The Theory of Fuzzy Sets: Beliefs and Realities

2015

On two important counts, the Zadehian theory of fuzzy sets urgently needs to be restructured. First, it can be established that for a normal fuzzy number N = [α, β, γ] with membership function Ψ 1 (x), if α ≤ x ≤ β, Ψ 2 (x), if β ≤ x ≤ γ, and 0, otherwise, Ψ 1 (x) is in fact the distribution function of a random variable defined in the interval [α, β], while Ψ 2 (x) is the complementary distribution function of another random variable defined in the interval [β, γ]. In other words, every normal law of fuzziness can be expressed in terms of two laws of randomness defined in the measure theoretic sense. This is how a normal fuzzy number should be constructed, and this is how partial presence of an element in a fuzzy set has to be defined. Hence the measure theoretic matters with reference to fuzziness have to be studied accordingly. Secondly, the field theoretic matters related to fuzzy sets are required to be revised all over again because in the current definition of the complement of a fuzzy set, fuzzy membership function and fuzzy membership value had been taken to be the same, which led to the conclusion that the fuzzy sets do not follow the set theoretic axioms of exclusion and contradiction. For the complement of a normal fuzzy set, fuzzy membership function and fuzzy membership value are two different things, and the complement of a normal fuzzy set has to be defined accordingly. We shall further show how fuzzy randomness should be explained with reference to two laws of randomness defined for every fuzzy observation so as to make fuzzy statistical conclusions. Finally, we shall explain how randomness can be viewed as a special case of fuzziness defined in our perspective with reference to normal fuzzy numbers of the type [α, β, β]. Indeed every probability distribution function is a Dubois-Prade left reference function, and probability can be viewed in that way too.

Applied Mathematics, 2014

If sample realizations are intervals, if the upper and the lower boundaries of such intervals are realizations of two independently distributed random variables, the two probability laws together lead to some interesting assertions. In this article, we shall attempt to remove certain confusions regarding the relationship between probability theory and fuzzy mathematics.

SpringerBriefs in Applied Sciences and Technology, 2014

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Journal of Process Management. New Technologies

Fuzzy randomness leads to fuzzy conclusions. Such fuzzy conclusions can indeed be made in terms of probability. In this article, the concept of fuzzy randomness has been discussed using the mathematics of partial presence. Two important points have been suggested in this article. First, fuzzy randomness should be explained with reference to the Randomness-Fuzziness Consistency Principle, and only then the mathematical explanations of fuzzy randomness would actually be complete. Secondly, in every case of fuzzy statistical hypothesis testing, the alternative hypotheses must necessarily be properly defined. The authors in this article have described fuzzy randomness with reference to a numerical example of using the Student's t-test statistic.

Journal of Mathematical Analysis and Applications, 1982

A general framework for a theory is presented that encompasses both statistical uncertainty. which falls within the province of probability theory, and nonstatistical uncertamty. which relates to the concept of a fuzzy set and possibility theory [L. A. Zadeh, J. FUZZJ Sers I (1978). 3-281. The concept of a fuzzy integral ts used to define the expected value of a random vartable. Properties of the fuzzy expectation are stated and a mean-value theorem for the fuzzy integral is proved. Comparisons between the fuzzy and the Lebesgue integral are presented. After a new concept of dependence IS formulated, various convergence concepts are defined and their relationshtps are studied by using a Chebyshevlike inequality for the fuzzy Integral. The possibility of using this theory m Bayestan estimation with fuzzy prior mformation IS explored.

Statistics & Probability Letters, 1992

Studies relating Probability and Fuzzy Set Theories have been developed in the literature. In this paper we prove that any fuzzy number determines an intuitive corresponding random interval, and discuss advantages and inconveniences of treating fuzzy numbers as random intervals. Keywords: Fuzzy numbers, membership function, one-point coverage function, operations for fuzzy numbers, operations for random intervals, random intervals.

International Journal of Information Engineering and Electronic Business, 2013

In this art icle, we would like to revisit and comment on the definit ion of co mplementation of fu zzy sets and also on some of the theories and formulas associated with this. Furthermore, the existing probability-possibility consistency principles are also revisited and related results are v iewed fro m the standpoint of the Randomness-Fuzziness consistency principles. It is found that the existing definition of complementation as well as the probability-possibility consistency principles is not well defined. Consequently the results obtained from these would be inappropriate fro m our standpoints. Hence we would like to suggest some new defin itions for so me of the terms often used in the theory of fuzzy sets whenever possible.

Applied Mathematical Modelling, 2014

Characterizing the distribution of random elements is valuable for different purposes. Among them, inferential conclusions about the population distribution can be drawn on the basis of the sample one. When one deals with real-valued random variables this characterization is usually made through the distribution function or other ones, like the moment-generating or the characteristic functions. In case of dealing with random elements taking on fuzzy number values, the distribution function cannot be adequately defined in terms of a total ordering since there is no universally acceptable one for fuzzy numbers. This paper introduces a characterization of the distribution of these random elements by extending the moment-generating function. Properties of this extension are examined, and the notion is illustrated by means of some examples.

2011

Two laws of randomness are necessary and sufficient to describe a normal fuzzy number. Trying to impose one single probability law on an interval on which a possibility law has been defined is absolutely illogical. This is the reason why the attempts of establishing a principle of consistency between probability and possibility have not yielded any fruitful result till this day. We are hereby nullifying all heuristic works that had been published in the last forty five years in this context the world over. For a normal fuzzy number [a, b, c], the partial presence of an element x in the interval [a, c] is either a probability distribution function F (x) defined in a ≤ x < b, or a complementary probability distribution function (1−G(x)) where G(x) is a probability distribution function defined in b ≤ x < c. In other words, possibility is indeed probability in disguise. Thus fuzziness is measure theoretic on its own right. We need not define a fuzzy measure, in the entire interval [a, c], which is not actually a measure in the classical sense. We need to correct this mathematical blunder as early as possible to fetch the mathematics of fuzziness back into the right path.

2014

Random elements of non-Euclidean spaces have reached the forefront of statistical research with the extension of continuous process monitoring, leading to a lively interest in functional data. A fuzzy set is a generalized set for which membership degrees are identified by a [0, 1]-valued function. The aim of this review is to present random fuzzy sets (also called fuzzy random variables) as a mathematical formalization of data-generating processes yielding fuzzy data. They will be contextualized as Borel measurable random elements of metric spaces endowed with a special convex cone structure. That allows one to construct notions of distribution, independence, expectation, variance, and so on, which mirror and generalize the literature of random variables and random vectors. The connections and differences between random fuzzy sets and random elements of classical function spaces (functional data) will be underlined. The paper also includes some bibliometric remarks, comments on the ...