A fuzzy representation of random variables: An operational tool in exploratory analysis and hypothesis testing

doi:10.1016/j.csda.2006.04.006

Computational Statistics & Data Analysis

Volume 51, Issue 1, 1 November 2006, Pages 163-176

https://doi.org/10.1016/j.csda.2006.04.006 Get rights and content

Abstract

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.

Introduction

Fuzzy random variables (hereafter FRVs for short) were introduced as a useful and well-formalized model for random elements taking on fuzzy set values. FRVs include random variables/vectors/sets as special cases. For the last two decades, FRVs have received a wide attention and several studies have been developed to analyze probabilistic aspects of these random elements. Among them, we can remark those connected with the formalization of the measurability (see Puri and Ralescu, 1986, Klement et al., 1986, Colubi et al., 2002, etc.), and the laws of large numbers which strengthen the suitability of the fuzzy mean (cf. Colubi et al., 1999, Molchanov, 1999, Proske and Puri, 2002, etc.).

Nevertheless, statistical aspects and applications of FRVs have hardly been examined in the literature. Although some limit theorems of the central type have been also stated (see, for instance, Li et al., 2003), they do not have natural statistical implications in contrast to the real-valued case.

Quite recently, a few research has been done on testing statistical hypotheses about the fuzzy means of FRVs (see Körner, 2000, Montenegro et al., 2001, Montenegro et al., 2004, Gil et al., 2006, González-Rodríguez et al., 2006) by considering some generalized operational metrics for fuzzy data.

In Montenegro et al., 2004, Montenegro et al., 2005 and Gil et al. (2006) an interesting fact has been revealed: in analyzing empirically how the presence of fuzziness affects the power of tests about means, several simulation studies have been carried out to compare tests based on FRVs with their particularization to the real-valued random variables obtained by ‘defuzzifying’ the fuzzy ones; these comparative empirical studies have led to assert that the power for most of the fuzzy-valued cases is appreciably greater than the power for the corresponding real-valued ones. A tentative explanation for this is that the greater the complexity of data used, the greater the reliability of the results.

This last conclusion has inspired the key ideas guiding this work. More precisely, the aim of this paper is to introduce a functional representation $γ$ ‘fuzzifying’ values of a real-valued random variable X to later handle the information contained in the (fuzzy) expected value of the FRV $γ \circ X$ . This representation will be defined in terms of some functions we can choose depending on different targets (for instance, the easy visualization of summary measures of the distribution of X, or the characterization of the distribution of X).

In this paper we first recall some preliminary concepts and introduce the $γ$ - fuzzy representation of a random variable X. Then, we will consider some special choices for which the fuzzy mean of the FRV $γ \circ X$ captures either visual information of some summary measures of the distribution of X, or the whole information on the distribution of X. Some immediate applications for statistical exploratory and inferential studies are presented. Finally, some implications to test the goodness of fit, or the equality of distributions, of random variables are derived and related future directions are suggested.

Section snippets

Preliminaries and $γ$ -fuzzy representation of random variables

FRVs (also referred to in the literature as random fuzzy sets or random upper semicontinuous functions) were introduced as an extension of random sets to model the situations in which the value assigned to each and every possible outcome in the performance of a random experiment is a fuzzy set.

Let $K (R^{p})$ be the class of non-empty compact subsets of the Euclidean space $R^{p}$ , and let $F (R^{p})$ denote the class of normal upper semicontinuous fuzzy sets of $R$ with bounded closure of the support; that is, $F (R^{p}) = \{\tilde{B}\}$

Capturing visual information about summary measures of the distribution of random variables by means of some $γ$ -fuzzy representations: application to exploratory analysis

As we have already commented, the choice of suitable functions defining the fuzzification operator $γ$ allows us to capture relevant information on the distribution of the original random variables X.

When one is interested in the immediate visualization of summary measures like the mean $E (X | P)$ and variance $Var (X | P)$ of the distribution of X, the family of fuzzy operators introduced in the next result offers an operational way to achieve this purpose.

Theorem 3.1

Consider a probability space $(Ω, A, P)$ and let $X : Ω$

Capturing the whole information about the distribution of random variables by means of some $γ$ -fuzzy representations: immediate inferential application

When one is interested in characterizing the distribution of X (irrespectively of this distribution having or not finite moments), next result provides us with a possible way to do it through the expected value of certain $γ$ -fuzzy representations of X.

Theorem 4.1

Consider a probability space $(Ω, A, P)$ . Let $γ^{C} : R \to F_{c} (R)$ the mapping transforming each value $x \in R$ into the fuzzy number ${(γ^{C} (x))}_{α} = [f_{L} (x) - (1 - α)^{1 / h_{L} (x)}, f_{R} (x) + (1 - α)^{1 / h_{R} (x)}]$ for all $α \in [0, 1]$ where $f_{L} : R \to R, f_{R} : R \to R, f_{L}, f_{R} \in L^{1} (Ω, A, P), f_{L} (x) ⩽ f_{R} (x) for all x \in R,$ and $h_{L} : R \to (0, + \infty$

Other inferential implications for random variables

The fact that $γ^{C}$ -fuzzy representations determine fuzzy means characterizing the distribution of the original random variables entails a relevant interesting application of the fuzzification operator above $γ^{C}$ which deserves a future deeper analysis.

This application is the one obtained by combining the $γ^{C}$ -fuzzy operators with the methodology for testing about fuzzy means of FRVs we have developed in previous papers (see, for instance, Montenegro et al., 2001, Montenegro et al., 2004, Gil et al.,

Concluding remarks

One of the most immediate future directions based on the ideas in this paper is that of discussing about the application to test either the goodness of fit of a completely specified distribution or the equality of distributions. A generalization of the first test to the goodness of fit for hypothesis involving unknown parameters would be also desirable. In this respect, it would be interesting to analyze the effects of the choice of the $γ^{C}$ -fuzzy operator as well as that of measures $W$ and $ϕ$ in

Acknowledgments

Authors are very grateful to Professors Renato Coppi and Pierpaolo D’Urso as well as to Dr. Paolo Giordani for their valuable comments and suggestions. The research in this paper has been partially supported by the Spanish Ministry of Education and Science Grants BFM2002-01057 and MAT2005-00045. Their financial support is gratefully acknowledged.

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A fuzzy representation of random variables: An operational tool in exploratory analysis and hypothesis testing

Abstract

Introduction

Section snippets

Preliminaries and γ-fuzzy representation of random variables

Capturing visual information about summary measures of the distribution of random variables by means of some γ-fuzzy representations: application to exploratory analysis

Capturing the whole information about the distribution of random variables by means of some γ-fuzzy representations: immediate inferential application

Other inferential implications for random variables

Concluding remarks

Acknowledgments

J. Statist. Plann. Inference

J. Math. Anal. Appl.

J. Math. Anal. Appl.

Inform. Sci

J. Math. Anal. Appl.

Inform. Sci.

On a new class of distances between fuzzy numbers

Mathware Soft Comput.

A generalized strong law of large numbers

Probab. Theory Related Fields

Preliminaries and $γ$ -fuzzy representation of random variables

Capturing visual information about summary measures of the distribution of random variables by means of some $γ$ -fuzzy representations: application to exploratory analysis

Capturing the whole information about the distribution of random variables by means of some $γ$ -fuzzy representations: immediate inferential application