Stochastic Geometry and Its Applications

by Chiu, Sung Nok; Stoyan, Dietrich; Kendall, Wilfrid S.; Mecke, Joseph

• ISBN13: 9780470664810
• ISBN10: 0470664819
• Edition: 3rd
• Format: Hardcover
• Copyright: 2013-09-10
• Publisher: Wiley

Summary

An extensive update to a classic text

Stochastic geometry and spatial statistics play a fundamental role in many modern branches of physics, materials sciences, engineering, biology and environmental sciences. They offer successful models for the description of random two- and three-dimensional micro and macro structures and statistical methods for their analysis.

The previous edition of this book has served as the key reference in its field for over 18 years and is regarded as the best treatment of the subject of stochastic geometry, both as a subject with vital applications to spatial statistics and as a very interesting field of mathematics in its own right.

This edition:

• Presents a wealth of models for spatial patterns and related statistical methods.
• Provides a great survey of the modern theory of random tessellations, including many new models that became tractable only in the last few years.
• Includes new sections on random networks and random graphs to review the recent ever growing interest in these areas.
• Provides an excellent introduction to theory and modelling of point processes, which covers some very latest developments.
• Illustrate the forefront theory of random sets, with many applications.
• Adds new results to the discussion of fibre and surface processes.
• Offers an updated collection of useful stereological methods.
• Includes 700 new references.
• Is written in an accessible style enabling non-mathematicians to benefit from this book.
• Provides a companion website hosting information on recent developments in the field www.wiley.com/go/cskm

Stochastic Geometry and its Applications is ideally suited for researchers in physics, materials science, biology and ecological sciences as well as mathematicians and statisticians. It should also serve as a valuable introduction to the subject for students of mathematics and statistics.

Author Biography

Sung Nok Chiu, Department of Mathematics, Hong Kong Baptist University, Hong Kong

Dietrich Stoyan, Institute of Stochastics, TU Bergakademie Freiberg, Germany

Wilfrid S. Kendall, Department of Statistics, University of Warwick, UK

Joseph Mecke, Faculty of Mathematics and Computer Science, Friedrich-Schiller-Universität Jena, Germany

Table of Contents

Foreword to the First Edition

From the Preface to the First Edition

Preface to the Second Edition

Preface to the Third Edition

1 Mathematical Foundations

1.1 Set theory

1.2 Topology in Euclidean spaces

1.3 Operations on subsets of Euclidean space

1.4 Mathematical morphology and image analysis

1.5 Euclidean isometries

1.6 Convex sets in Euclidean spaces

1.7 Functions describing convex sets

1.7.1 General

1.7.2 Set covariance

1.7.3 Chord length distribution

1.7.4 Erosion–dilation functions

1.8 Polyconvex sets

1.9 Measure and integration theory

2 Point Processes I | The Poisson Point Process

2.1 Introduction

2.2 The binomial point process

2.2.1 Introduction

2.2.2 Basic properties

2.2.3 Simulation

2.3 The homogeneous Poisson point process

2.3.1 Definition and defining properties

2.3.2 Characterisation of the homogeneous Poisson point process

2.3.3 Moments and moment measures

2.3.4 The Palm distribution of a homogeneous Poisson point process

2.4 The inhomogeneous and general Poisson point process

2.5 Simulation of Poisson point processes

2.5.1 Simulation of a homogeneous Poisson point process

2.5.2 Simulation of an inhomogeneous Poisson point process

2.6 Statistics for the homogeneous Poisson point process

2.6.1 Introduction

2.6.2 Estimating the intensity

2.6.3 Testing the hypothesis of homogeneity

2.6.4 Testing the Poisson process hypothesis

3 Random Closed Sets I | The Boolean Model

3.1 Introduction and basic properties

3.1.1 Model description

3.1.2 Applications

3.1.3 Stationarity and isotropy

3.1.4 Simulation

3.1.5 The capacity functional

3.1.6 Basic characteristics

3.1.7 Contact distribution functions

3.2 The Boolean model with convex grains

3.2.1 The simplified formula for the capacity functional

3.2.2 Intensities or densities of intrinsic volumes

3.2.3 Contact distribution functions

3.2.4 Morphological functions

3.2.5 Intersections with linear subspaces

3.2.6 Formulae for some special Boolean models with isotropic convex

grains

3.3 Coverage and connectivity

3.3.1 Coverage probabilities

3.3.2 Clumps

3.3.3 Connectivity

3.3.4 Percolation

3.3.5 Vacant regions

3.4 Statistics

3.4.1 General remarks

3.4.2 Testing model assumptions

3.4.3 Estimation of model parameters

3.5 Generalisations and variations

3.6 Hints for practical applications

4 Point Processes II | General Theory

4.1 Basic properties

4.1.1 Introduction

4.1.2 The distribution of a point process

4.1.3 Notation

4.1.4 Stationarity and isotropy

4.1.5 Intensity measure and intensity

4.1.6 Ergodicity and central limit theorem

4.1.7 Contact distributions

4.2 Marked point processes

4.2.1 Fundamentals

4.2.2 Intensity and mark distribution

4.3 Moment measures and related quantities

4.3.1 Moment measures

4.3.2 Factorial moment measures

4.3.3 Product densities

4.3.4 The Campbell measure

4.3.5 The mark correlation function

4.3.6 The probability generating functional

4.4 Palm distributions

4.4.1 Heuristic introduction

4.4.2 The Palm distribution: first definition

4.4.3 The Palm distribution: second definition

4.4.4 Reduced Palm distributions

4.4.5 Isotropy of Palm distribution

4.4.6 Inversion formulae

4.4.7 n-fold Palm distributions

4.4.8 Palm distributions for marked point processes

4.4.9 Point-stationarity

4.4.10 Stationary and balanced partitions

4.5 The second moment measure

4.6 Summary characteristics

4.7 Introduction to statistics for spatial point processes

4.7.1 General remarks

4.7.2 Edge-corrections

4.7.3 Estimation of the intensity λ

4.7.4 Estimation of the reduced second moment measure

4.7.5 Estimation of the spherical contact distribution and of the probability

generating functional

4.7.6 Estimation of the nearest-neighbour distance distribution function

4.7.7 Estimation of Palm characteristics and mark distributions

4.7.8 Parameter estimation

4.7.9 Representative windows, representative volume elements

4.7.10 Hypotheses testing

4.8 General point processes

5 Point Processes III { Models

5.1 Operations on point processes

5.2 Doubly stochastic Poisson processes (Cox processes)

5.2.1 Introduction

5.2.2 Examples of Cox processes

5.2.3 Formulae for characteristics of Cox processes

5.3 Neyman–Scott processes

5.4 Hard-core point processes

5.5 Gibbs point processes

5.5.1 Introduction

5.5.2 Gibbs point processes in bounded regions

5.5.3 Stationary Gibbs point processes

5.5.4 Spatial birth-and-death processes

5.5.5 Simulation of stationary Gibbs processes

5.6 Shot-noise fields

5.6.1 Definition and examples

5.6.2 Moment formulae for stationary shot-noise fields

 

 

6 Random Closed Sets II | The General Case

6.1 Basic properties

6.1.1 Introduction

6.1.2 Random set definition

6.1.3 Capacity functional and Choquet theorem

6.1.4 Distributional properties

6.1.5 Miscellany

6.2 Random compact sets

6.2.1 Definition of means

6.2.2 Mean-value formulae for convex random sets

6.3 Characteristics for stationary and isotropic random closed sets

6.3.1 The area or volume fraction

6.3.2 The covariance

6.3.3 Contact distribution functions

6.3.4 Chord length distributions

6.3.5 Directional analysis of random closed sets .

6.3.6 Intensities or densities of random closed sets

6.4 Nonparametric statistics for stationary random closed sets

6.4.1 Introduction

6.4.2 Estimation of the area or volume fraction p

6.4.3 Estimation of the covariance

6.4.4 Second-order analysis with random fields

6.4.5 Estimation of contact distributions HB(r)

6.4.6 Representative volume elements

6.5 Germ–grain models

6.5.1 Basic facts

6.5.2 Formulae for p and C(r)

6.5.3 Models of mutually non-overlapping balls

6.5.4 Shot-noise germ–grain models

6.5.5 Weighted grain distributions

6.5.6 Intersection formulae

6.5.7 Statistics for motion-invariant germ–grain models

6.6 Other random closed set models

6.6.1 Gibbs discrete random sets

6.6.2 Dilated fibre and surface processes

6.6.3 Excursion sets

6.6.4 Birth-and-growth processes

6.7 Stochastic reconstruction of random sets

7 Random Measures

7.1 Fundamentals

7.1.1 Introduction

7.1.2 Definitions and facts

7.1.3 Palm distributions

7.1.4 Marked random measures

7.2 Moment measures and related characteristics

7.2.1 The Laplace functional

7.2.2 Moment measures

7.3 Examples of random measures

7.3.1 Random measures constructed from point processes

7.3.2 Random measures constructed from random fields

7.3.3 Completely random measures

7.3.4 Random measures generated by random closed sets: curvature measures

8 Line, Fibre and Surface Processes

8.1 Introduction

8.2 Flat processes

8.2.1 Introduction

8.2.2 Planar line processes

8.2.3 Spatial line and plane processes

8.2.4 Applications of line and plane processes

8.3 Planar fibre processes

8.3.1 Fundamentals

8.3.2 Intersections of fibre processes

8.3.3 Basic statistical methods for planar fibre processes

8.4 Spatial fibre processes

8.5 Surface processes

8.5.1 Plane processes

8.5.2 General surface processes

8.6 Marked fibre and surface processes

9 Random Tessellations, Geometrical Networks and Graphs

9.1 Introduction and definitions

9.2 Mathematical models for random tessellations

9.3 General ideas and results for stationary planar tessellations

9.3.1 Point processes related to tessellations

9.3.2 Typical vertex, edge and cell

9.3.3 Zero cell

9.3.4 Mean-value relationships for stationary planar tessellations

9.3.5 The neighbourhood of the typical cell .

9.4 Mean-value formulae for stationary spatial tessellations

9.5 Poisson line and plane tessellations

9.5.1 Poisson line tessellations

9.5.2 Poisson plane tessellations

9.6 STIT tessellations

9.7 Poisson-Voronoi and Delaunay tessellations

9.7.1 General

9.7.2 Planar Poisson-Voronoi tessellations (Poisson-Dirichlet tessellations)

9.7.3 Spatial Poisson-Voronoi tessellations .

9.7.4 Poisson-Delaunay tessellations

9.8 Laguerre tessellations

9.9 Johnson–Mehl tessellations

9.10 Statistics for stationary tessellations

9.10.1 Reconstruction

9.10.2 Summary characteristics

9.10.3 Statistics for planar tessellations

9.10.4 Statistics for Voronoi, Laguerre and Johnson–Mehl tessellations

9.11 Random geometrical networks

9.11.1 Introduction

9.11.2 Formal definition of random geometrical networks

9.11.3 Summary characteristics of stationary random geometrical networks

9.11.4 Statistics for networks

9.11.5 Models of random geometrical networks

9.12 Random graphs

9.12.1 Introduction

9.12.2 Random graph models and their properties

10 Stereology

10.1 Introduction

10.2 The fundamental mean-value formulae of stereology

10.2.1 Notation

10.2.2 Planar and linear sections

10.2.3 Thick sections

10.2.4 Stereology for excursion sets

10.2.5 On the precision of stereological estimators

10.3 Stereological mean-value formulae for germ–grain models

10.3.1 Planar sections

10.3.2 Thick sections of spatial germ–grain models

10.3.3 Tubular structures and membranes

10.4 Stereological methods for spatial systems of balls

10.4.1 Introduction

10.4.2 Planar sections and the Wicksell corpuscle problem

10.4.3 Linear sections

10.4.4 Thick sections

10.4.5 Sieving distributions for balls

10.5 Stereological problems for non-spherical grains (shape-and-size problems)

10.5.1 General remarks

10.5.2 Two particular grain shapes

10.6 Stereology for spatial tessellations

10.7 Second-order characteristics and directional distributions

10.7.1 Introduction

10.7.2 Stereological determination of the pair correlation function of a system of ball centres

10.7.3 Second-order analysis for spatial fibre systems

10.7.4 Determination of directional distributions

References

Author Index

Subject Index