An introduction to random sets

Hung T. Nguyen158488519X, 9781584885191, 9781420010619

The study of random sets is a large and rapidly growing area with connections to many areas of mathematics and applications in widely varying disciplines, from economics and decision theory to biostatistics and image analysis. The drawback to such diversity is that the research reports are scattered throughout the literature, with the result that in science and engineering, and even in the statistics community, the topic is not well known and much of the enormous potential of random sets remains untapped. An Introduction to Random Sets provides a friendly but solid initiation into the theory of random sets. It builds the foundation for studying random set data, which, viewed as imprecise or incomplete observations, are ubiquitous in today’s technological society. The author, widely known for his best-selling A First Course in Fuzzy Logic text as well as his pioneering work in random sets, explores motivations, such as coarse data analysis and uncertainty analysis in intelligent systems, for studying random sets as stochastic models. Other topics include random closed sets, related uncertainty measures, the Choquet integral, the convergence of capacity functionals, and the statistical framework for set-valued observations. An abundance of examples and exercises reinforce the concepts discussed. Designed as a textbook for a course at the advanced undergraduate or beginning graduate level, this book will serve equally well for self-study and as a reference for researchers in fields such as statistics, mathematics, engineering, and computer science.

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Table of contents :
An Introduction to Random Sets……Page 2
Contents……Page 4
Preface……Page 6
About the Author……Page 8
1.1 Survey Sampling Revisited……Page 9
1.2 Mathematical Models for Random Phenomena……Page 11
1.3 Random Elements……Page 13
1.4 Distribution Functions of Random Variables……Page 14
1.5 Distribution Functions of Random Vectors……Page 18
1.6 Exercises……Page 20
2.1 Probability Sampling Designs……Page 23
2.2 Confidence Regions……Page 26
2.3 Robust Bayesian Statistics……Page 29
2.4 Probability Density Estimation……Page 31
2.5 Coarse Data Analysis……Page 32
2.6 Perception-Based Information……Page 34
2.7 Stochastic Point Processes……Page 35
2.8 Exercises……Page 38
3.1 Random Sets and Their Distributions……Page 42
3.2 Set-Valued Observations……Page 49
3.3 Imprecise Probabilities……Page 58
3.4 Decision Making with Random Sets……Page 62
3.5 Exercises……Page 75
4.1 Some Set Functions……Page 78
4.2 Incidence Algebras……Page 93
4.3 Cores of Capacity Functionals……Page 104
4.4 Exercises……Page 112
5.1 Introduction……Page 116
5.2 The Hit-or-Miss Topology……Page 117
5.3 Capacity Functionals……Page 119
5.4 Notes on the Choquet Theorem on Polish Spaces (optional)……Page 129
5.5 Exercises……Page 136
6.1 Some Motivations……Page 138
6.2 The Choquet Integral……Page 141
6.3 Radon-Nikodym Derivatives……Page 149
6.4 Exercises……Page 160
7.1 Stochastic Convergence of Random Sets……Page 163
7.2 Convergence in Distribution……Page 170
7.3 Weak Convergence of Capacity Functionals……Page 175
7.4 Exercises……Page 187
8.1 Expectations and Limit Theorems……Page 189
8.2 A Statistical Inference Framework for Coarse Data……Page 192
8.3 A Related Statistical Setting……Page 207
8.4 A Variational Calculus of Set Functions……Page 212
8.5 Exercises……Page 217
A.1 Probability Spaces……Page 220
A.2 Topological Spaces……Page 222
A.3 Expectation of a Random Variable……Page 228
A.4 Convergence of Random Elements……Page 233
A.5 Convergence in Distribution……Page 234
A.6 Radon-Nikodym Theorem……Page 237
A.7 Large Deviations……Page 241
A.7.1 Some motivations……Page 242
A.7.2 Formulation of large deviations principles……Page 246
A.7.3 Large deviations techniques……Page 250
References……Page 252