Theory of Fuzzy Sets: Beliefs and Realities

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 ...

The aim of this work is to compare between what seems to be entirely different two highly developing "fuzzy probability" theories. The first theory had been developed firstly by statisticians and the other separately by physicists. We start by indicating the needs to develop such theories and what helped to develop each, then we will establish the basis of the two theories and illustrate that each indeed extends classical probability theory. Moreover, we will try to see whether or not any of the two theory can be embedded into the other.

This chapter presents a rigorous theory of random fuzzy sets in its most general form. Some applications are included.

Studies in Fuzziness and Soft Computing, 2006

It is well known that in decision making under uncertainty, while we are guided by a general (and abstract) theory of probability and of statistical inference, each specific type of observed data requires its own analysis. Thus, while textbook techniques treat precisely observed data in multivariate analysis, there are many open research problems when data are censored (e.g., in medical or bio-statistics), missing, or partially observed (e.g., in bioinformatics). Data can be imprecise due to various reasons, e.g., due to fuzziness of linguistic data. Imprecise observed data are usually called coarse data. In this chapter, we consider coarse data which are both random and fuzzy.

Many real-life random experiments involve variables which are associated with judgements, opinions, perceptions, ratings, and so on. 'Values' for these variables are usually non-numerical, but they correspond to imprecise values or categories. A well-known example of this type of experiments is the one corresponding to most of the usual questionnaires and surveys with a pre-specified response format, in which people are asked to respond to a series of questions and variable values are the different answers from respondents.

Information Sciences, 1984

We discuss some relationships between probability theory and statistics on one hand, and the theory of fuzzy sets on the other hand. We develop various statistical techniques for the analysis of imprecise data and for inference based on imprecise data. We study the estimation of fuzzy parameters, testing problems with fuzzy probabilities, and confidence estimation under uncertainty.