1st Edition

An Introduction to Random Sets

By Hung T. Nguyen Copyright 2006
    272 Pages 1 B/W Illustrations
    by Chapman & Hall

    272 Pages 1 B/W Illustrations
    by Chapman & Hall

    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 PROBABILITY
    Survey Sampling Revisited
    Mathematical Models for Random Phenomena
    Random Elements
    Distribution Functions of Random Variables
    Distribution 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

    Biography

    Nguyen, Hung T.