Statistics with fuzzy data by using random fuzzy sets

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.

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.

The notion of Fuzzy Random Variable has been introduced to model random mechanisms generating imprecisely-valued data which can be properly described by means of fuzzy sets. Probabilistic aspects of these random elements have been deeply discussed in the literature. However, statistical analysis of fuzzy random variables has not received so much attention, in spite that implications of this analysis range over many fields, including Medicine, Sociology, Economics, and so on. A summary of the fundamentals of fuzzy random variables is presented. Then, some related "parameters" associated with the distribution of these variables are defined. Inferential procedures concerning these "parameters" are described. Some recent results related to linear models for fuzzy data are finally reviewed.

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.

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 ...

SpringerBriefs in Applied Sciences and Technology, 2014

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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.

Computational Statistics & Data Analysis, 2006

For the last decades, research studies have been developed in which a coalition of Fuzzy Sets Theory and Statistics has been established with different purposes. These namely are: (i) to introduce new data analysis problems in which the objective involves either fuzzy relationships or fuzzy terms; (ii) to establish well-formalized models for elements combining randomness and fuzziness; (iii) to develop uni-and multivariate statistical methodologies to handle fuzzy-valued data; and (iv) to incorporate fuzzy sets to help in solving traditional statistical problems with non-fuzzy data. In spite of a growing literature concerning the development and application of fuzzy techniques in statistical analysis, the need is felt for a more systematic insight into the potentialities of cross fertilization between Statistics and Fuzzy Logic. In line with the synergistic spirit of Soft Computing, some instances of the existing research activities on the topic are recalled. Particular attention is paid to summarize the papers gathered in this Special Issue, ranging from the position paper on the theoretical management of uncertainty by the "father" of Fuzzy Logic to a wide diversity of topics concerning foundational/methodological/applied aspects of the integration of Fuzzy Sets and Statistics.

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.

Computational Statistics & Data Analysis, 2006

A family of fuzzy representations of random variables is presented. Each representation transforms a real-valued random variable into a fuzzy-valued one. These representations can be chosen so that they lead to fuzzy random variables whose means capture different relevant information on the probability distribution of the original real-valued random variable. In this way, the means of the transformed fuzzy random variables can capture, for instance, immediate visual information about some key parameters, and even the whole information about the distribution of the original random variable. Representations capturing visual information on parameters of the original random variable may be considered for statistical descriptive/exploratory purposes. Representations for which the fuzzy mean characterizes the distribution of the original random variable will be mainly valuable to develop statistical inferences on this variable. Some interesting inferential applications for classical random variables based on the last fuzzy representations are commented, and an example illustrates one of them empirically and motivate future directions and discussions.