An Introduction to Random Sets

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.

GENERALITIES ON PROBABILITYSurvey Sampling RevisitedMathematical Models for Random PhenomenaRandom ElementsDistribution Functions of Random VariablesDistribution Functions of Random Vectors Exercises SOME RANDOM SETS IN STATISTICS Probability Sampling Designs Confidence Regions Robust Bayesian Statistics Probability Density Estimation Coarse Data Analysis Perception-Based Information Stochastic Point Processes Exercises FINITE RANDOM SETS Random Sets and Their Distributions Set-Valued Observations Imprecise Probabilities Decision Making with Random Sets Exercises RANDOM SETS AND RELATED UNCERTAINTY MEASURES Some Set Functions Incidence Algebras Cores of Capacity Functionals Exercises RANDOM CLOSED SETS Introduction The Hit-or-Miss Topology Capacity Functionals Notes on the Choquet Theorem on Polish Spaces Exercises THE CHOQUET INTEGRAL Some Motivations The Choquet Integral Radon-Nikodym Derivatives Exercises CHOQUET WEAK CONVERGENCE Stochastic Convergence of Random Sets Convergence in Distribution Weak Convergence of Capacity Functionals Exercises SOME ASPECTS OF STATISTICAL INFERENCE WITH COARSE DATA Expectations and Limit Theorems A Statistical Inference Framework for Coarse Data A Related Statistical Setting A Variational Calculus of Set Functions Exercises APPENDIX: BASIC CONCEPTS AND RESULTS OF PROBABILITY THEORY References Index

Nguyen, Hung T.