Building Fuzzy Models of Stochastic Processes to Determine Probabilistic and Linguistic Types of Uncertainty

Building Fuzzy Models of Stochastic Processes to Determine Probabilistic and Linguistic Types of Uncertainty

Anna Walaszek-Babiszewska
Copyright: © 2016 |Pages: 16
DOI: 10.4018/978-1-5225-0044-5.ch014
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Abstract

In the chapter an advanced fuzzy modeling methods have been presented which joints fuzzy and probabilistic approaches. The Zadeh's notions of probability of imprecise events has served as a basis for determining probability distributions of linguistic random variable, stochastic process with fuzzy states, and fuzzy-valued stochastic Markov process. Rule based fuzzy representations have been also presented. Exemplary calculations illustrate the way of building fuzzy models, using empirical data or theoretical probability distribution.
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Basic Definitions

Fuzzy Set

The notion of a fuzzy set is a generalization of a classic set, defined in mathematics by a characteristic function, 978-1-5225-0044-5.ch014.m01, which assigns number 1 to every element x from X, if x belongs to A, and 0 if x does not belong to A.

Fuzzy set A, as a subset of a certain space of consideration X, is determined by a membership function, 978-1-5225-0044-5.ch014.m02, which assigns a real number in the interval [0,1] to every element 978-1-5225-0044-5.ch014.m03, 978-1-5225-0044-5.ch014.m04. The values of function 978-1-5225-0044-5.ch014.m05 represent membership degrees of particular elements x in A (Zadeh, 1965).

Fuzzy set A in a space of consideration, 978-1-5225-0044-5.ch014.m06, is denoted by a set of pairs

978-1-5225-0044-5.ch014.m07
(1) where the membership function can be given in some analytic form, e.g.

978-1-5225-0044-5.ch014.m08
(2)

When the space of consideration is a finite set, 978-1-5225-0044-5.ch014.m09, fuzzy set A is denoted as a sum of elements with the membership degrees, in the following form

978-1-5225-0044-5.ch014.m10
. (3)

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