Abstract
Fuzzy set theory is an effective alternative to probability theory in solving many problems of studying processes and systems under conditions of uncertainty. The application of this theory is especially in demand in situations where the system under study operates under conditions of rapidly changing influencing parameters or characteristics of the environment. In these cases, the use of solutions obtained by standard methods of the probability theory is not quite correct. At the same time, the conceptual, methodological and hardware base of the alternative fuzzy set theory is not sufficiently developed. The paper attempts to fill existing gaps in the fuzzy set theory in some important areas. For continuous fuzzy quantities, the concept of distribution density of these quantities is introduced. Using this concept, a method for calculating the main numerical characteristics of fuzzy quantities, as well as a technology for calculating membership functions for fuzzy values of functions from these fuzzy quantities and their moments is proposed. The introduction of these formalisms significantly extends the capabilities of the fuzzy set theory for solving many real problems of computational mathematics. Using these formalisms, a large number of practical problems can be solved: fuzzy regression and clustering, fuzzy multivariate discriminant analysis, differentiation and integration of functions of fuzzy arguments, state diagnostics in a situation where the initial data are fuzzy, methods for solving problems of unconditional and conditional optimization, etc. The proof of the central limit theorem for the sum of a large number of fuzzy quantities is obtained. This proof is based on the characteristic functions of fuzzy quantities introduced in the work and described at the formal level. The concepts of independence and dependence for fuzzy quantities are introduced. The method for calculating the correlation coefficient for fuzzy numbers is proposed. Examples of problem solving are considered
Highlights
The recent increase in the number of publications on solving problems in conditions of uncertainty is objectively motivated
probability theory (PT) has significant advantages over fuzzy set theory (FST), which are determined by the presence of deep and constructive probabilistic formalisms [2]: moment theory arising from the existence of distribution densities of random variables (RV); the possibility to calculate the probabilities of falling of RV in given subsets of a set of possible values; calculation of numerical characteristics of functions random variable functions; use of characteristic functions of random variables; the existence of the central limit theorem, etc
Some important issues of the fuzzy measure theory remain insufficiently studied. In this regard, using canonized approaches implemented in the probability theory, it is advisable to formulate the problem of developing a system of conceptual, axiomatic and framework formalisms in order to improve the methodological and model base, as well as the analytical framework of the fuzzy set theory
Summary
The recent increase in the number of publications on solving problems in conditions of uncertainty is objectively motivated. The fuzzy set theory (FST) [1,2,3,4,5] is a successful and productive alternative to the probability theory (PT) in situations where the mechanism of formation of random variables changes unpredictably This precludes the use of PT methods for the analytical construction of appropriate distribution densities or reconstructing them from experimental data in these cases. PT has significant advantages over FST, which are determined by the presence of deep and constructive probabilistic formalisms [2]: moment theory arising from the existence of distribution densities of random variables (RV); the possibility to calculate the probabilities of falling of RV in given subsets of a set of possible values; calculation of numerical characteristics of functions random variable functions; use of characteristic functions of random variables; the existence of the central limit theorem, etc This circumstance stimulates attempts to formally extend the axiomatic base of FST in order. Methods for solving problems of extending the conceptual and analytical framework of the fuzzy set theory
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