An Introduction to Random Sets: 1st Edition (Hardback) book cover

An Introduction to Random Sets

1st Edition

By Hung T. Nguyen

Chapman and Hall/CRC

272 pages | 1 B/W Illus.

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Description

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.

Table of Contents

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

About the Author

Nguyen, Hung T.

Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT028000
MATHEMATICS / Set Theory
MAT029010
MATHEMATICS / Probability & Statistics / Bayesian Analysis
SCI040000
SCIENCE / Mathematical Physics