An Introduction to Random Sets

Random sets as models for set-valued observations, are a new type of data proving useful in areas such as survey sampling, biostatistics, and intelligent systems. This is the first text to explore the topic in depth, using extended probability theory to provide a framework and tools for statistical analysis of random sets. With an abundance of examples, it highlights the basic role random sets play in a variety of statistical settings, links their study to fuzzy logic, fully develops the theory, and concludes with a variety of applications. Written by an author of the best-selling A First Course in Fuzzy Logic, this book is rigorous yet readable and fills a significant need for a textbook treatment of the subject.

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.