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This short book provides a unified view of the history and theory of random sets and fuzzy random variables, with special emphasis on its use for representing higher-order non-statistical uncertainty about statistical experiments. The authors lay bare the existence of two streams of works using the same mathematical ground, but differing form their use of sets, according to whether they represent objects of interest naturally taking the form of sets, or imprecise knowledge about such objects. Random (fuzzy) sets can be used in many fields ranging from mathematical morphology, economics, artificial intelligence, information processing and statistics per se, especially in areas where the outcomes of random experiments cannot be observed with full precision. This book also emphasizes the link between random sets and fuzzy sets with some techniques related to the theory of imprecise probabilities. This small book is intended for graduate and doctoral students in mathematics or engineering, but also provides an introduction for other researchers interested in this area. It is written from a theoretical perspective. However, rather than offering a comprehensive formal view of random (fuzzy) sets in this context, it aims to provide a discussion of the meaning of the proposed formal constructions based on many concrete examples and exercises. This book should enable the reader to understand the usefulness of representing and reasoning with incomplete information in statistical tasks.  Each chapter ends with a list of exercises.

Engineering. --- Artificial intelligence. --- Statistics. --- Computational intelligence. --- Computational Intelligence. --- Artificial Intelligence (incl. Robotics). --- Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences. --- Intelligence, Computational --- Artificial intelligence --- Soft computing --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Mathematics --- Econometrics --- AI (Artificial intelligence) --- Artificial thinking --- Electronic brains --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Electronic data processing --- Logic machines --- Machine theory --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Construction --- Industrial arts --- Technology --- Random sets. --- Fuzzy sets. --- Sets, Fuzzy --- Fuzzy mathematics --- Set theory --- Geometric probabilities --- Artificial Intelligence. --- Statistics . --- Engineering --- Fuzzy sets --- Random sets

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This carefully edited book presents an up-to-date state of current research in the use of fuzzy sets and their extensions, paying attention to foundation issues and to their application to four important areas where fuzzy sets are seen to be an important tool for modelling and solving problems. The book contains 34 chapters divided into two parts. The first part is divided into two sections. Section 1 contains four review papers introducing some non standard representations that extend fuzzy sets (type-2 fuzzy sets, Atanassov’s IFS, fuzzy rough sets and computing with words under the fuzzy sets perspective). Section 2 reviews different aggregation issues from a theoretical and practical point of view; this second part is divided into four sections. Section 3 is devoted to decision making, with seven papers that show how fuzzy sets and their extensions are an important tool for modelling choice problems. Section 4 includes eight papers that cover different aspects on the use of fuzzy sets and their extensions in data mining, giving an illustrative review of the state of the art on the topic. Section 5 is devoted to the emergent topic of web intelligence and contains four papers that show the use of fuzzy sets theory in some problems that can be tackled in this topic. Section 6 is devoted to the use of fuzzy sets and their extensions in the field of computer vision, suggesting how these can be an useful tool in this area. This volume will be extremely useful to any non-expert reader who is keen to get a good overview on the latest developments in this research field. It will also support those specialists who wish to discover the latest results and trends in the abovementioned areas.

Fuzzy sets --- Fuzzy logic --- Decision making --- Data mining --- World Wide Web --- Computer vision --- Soft computing --- Ensembles flous --- Logique floue --- Prise de décision --- Exploration de données (Informatique) --- Web --- Vision par ordinateur --- Informatique douce --- Congresses. --- Mathematical models --- Mathematics --- Congrès --- Modèles mathématiques --- Mathématiques --- Applied Mathematics --- Civil Engineering --- Computer Science --- Engineering & Applied Sciences --- Civil & Environmental Engineering --- Soft computing. --- Fuzzy sets. --- Data mining. --- Algorithmic knowledge discovery --- Factual data analysis --- KDD (Information retrieval) --- Knowledge discovery in data --- Knowledge discovery in databases --- Mining, Data --- Sets, Fuzzy --- Computer science. --- Computers. --- Artificial intelligence. --- Applied mathematics. --- Engineering mathematics. --- Computer Science. --- Theory of Computation. --- Appl.Mathematics/Computational Methods of Engineering. --- Artificial Intelligence (incl. Robotics). --- Database searching --- Fuzzy mathematics --- Set theory --- Cognitive computing --- Electronic data processing --- Computational intelligence --- Information theory. --- Mathematical and Computational Engineering. --- Artificial Intelligence. --- AI (Artificial intelligence) --- Artificial thinking --- Electronic brains --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Logic machines --- Machine theory --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Engineering --- Engineering analysis --- Mathematical analysis --- Communication theory --- Communication --- Cybernetics --- Automatic computers --- Automatic data processors --- Computer hardware --- Computing machines (Computers) --- Electronic calculating-machines --- Electronic computers --- Hardware, Computer --- Computer systems --- Calculators --- Cyberspace

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Probability theory has been the only well-founded theory of uncertainty for a long time. It was viewed either as a powerful tool for modelling random phenomena, or as a rational approach to the notion of degree of belief. During the last thirty years, in areas centered around decision theory, artificial intelligence and information processing, numerous approaches extending or orthogonal to the existing theory of probability and mathematical statistics have come to the front. The common feature of those attempts is to allow for softer or wider frameworks for taking into account the incompleteness or imprecision of information. Many of these approaches come down to blending interval or fuzzy interval analysis with probabilistic methods. This book gathers contributions to the 4th International Conference on Soft methods in Probability and Statistics. Its aim is to present recent results illustrating such new trends that enlarge the statistical and uncertainty modeling traditions, towards the handling of incomplete or subjective information. It covers a broad scope ranging from philosophical and mathematical underpinnings of new uncertainty theories, with a stress on their impact in the area of statistics and data analysis, to numerical methods and applications to environmental risk analysis and mechanical engineering. A unique feature of this collection is to establish a dialogue between fuzzy random variables and imprecise probability theories.

Soft computing --- Fuzzy sets --- Probabilities --- Mathematical statistics --- Distribution (Probability theory. --- Computer aided design. --- Artificial intelligence. --- Engineering mathematics. --- Statistics. --- Information technology. --- Probability Theory and Stochastic Processes. --- Computer-Aided Engineering (CAD, CAE) and Design. --- Artificial Intelligence. --- Mathematical and Computational Engineering. --- Statistics, general. --- IT in Business. --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Mathematics --- Econometrics --- Engineering --- Engineering analysis --- Mathematical analysis --- AI (Artificial intelligence) --- Artificial thinking --- Electronic brains --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Electronic data processing --- Logic machines --- Machine theory --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- CAD (Computer-aided design) --- Computer-assisted design --- Computer-aided engineering --- Design --- IT (Information technology) --- Technology --- Telematics --- Information superhighway --- Knowledge management --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities. --- Computer-aided engineering. --- Applied mathematics. --- Statistics . --- Business—Data processing. --- CAE --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Risk --- Data processing