Distributions of numbers of points

The binomial distribution

The Poisson distribution

Compound distributions

Generalised distributions

Intensity functions and their estimation

Parametric statistics for the intensity function

Non-parametric estimation of the intensity function

Estimating the point density distribution function

Inhomogeneous Poisson process and finite Cox process

The inhomogeneous Poisson process

The finite Cox process

Summary characteristics for finite point processes

Nearest-neighbour distances

Dilation function

Graph-theoretic statistics

Second-order characteristics

Finite Gibbs processes

Introduction

Gibbs processes with fixed number of points

Gibbs processes with a random number of points

Second-order summary characteristics of finite Gibbs processes

Further discussion

Statistical inference for finite Gibbs processes

Stationary point processes

Basic definitions and notation

Summary characteristics for stationary point processes

Introduction

Edge-correction methods

The intensity ¿

Indices as summary characteristics

Empty-space statistics and other morphological summaries

The nearest-neighbour distance distribution function

The J-function

Second-order characteristics

The three functions: K, L and g

Theoretical foundations of second-order characteristics

Estimators of the second-order characteristics

Interpretation of pair correlation functions

Higher-order and topological characteristics

Introduction

Third-order characteristics

Delaunay tessellation characteristics

The connectivity function

Orientation analysis for stationary point processes

Introduction

Nearest-neighbour orientation distribution

Second-order orientation analysis

Outliers, gaps and residuals

Introduction

Simple outlier detection

Simple gap detection

Model-based outliers

Residuals

Replicated patterns

Introduction

Aggregation recipes

Choosing appropriate observation windows

General ideas

Representative windows

Multivariate analysis of series of point patterns

Summary characteristics for the non-stationary case

Formal application of stationary characteristics and estimators

Intensity reweighting

Local rescaling

Stationary marked point processes

Basic definitions and notation

Introduction

Marks and their properties

Marking models

Stationarity

First-order characteristics

Mark-sum measure

Palm distribution

Summary characteristics

Introduction

Intensity and mark-sum intensity

Mean mark, mark d.f. and mark probabilities

Indices for stationary marked point processes

Nearest-neighbour distributions

Second-order characteristics for marked point processes

Introduction

Definitions for qualitative marks

Definitions for quantitative marks

Estimation of second-order characteristics

Orientation analysis for marked point processes

Introduction

Orientation analysis for non-isotropic processes with angular marks

Orientation analysis for isotropic processes with angular marks

Orientation analysis with constructed marks

Modelling and simulation of stationary point processes

Introduction

Operations with point processes

Thinning

Clustering

Superposition

Cluster processes

General cluster processes

Neyman-Scott processes

Stationary Cox processes

Introduction

Properties of stationary Cox processes

Hard-core point processes

Introduction

Matérn hard-core processes

The dead leaves model

The RSA model

Random dense packings of hard spheres

Stationary Gibbs processes

Basic ideas and equations

Simulation of stationary Gibbs processes

Statistics for stationary Gibbs processes

Reconstruction of point patterns

Reconstructing point patterns without a specified model

An example: reconstruction of Neyman-Scott processes

Practical application of the reconstruction algorithm

Formulas for marked point process models

Introduction

Independent marks

Random field model

Intensity-weighted marks

Moment formulas for stationary shot-noise fields

Space-time point processes

Introduction

Space-time Poisson processes

Second-order statistics for completely stationary event processes

Two examples of space-time processes

Correlations between point processes and other random structures

Introduction

Correlations between point processes and random fields

Correlations between point processes and fibre processes

Fitting and testing point process models

Choice of model

Parameter estimation

Maximum likelihood method

Method of moments

Trial-and-error estimation

Variance estimation by bootstrap

Goodness-of-fit tests

Envelope test

Deviation test

Testing mark hypotheses

Introduction

Testing independent marking, test of association

Testing geostatistical marking

Bayesian methods for point pattern analysis

Fundamentals of statistics

Geometrical characteristics of sets

Fundamentals of geostatistics

References

Notation index

Author index

Subject index

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