An Introduction to Random Sets
Features
Summary
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
