Stochastic Geometry: Likelihood and Computation

by Kendall; Wilfrid S.

• ISBN13: 9780849303968
• ISBN10: 0849303966
• Format: Hardcover
• Copyright: 1998-10-20
• Publisher: Chapman & Hall/

Summary

Stochastic geometry involves the study of random geometric structures, and blends geometric, probabilistic, and statistical methods to provide powerful techniques for modeling and analysis. Recent developments in computational statistical analysis, particularly Markov chain Monte Carlo, have enormously extended the range of feasible applications. Stochastic Geometry: Likelihood and Computation provides a coordinated collection of chapters on important aspects of the rapidly developing field of stochastic geometry, including:o a "crash-course" introduction to key stochastic geometry themeso considerations of geometric sampling bias issueso tesselationso shapeo random setso image analysiso spectacular advances in likelihood-based inference now available to stochastic geometry through the techniques of Markov chain Monte Carlo

Table of Contents

Contributors	xi	(2)
Preface	xiii
1 Crash course in stochastic geometry	1	(36)
1.1 Introduction	1	(2)
1.2 Hitting probabilities and geometrical measures	3	(3)
1.3 Linearity of expectation	6	(5)
1.4 Integral geometry	11	(2)
1.5 Independence, Poisson processes and Boolean models	13	(4)
1.6 Incidence and vacancy	17	(2)
1.7 Distances and waiting times	19	(1)
1.8 Weighted distributions	19	(7)
1.9 Conditioning and variational methods	26	(11)
2 Sampling and censoring	37	(42)
2.1 Spatial sampling bias	39	(5)
2.2 Unbiased sampling rules	44	(3)
2.3 Additive functionals	47	(1)
2.4 Horvitz-Thompson estimators	48	(2)
2.5 Sampling bias for point processes	50	(9)
2.6 Censoring effects for point processes	59	(10)
2.7 Line segment processes	69	(1)
2.8 Censoring of random sets	70	(9)
3 Likelihood inference for spatial point processes	79	(62)
3.1 Introduction	79	(1)
3.2 Families of unnormalized densities	80	(1)
3.3 Examples of families of unnormalized densities	81	(4)
3.4 Markov chain Monte Carlo	85	(8)
3.5 Finite point processes	93	(6)
3.6 Simulating finite point processes	99	(3)
3.7 Stability conditions	102	(1)
3.8 Markov Chain Convergence	103	(7)
3.9 Two New Point Processes	110	(3)
3.10 Monte Carlo Likelihood Inference	113	(3)
3.11 Stochastic Approximation	116	(1)
3.12 Reverse Logistic Regression	117	(2)
3.13 Fitting the Saturation Model	119	(6)
3.14 Hypothesis Tests	125	(2)
3.15 Missing Data and Edge Effects	127	(4)
3.16 Fitting the Triplets Process	131	(2)
3.17 Comparing the Saturated and Triplets Models	133	(4)
3.18 Conclusion	137	(4)
4 Markov chain Monte Carlo and spatial point processes	141	(32)
4.1 Introduction	141	(2)
4.2 Setup	143	(2)
4.3 Metropolis-Hastings algorithms for finite point processes	145	(5)
4.4 Markovian models for clustered and regular patterns	150	(3)
4.5 Ripley-Kelly Markov point processes	153	(20)
5 Topics in Voronoi and Johnson-Mehl tessellations	173	(26)
5.1 Introduction	173	(4)
5.2 Geometric structure	177	(3)
5.3 Stationary Voronoi and Johnson-Mehl tessellations	180	(6)
5.4 Poisson models	186	(5)
5.5 A special class of Poisson models	191	(8)
6 Mathematical morphology	199	(86)
6.1 Introduction	199	(2)
6.2 Brief review of mathematical morphology	201	(14)
6.3 Advanced morphological segmentation techniques	215	(36)
6.4 Granulometries: applications and algorithms	251	(29)
6.5 Conclusion	280	(5)
7 Random closed sets	285	(48)
7.1 Introduction	285	(1)
7.2 Distributions of random sets	286	(7)
7.3 Convergence of random set distributions	293	(7)
7.4 Limit theorems	300	(11)
7.5 Statistical inference for random sets	311	(12)
7.6 Concluding remarks	323	(10)
8 General shape and registration analysis	333	(32)
8.1 Introduction	333	(1)
8.2 Matching with regression	334	(10)
8.3 Estimation by matching	344	(5)
8.4 Distributions and inference	349	(4)
8.5 Robust matching	353	(4)
8.6 Smoothed matching	357	(2)
8.7 Discussion	359	(6)
9 Nash inequalities	365	(36)
9.1 Introduction	365	(9)
9.2 Poincare inequalities	374	(15)
9.3 Nash inequalities	389	(9)
9.4 Conclusion	398	(3)
Index	401