Stochastic Geometry and Its Applications
Wiley-Blackwell (Verlag)
978-0-470-66481-0 (ISBN)
Extensively updated, this mew edition includes new sections on analytical and numerically tractable results and applications of Voronoi tessellations; introduces models such as Laguerre and iterated tessellations; and presents theoretical results.
Statistics for planar point processes are introduced, and the text also includes a new section on random geometrical graphs and random networks.
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.
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-Universitat Jena, Germany
Foreword to the first edition xiii
From the preface to the first edition xvii
Preface to the second edition xix
Preface to the third edition xxi
Notation xxiii
1 Mathematical foundations 1
1.1 Set theory 1
1.2 Topology in Euclidean spaces 3
1.3 Operations on subsets of Euclidean space 5
1.4 Mathematical morphology and image analysis 7
1.5 Euclidean isometries 9
1.6 Convex sets in Euclidean spaces 10
1.7 Functions describing convex sets 17
1.7.1 General 17
1.7.2 Set covariance 17
1.7.3 Chord length distribution 20
1.7.4 Erosion-dilation functions 24
1.8 Polyconvex sets 24
1.9 Measure and integration theory 27
2 Point processes I - The Poisson point process 35
2.1 Introduction 35
2.2 The binomial point process 36
2.2.1 Introduction 36
2.2.2 Basic properties 37
2.2.3 Simulation 38
2.3 The homogeneous Poisson point process 41
2.3.1 Definition and defining properties 41
2.3.2 Characterisation of the homogeneous Poisson point process 43
2.3.3 Moments and moment measures 44
2.3.4 The Palm distribution of a homogeneous Poisson point process 48
2.4 The inhomogeneous and general Poisson point process 51
2.5 Simulation of Poisson point processes 53
2.5.1 Simulation of a homogeneous Poisson point process 53
2.5.2 Simulation of an inhomogeneous Poisson point process 54
2.6 Statistics for the homogeneous Poisson point process 55
2.6.1 Introduction 55
2.6.2 Estimating the intensity 55
2.6.3 Testing the hypothesis of homogeneity 56
2.6.4 Testing the Poisson process hypothesis 56
3 Random closed sets I - The Boolean model 64
3.1 Introduction and basic properties 64
3.1.1 Model description 64
3.1.2 Applications 66
3.1.3 Stationarity and isotropy 69
3.1.4 Simulation 71
3.1.5 The capacity functional 71
3.1.6 Basic characteristics 73
3.1.7 Contact distribution functions 76
3.2 The Boolean model with convex grains 78
3.2.1 The simplified formula for the capacity functional 78
3.2.2 Intensities or densities of intrinsic volumes 79
3.2.3 Contact distribution functions 81
3.2.4 Morphological functions 84
3.2.5 Intersections with linear subspaces 84
3.2.6 Formulae for some special Boolean models with isotropic
convex grains 87
3.3 Coverage and connectivity 89
3.3.1 Coverage probabilities 89
3.3.2 Clumps 91
3.3.3 Connectivity 91
3.3.4 Percolation 92
3.3.5 Vacant regions 94
3.4 Statistics 95
3.4.1 General remarks 95
3.4.2 Testing model assumptions 96
3.4.3 Estimation of model parameters 98
3.5 Generalisations and variations 103
3.6 Hints for practical applications 106
4 Point processes II - General theory 108
4.1 Basic properties 108
4.1.1 Introduction 108
4.1.2 The distribution of a point process 110
4.1.3 Notation 110
4.1.4 Stationarity and isotropy 112
4.1.5 Intensity measure and intensity 112
4.1.6 Ergodicity and central limit theorem 114
4.1.7 Contact distributions 115
4.2 Marked point processes 116
4.2.1 Fundamentals 116
4.2.2 Intensity and mark distribution 118
4.3 Moment measures and related quantities 120
4.3.1 Moment measures 120
4.3.2 Factorial moment measures 121
4.3.3 Product densities 122
4.3.4 The Campbell measure 123
4.3.5 The mark correlation function 123
4.3.6 The probability generating functional 125
4.4 Palm distributions 127
4.4.1 Heuristic introduction 127
4.4.2 The Palm distribution: First definition 129
4.4.3 The Palm distribution: Second definition 130
4.4.4 Reduced Palm distributions 131
4.4.5 Isotropy of Palm distribution 133
4.4.6 Inversion formulae 134
4.4.7 n-fold Palm distributions 134
4.4.8 Palm distributions for marked point processes 134
4.4.9 Point-stationarity 136
4.4.10 Stationary and balanced partitions 137
4.5 The second moment measure 139
4.6 Summary characteristics 143
4.7 Introduction to statistics for stationary spatial point processes 145
4.7.1 General remarks 145
4.7.2 Edge-corrections 145
4.7.3 Estimation of the intensity lambda 147
4.7.4 Estimation of the reduced second moment measure 148
4.7.5 Estimation of the spherical contact distribution and of the
probability generating functional 150
4.7.6 Estimation of the nearest-neighbour distance distribution function 151
4.7.7 Estimation of Palm characteristics and mark distributions 152
4.7.8 Parameter estimation 153
4.7.9 Representative windows, representative volume elements 154
4.7.10 Hypotheses testing 155
4.8 General point processes 156
5 Point processes III - Models 158
5.1 Operations on point processes 158
5.2 Doubly stochastic Poisson processes (Cox processes) 166
5.2.1 Introduction 166
5.2.2 Examples of Cox processes 166
5.2.3 Formulae for characteristics of Cox processes 168
5.3 Neyman-Scott processes 171
5.4 Hard-core point processes 176
5.5 Gibbs point processes 178
5.5.1 Introduction 178
5.5.2 Gibbs point processes in bounded regions 180
5.5.3 Stationary Gibbs point processes 186
5.5.4 Spatial birth-and-death processes 192
5.5.5 Simulation of stationary Gibbs processes 194
5.6 Shot-noise fields 200
5.6.1 Definition and examples 200
5.6.2 Moment formulae for stationary shot-noise fields 202
6 Random closed sets II - The general case 205
6.1 Basic properties 205
6.1.1 Introduction 205
6.1.2 Random set definition 206
6.1.3 Capacity functional and Choquet theorem 207
6.1.4 Distributional properties 208
6.1.5 Miscellany 211
6.2 Random compact sets 213
6.2.1 Definition of means 213
6.2.2 Mean-value formulae for convex random sets 215
6.3 Characteristics for stationary and isotropic random closed sets 216
6.3.1 The area or volume fraction 216
6.3.2 The covariance 217
6.3.3 Contact distribution functions 223
6.3.4 Chord length distributions 226
6.3.5 Directional analysis of random closed sets 228
6.3.6 Intensities or densities of random closed sets 229
6.4 Nonparametric statistics for stationary random closed sets 230
6.4.1 Introduction 230
6.4.2 Estimation of the area or volume fraction p 230
6.4.3 Estimation of the covariance 233
6.4.4 Second-order analysis with random fields 234
6.4.5 Estimation of contact distributions 235
6.4.6 Representative volume elements 236
6.5 Germ-grain models 237
6.5.1 Basic facts 237
6.5.2 Formulae for p and C(r) 238
6.5.3 Models of mutually non-overlapping balls 239
6.5.4 Shot-noise germ-grain models 245
6.5.5 Weighted grain distributions 247
6.5.6 Intersection formulae 248
6.5.7 Statistics for motion-invariant germ-grain models 250
6.6 Other random closed set models 255
6.6.1 Gibbs discrete random sets 255
6.6.2 Dilated fibre and surface processes 259
6.6.3 Excursion sets 259
6.6.4 Birth-and-growth processes 269
6.7 Stochastic reconstruction of random sets 276
7 Random measures 279
7.1 Fundamentals 279
7.1.1 Introduction 279
7.1.2 Definitions and facts 280
7.1.3 Palm distributions 281
7.1.4 Marked random measures 283
7.2 Moment measures and related characteristics 284
7.2.1 The Laplace functional 284
7.2.2 Moment measures 284
7.3 Examples of random measures 286
7.3.1 Random measures constructed from point processes 286
7.3.2 Random measures constructed from random fields 286
7.3.3 Completely random measures 286
7.3.4 Random measures generated by random closed sets:
Curvature measures 287
8 Line, fibre and surface processes 297
8.1 Introduction 297
8.2 Flat processes 302
8.2.1 Introduction 302
8.2.2 Planar line processes 302
8.2.3 Spatial line and plane processes 311
8.2.4 Applications of line and plane processes 313
8.3 Planar fibre processes 314
8.3.1 Fundamentals 314
8.3.2 Intersections of fibre processes 322
8.3.3 Basic statistical methods for planar fibre processes 327
8.4 Spatial fibre processes 330
8.5 Surface processes 333
8.5.1 Plane processes 333
8.5.2 General surface processes 335
8.6 Marked fibre and surface processes 339
9 Random tessellations, geometrical networks and graphs 343
9.1 Introduction and definitions 343
9.2 Mathematical models for random tessellations 346
9.3 General ideas and results for stationary planar tessellations 357
9.3.1 Point processes related to tessellations 357
9.3.2 Typical vertex, edge and cell 358
9.3.3 Zero cell 359
9.3.4 Mean-value relationships for stationary planar tessellations 359
9.3.5 The neighbourhood of the typical cell 366
9.4 Mean-value formulae for stationary spatial tessellations 367
9.5 Poisson line and plane tessellations 370
9.5.1 Poisson line tessellations 371
9.5.2 Poisson plane tessellations 373
9.6 STIT tessellations 375
9.7 Poisson-Voronoi and Delaunay tessellations 376
9.7.1 General 376
9.7.2 Planar Poisson-Voronoi tessellations (Poisson-Dirichlet
tessellations) 377
9.7.3 Spatial Poisson-Voronoi tessellations 380
9.7.4 Poisson-Delaunay tessellations 383
9.8 Laguerre tessellations 386
9.9 Johnson-Mehl tessellations 388
9.10 Statistics for stationary tessellations 390
9.10.1 Reconstruction 390
9.10.2 Summary characteristics 390
9.10.3 Statistics for planar tessellations 391
9.10.4 Statistics for Voronoi, Laguerre and Johnson-Mehl
tessellations 393
9.11 Random geometrical networks 397
9.11.1 Introduction 397
9.11.2 Formal definition of random geometrical networks 398
9.11.3 Summary characteristics of stationary random
geometrical networks 399
9.11.4 Statistics for networks 400
9.11.5 Models of random geometrical networks 401
9.12 Random graphs 402
9.12.1 Introduction 402
9.12.2 Random graph models and their properties 403
10 Stereology 411
10.1 Introduction 411
10.2 The fundamental mean-value formulae of stereology 413
10.2.1 Notation 413
10.2.2 Planar and linear sections 414
10.2.3 Thick sections 419
10.2.4 Stereology for excursion sets 420
10.2.5 On the precision of stereological estimators 421
10.3 Stereological mean-value formulae for germ-grain models 421
10.3.1 Planar sections 421
10.3.2 Thick sections of spatial germ-grain models 424
10.3.3 Tubular structures and membranes 425
10.4 Stereological methods for spatial systems of balls 425
10.4.1 Introduction 425
10.4.2 Planar sections and the Wicksell corpuscle problem 426
10.4.3 Linear sections 433
10.4.4 Thick sections 435
10.4.5 Sieving distributions for balls 436
10.5 Stereological problems for nonspherical grains (shape-and-size
problems) 436
10.5.1 General remarks 436
10.5.2 Two particular grain shapes 439
10.6 Stereology for spatial tessellations 440
10.7 Second-order characteristics and directional distributions 444
10.7.1 Introduction 444
10.7.2 Stereological determination of the pair correlation function
of a system of ball centres 445
10.7.3 Second-order analysis for spatial fibre systems 448
10.7.4 Determination of directional distributions 451
References 453
Author index 507
Subject index 521
| Erscheint lt. Verlag | 30.8.2013 |
|---|---|
| Reihe/Serie | Wiley Series in Probability and Statistics |
| Zusatzinfo | Illustrations |
| Verlagsort | Chicester |
| Sprache | englisch |
| Maße | 177 x 252 mm |
| Gewicht | 1038 g |
| Einbandart | gebunden |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Geometrie / Topologie |
| Mathematik / Informatik ► Mathematik ► Statistik | |
| Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik | |
| ISBN-10 | 0-470-66481-9 / 0470664819 |
| ISBN-13 | 978-0-470-66481-0 / 9780470664810 |
| Zustand | Neuware |
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