An Introduction to Random Sets
CRC Press, 27 mars 2006 - 272 pages
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.
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Some Random Sets in Statistics
Finite Random Sets
Random Sets and Related Uncertainty Measures
Random Closed Sets
The Choquet Integra
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alternating of infinite B(Rd belief functions capacity functional Chapter Choquet integral Choquet theorem coarsening comonotonic compact sets concept consider convergence in distribution convergence of probability convex core(T defined density f distribution function example exists finite set function F Hausdorff metric hence hit-or-miss topology infinite order large deviations Lemma Let f Let Q lim sup maximum entropy maxitive measurable space metric space Möbius Möbius transform monotone of infinite monotone of Order nonempty random nonnegative Note numbers observations open sets polish spaces power set probability density probability law probability measures probability space problem Proof properties random closed sets random elements random variable random vector rate function respect satisfies second countable sequence set function sets of Rd Show space Q Specifically statistical subsets t)dt taking values topological space weak convergence