An Introduction to Random Sets

by Nguyen; Hung T.

• ISBN13: 9781584885191
• ISBN10: 158488519X
• Edition: 1st
• Format: Hardcover
• Copyright: 2006-03-27
• Publisher: Chapman & Hall/

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.

Author Biography

Hung T. Nguyen is a professor in the Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico, U.S.A.

Table of Contents

Preface	vii
About the Author	ix
Generalities on Probability	1	(14)
Survey Sampling Revisited	1	(2)
Mathematical Models for Random Phenomena	3	(2)
Random Elements	5	(1)
Distribution Functions of Random Variables	6	(4)
Distribution Functions of Random Vectors	10	(2)
Exercises	12	(3)
Some Random Sets in Statistics	15	(20)
Probability Sampling Designs	15	(3)
Confidence Regions	18	(3)
Robust Bayesian Statistics	21	(2)
Probability Density Estimation	23	(1)
Coarse Data Analysis	24	(2)
Perception-Based Information	26	(1)
Stochastic Point Processes	27	(3)
Exercises	30	(5)
Finite Random Sets	35	(36)
Random Sets and Their Distributions	35	(7)
Set-Valued Observations	42	(9)
Imprecise Probabilities	51	(4)
Decision Making with Random Sets	55	(13)
Exercises	68	(3)
Random Sets and Related Uncertainty Measures	71	(38)
Some Set Functions	71	(15)
Incidence Algebras	86	(11)
Cores of Capacity Functionals	97	(8)
Exercises	105	(4)
Random Closed Sets	109	(22)
Introduction	109	(1)
The Hit-or-Miss Topology	110	(2)
Capacity Functionals	112	(10)
Notes on the Choquet Theorem on Polish Spaces (optional)	122	(7)
Exercises	129	(2)
The Choquet Integral	131	(26)
Some Motivations	131	(3)
The Choquet Integral	134	(8)
Radon-Nikodym Derivatives	142	(11)
Exercises	153	(4)
Choquet Weak Convergence	157	(26)
Stochastic Convergence of Random Sets	157	(7)
Convergence in Distribution	164	(5)
Weak Convergence of Capacity Functionals	169	(12)
Exercises	181	(2)
Some Aspects of Statistical Inference with Coarse Data	183	(32)
Expectations and Limit Theorems	183	(3)
A Statistical Inference Framework for Coarse Data	186	(15)
A Related Statistical Setting	201	(5)
A Variational Calculus of Set Functions	206	(5)
Exercises	211	(4)
Appendix: Basic Concepts and Results of Probability Theory	215	(32)
Probability Spaces	215	(2)
Topological Spaces	217	(6)
Expectation of a Random Variable	223	(5)
Convergence of Random Elements	228	(1)
Convergence in Distribution	229	(3)
Radon-Nikodym Theorem	232	(4)
Large Deviations	236	(11)
Some motivations	237	(4)
Formulation of large deviations principles	241	(4)
Large deviations techniques	245	(2)
References	247	(8)
Index	255