María Ángeles Gil

Overview on the development of fuzzy random variables

This paper presents a backward analysis on the interpretation, modelling and impact of the concept of fuzzy random variable. After some preliminaries, the situations modelled by means of fuzzy random variables as well as the main approaches to model them are explained. We also summarize briefly some of the probabilistic studies concerning this concept as well as some statistical applications.

Fuzzy data treated as functional data: A one-way ANOVA test approach

The use of the fuzzy scale of measurement to describe an important number of observations from real-life attributes or variables is first explored. In contrast to other well-known scales (like nominal or ordinal), a wide class of statistical measures and techniques can be properly applied to analyze fuzzy data. This fact is connected with the possibility of identifying the scale with a special subset of a functional Hilbert space. The identification can be used to develop methods for the statistical analysis of fuzzy data by considering techniques in functional data analysis and vice versa. In this respect, an approach to the FANOVA test is presented and analyzed, and it is later particularized to deal with fuzzy data. The proposed approaches are illustrated by means of a real-life case study.

A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread

One of the most important aspects of the (statistical) analysis of imprecise data is the usage of a suitable distance on the family of all compact, convex fuzzy sets, which is not too hard to calculate and which reflects the intuitive meaning of fuzzy sets. On the basis of expressing the metric of Bertoluzza et al. [C. Bertoluzza, N. Corral, A. Salas, On a new class of distances between fuzzy numbers, Mathware Soft Comput. 2 (1995) 71-84] in terms of the mid points and spreads of the corresponding intervals we construct new families of metrics on the family of all d-dimensional compact convex sets as well as on the family of all d-dimensional compact convex fuzzy sets. It is shown that these metrics not only fulfill many good properties, but also that they are easy to calculate and easy to manage for statistical purposes, and therefore, useful from the practical point of view.

Bootstrap approach to the multi-sample test of means with imprecise data

A bootstrap approach to the multi-sample test of means for imprecisely valued sample data is introduced. For this purpose imprecise data are modelled in terms of fuzzy values. Populations are identified with fuzzy-valued random elements, often referred to in the literature as fuzzy random variables. An example illustrates the use of the suggested method. Finally, the adequacy of the bootstrap approach to test the multi-sample hypothesis of means is discussed through a simulation comparative study.

Bootstrap techniques and fuzzy random variables: Synergy in hypothesis testing with fuzzy data

In previous studies we have stated that the well-known bootstrap techniques are a valuable tool in testing statistical hypotheses about the means of fuzzy random variables, when these variables are supposed to take on a finite number of different values and these values being fuzzy subsets of the one-dimensional Euclidean space. In this paper we show that the one-sample method of testing about the mean of a fuzzy random variable can be extended to general ones (more precisely, to those whose range is not necessarily finite and whose values are fuzzy subsets of finite-dimensional Euclidean space). This extension is immediately developed by combining some tools in the literature, namely, bootstrap techniques on Banach spaces, a metric between fuzzy sets based on the support function, and an embedding of the space of fuzzy random variables into a Banach space which is based on the support function.

On the use of Zadeh's probabilistic definition for testing statistical hypotheses from fuzzy information

Abstract A statistical hypothesis is an assertion about the distribution of an experiment. We consider the study of the problem of testing a statistical hypothesis (that is, the problem of concluding whether or not the hypothesis is correct) on the basis of data from the experiment, when its outcomes do not provide exact but rather fuzzy information. For establishing optimality criteria of testing we will use the definition of probability of a fuzzy event, given by Zadeh, in order to extend both Neyman-Pearson and Bayes theories, to the fuzzy framework. Then, we will analyze several properties for the new criteria. Particularly, the goodness of optimal procedures in both the fuzzy and the nonfuzzy situation, will be compared for each criterion. Finally, we will apply the extended criteria for testing simple hypotheses. This application leads us to prefer Bayesian procedures to Neyman-Pearson procedures in the fuzzy context.

Two-sample hypothesis tests of means of a fuzzy random variable

Abstract In this paper we will consider some two-sample hypothesis tests for means concerning a fuzzy random variable in two populations. For this purpose, we will make use of a generalized metric for fuzzy numbers, and we will develop an exact study for the case of normal fuzzy random variables and an asymptotic study for the case of simple general fuzzy random variables.

Soft methods in probability, statistics and data analysis

Part I: Introductory Papers.- Part II: Soft Methods in Probability - Fundamentals.- Part III: Soft Methods in Statistics - Fuzzy Stochastic Models.- Part IV: Soft Methods in Data Analysis - Fuzzy, Rough and Other Approaches.

Constructive definitions of fuzzy random variables

When we deal with a random experiment, we are often interested in functions of the experimental outcomes rather than the outcomes themselves. Fuzzy random variables formalize fuzzy-valued functions of the outcomes in a random experiment, that is, existing imprecise quantification processes. The concepts of fuzzy random variable and its fuzzy expected value, have been introduced by Puri and Ralescu by means of descriptive definitions. Nevertheless, constructive definitions of fuzzy random variables would play an essential role in the constructive definition of their integrals, which will be especially valuable to perform practical computations and to develop further results concerning the integration of these variables. In this paper we present constructive definitions of fuzzy random variables and integrably bounded fuzzy random variables based on the Hausdorff convergence. The use of the last definition to obtain a constructive definition of the fuzzy expected value of an integrably bounded fuzzy random variable is finally discussed.

Combining Soft Computing and Statistical Methods in Data Analysis

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