Random Fuzzy Sets

Computational Statistics & Data Analysis, 2006

The theoretical aspects of statistical inference with imprecise data, with focus on random sets, are considered. On the setting of coarse data analysis imprecision and randomness in observed data are exhibited, and the relationship between probability and other types of uncertainty, such as belief functions and possibility measures, is analyzed. Coarsening schemes are viewed as models for perception-based information gathering processes in which random fuzzy sets appear naturally. As an implication, fuzzy statistics is statistics with fuzzy data. That is, fuzzy sets are a new type of data and as such, complementary to statistical analysis in the sense that they enlarge the domain of applications of statistical science.

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.

Data obtained in association with many real-life random experiments from different fields cannot be perfectly/exactly quantified. Often the underlying imprecision can be suitably described in terms of fuzzy numbers/ values. For these random experiments, the scale of fuzzy numbers/values enables to capture more variability and subjectivity than that of categorical data, and more accuracy and expressiveness than that of numerical/vectorial data. On the other hand, random fuzzy numbers/sets model the random mechanisms generating experimental fuzzy data, and they are soundly formalized within the probabilistic setting. This paper aims to review a significant part of the recent literature concerning the statistical data analysis with fuzzy data and being developed around the concept of random fuzzy numbers/sets.

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 ...

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

SpringerBriefs in Applied Sciences and Technology, 2014

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Vulnerability, Uncertainty, and Risk, 2011

In the majority of decision models used in practice all input data are assumed to be precise. This assumption is made both for random results of measurements, and for constant parameters such as, e.g. costs related to decisions. In reality many of these values are reported in an imprecise way. When this imprecision cannot be related to randomness the fuzzy set theory yields tools for its description. It seems to be important to retain both types of uncertainty, random and fuzzy, while building mathematical models for making decisions. In the paper we propose a fuzzy-Bayesian model for making statistical decisions. In the proposed model the randomness of data is reflected in related risks, and fuzziness is described by possibility measures of dominance such as PSD (Possibility of Strict Dominance) and NSD (Necessity of Strict Dominance). The proposed model allows a decisionmaker to reflect in his/hers decisions different types of uncertainty.

Fuzzy Sets and Systems, 2007

A basic problem, at the present stage of the Information society, is how to manage cognitive processes while taking into account their intrinsic features of uncertainty, including imprecision and vagueness. This has both theoretical and practical implications in Technology, Economics, Bio-Medicine, and so on. In fact, real-life situations are the prime source of motivation for this management to be considered. Traditional Statistics has developed tools and procedures for coping with this problem, assuming that uncertainty is basically due to random mechanisms appropriately handled by means of models from Probability Theory. Fuzzy Sets Theory and its generalization to what may be called "Fuzzy thinking'' has widened the scope of Statistics enabling us to deal with other sources of uncertainty, such as vagueness and imprecision, pervading both empirical data and/or models for data analysis. In this respect, for the last decades many research studies have been developed in which a coalition of Fuzzy Sets Theory and Statistics has been established with different purposes, namely,

2011

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.

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.