Ana Colubi

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.

A _{}[0,1] representation of random upper semicontinuous functions

In this paper a representation of random upper semicontinuous functions in terms of D E [0, 1]-valued random elements is stated. This fact allows us to consider for the first time a complete and separable metric, the Skorohod one, on a wide class of upper semicontinuous functions. Finally, different relevant concepts of measurability for random upper semicontinuous functions are studied and the relationships between them are analyzed.

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.

On the formalization of fuzzy random variables

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

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.

Estimation of a simple linear regression model for fuzzy random variables

A generalized simple linear regression statistical/probabilistic model in which both input and output data can be fuzzy subsets of R^p is dealt with. The regression model is based on a fuzzy-arithmetic approach and it considers the possibility of fuzzy-valued random errors. Specifically, the least-squares estimation problem in terms of a versatile metric is addressed. The solutions are established in terms of the moments of the involved random elements by employing the concept of support function of a fuzzy set. Some considerations concerning the applicability of the model are made.

A linear regression model for imprecise response

A linear regression model with imprecise response and p real explanatory variables is analyzed. The imprecision of the response variable is functionally described by means of certain kinds of fuzzy sets, the LR fuzzy sets. The LR fuzzy random variables are introduced to model usual random experiments when the characteristic observed on each result can be described with fuzzy numbers of a particular class, determined by 3 random values: the center, the left spread and the right spread. In fact, these constitute a natural generalization of the interval data. To deal with the estimation problem the space of the LR fuzzy numbers is proved to be isometric to a closed and convex cone of R^3 with respect to a generalization of the most used metric for LR fuzzy numbers. The expression of the estimators in terms of moments is established, their limit distribution and asymptotic properties are analyzed and applied to the determination of confidence regions and hypothesis testing procedures. The results are illustrated by means of some case-studies.

Statistical inference about the means of fuzzy random variables: Applications to the analysis of fuzzy- and real-valued data

The expected value of a fuzzy random variable plays an important role as central summary measure, and for this reason, in the last years valuable statistical inferences about the means of the fuzzy random variables have been developed. Some of the main contributions in this topic are gathered and discussed. Concerning the hypothesis testing, the bootstrap techniques have empirically shown to be efficient and powerful. Algorithms to apply these techniques in practice and some illustrative real-life examples are included. On the other hand, it has been recently shown that the distribution of any real-valued random variable can be represented by means of a fuzzy set. The characterizing fuzzy sets correspond to the expected value of a certain fuzzy random variable based on a family of fuzzy-valued transformations of the original real-valued ones. They can be used for descriptive/exploratory or inferential purposes. This fact adds an extra-value to the fuzzy expected value and the preceding statistical procedures, that can be used in statistics about real distributions.

Least squares estimation of linear regression models for convex compact random sets

Simple and multiple linear regression models are considered between variables whose “values” are convex compact random sets in