Statistical Modeling, Analysis and Management of Fuzzy Data
The contributions in this book connect Probability Theory/Statistics and Fuzzy Set Theory in different ways. Some of these connections are either philosophical or theoretical in nature, but most of them state models and methods to work with fuzzy data (or
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Code D4223, SPAWARSYSCEN, San Diego, CA 92152-7446-USA Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003-800l-USA
Abstract. This paper presents a survey and some new results of the mathematical investigation into formal connections between two types of uncertainty: fuzziness and randomness.
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Introduction
In his pioneering work on random elements in metric spaces, Fn3chet (1948) pointed out that besides standard random objects (such as points, vectors, functions), nature, science and technology offer other random elements, some of which cannot be described mathematically. For example, to each population of humans, chosen at random, one might be interested in its "morality," its "political spiritj" to each town chosen at random, one might be interested in its "form," its "beauty," .... It is clear that when we use natural language to describe properties of things, we often run into such situations. A property p on a collection of objects il defines a subset A of il, namely those elements of il which possess the property p, provided that p is crisp, in the sense that each element of il has the property or does not have it. If a property p stands for "tall" in a human population, it is not clear how p determines a subset of il. Concepts such as "tall" are called fuzzy concepts. Thus, examples of random elements mentioned by Frechet are fuzzy concepts chosen at random. In 1965, Zadeh (1965) proposed a mathematical theory for modeling offuzzy concepts, in which fuzziness is a matter of degree and is described by membership functions. While it appears that fuzziness and randomness are two distinct types of uncertainty, it is of interest to find out whether or not there exist formal relations between them. This is similar to Potential Theory and Markov Processes where a formal connection between them is beneficial : Potential Theory provides powerful tools for studying Markov processes, Markov processes provide probability interpretations for various concepts in Potential Theory. This dual aspect is also reminiscent in many other areas of mathematics : Fourier or Laplace tranforms are considered as appropriate according to the need for a time domain analysis or a frequency domain analysisj or, as an analogue with complex analyis : using complex variable z = x + iy such as in residue representation theorems and complex Taylor series expansions, but alternatively, at times, considering the real part (x) and the imaginary part (y) such as in the Cauchy-Riemann criterion and in harmonic analysis.
C. Bertoluzza et al. (eds.), Statistical Modeling, Analysis and Management of Fuzzy Data © Springer-Verlag Berlin Heidelberg 2002
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I.R. Goodman & H.T. Nguyen
In subsequent sections we will investigate formal relationships between fuzziness and randomness and explore their consequences.
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Generalities on" fuzzy set: theory
A fuzzy (sub) set A of a set U is a map A : U ~ [0,1]. Note that we use the same symbol A to denote the fuzzy concept and its mathematical modeling by the membership
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