Estad́ıstica Random fuzzy sets : why , when , how 1

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.

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.

Fuzzy Sets and Systems, 2013

In this paper we deal with the problem of obtaining a random procedure for generating fuzzy measures. We use the fact that the polytope of fuzzy measures is an order polytope, so that it has special properties that allow to build a uniform algorithm. First, we derive an exact procedure based on an existing procedure to generate random linear extensions; then, we study the applicability of this algorithm to the polytope of fuzzy measures, showing that the complexity grows dramatically with the cardinality of the referential set. Next, we study other heuristics appearing in the literature for the polytope of fuzzy measures; our results seem to mean that these procedures cannot be applied to this case either. Finally, we propose another heuristic that reduces the complexity and could be used instead of the other procedures. We finish comparing the performance of this heuristic with the other possibilities, showing that our alternative seems to work better for the polytope of fuzzy measures.

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.

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.

Computational Statistics & Data Analysis, 2006

The concept of fuzzy random variable has been shown to be as a valuable model for handling fuzzy data in statistical problems. The theory of fuzzy-valued random elements provides a suitable formalization for the management of fuzzy data in the probabilistic setting. A concise overview of fuzzy random variables, focussed on the crucial aspects for data analysis, is presented.

Information Sciences, 2001

In this paper we develop a discussion on the mathematical formalization of the concept of fuzzy random variable. This discussion is mainly focused on ®nding an adequate notion of measurability to be coherent with the notions on the space these random elements take values. Ó

IEEE Transactions on Fuzzy Systems, 2000

The generation of fuzzy measures is an important question arising in the practical use of these operators. In this paper, we deal with the problem of developing a random generator of fuzzy measures. More concretely, we study some of the properties that any random generator should satisfy. These properties lead to some theoretical problems concerning the group of isometries that we tackle in this paper for some subfamilies of fuzzy measures.

SpringerBriefs in Applied Sciences and Technology, 2014

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