An Introduction to Random Sets

About this book

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.

Contents

GENERALITIES ON PROBABILITYSurvey Sampling RevisitedMathematical Models for Random PhenomenaRandom ElementsDistribution Functions of Random VariablesDistribution Functions of Random VectorsExercisesSOME RANDOM SETS IN STATISTICSProbability Sampling DesignsConfidence RegionsRobust Bayesian StatisticsProbability Density EstimationCoarse Data AnalysisPerception-Based InformationStochastic Point ProcessesExercisesFINITE RANDOM SETSRandom Sets and Their DistributionsSet-Valued ObservationsImprecise ProbabilitiesDecision Making with Random SetsExercisesRANDOM SETS AND RELATED UNCERTAINTY MEASURESSome Set FunctionsIncidence AlgebrasCores of Capacity FunctionalsExercisesRANDOM CLOSED SETSIntroductionThe Hit-or-Miss TopologyCapacity FunctionalsNotes on the Choquet Theorem on Polish SpacesExercisesTHE CHOQUET INTEGRALSome MotivationsThe Choquet IntegralRadon-Nikodym DerivativesExercisesCHOQUET WEAK CONVERGENCEStochastic Convergence of Random SetsConvergence in DistributionWeak Convergence of Capacity FunctionalsExercisesSOME ASPECTS OF STATISTICAL INFERENCE WITH COARSE DATAExpectations and Limit TheoremsA Statistical Inference Framework for Coarse DataA Related Statistical SettingA Variational Calculus of Set FunctionsExercisesAPPENDIX: BASIC CONCEPTS AND RESULTS OF PROBABILITY THEORYReferencesIndex

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