Miguel López-Díaz

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.

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.

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.

The l-average value and the fuzzy expectation of a fuzzy random variable

Abstract The concepts of fuzzy random variable, and the associated fuzzy expected value, have been introduced by Puri and Ralescu as an extension of measurable set-valued functions (random sets), and of the Aumann integral of these functions, respectively. On the other hand, the λ-average function has been suggested by Campos and Gonzalez as an appropriate function to rank fuzzy numbers. In this paper we are going to analyze some useful properties concerning the λ-average value of the expectation of a fuzzy random variable, and some practical implications of these properties are also commented on.

The l a -mean squared dispersion associated with a fuzzy random variable

In this paper we introduce a parameterized real-valued measure of the mean dispersion of a fuzzy random variable with respect to an arbitrary fuzzy number. This measure extends the second moment of a classical random variable, and is based on a parameterized distance between fuzzy numbers. Properties of the measure presented are analyzed, and the extension of the variance of a classical random variable, which particularizes the mean squared dispersion, is also examined. Some examples illustrating the computation and possible applications of the measure are included. Finally, a brief discussion about the interest of using a parameterized distance, and about some future directions of this study, is developed.

Fundamentals and Bayesian analyses of decision problems with fuzzy-valued utilities

Abstract A manageable model to deal with general single-stage statistical decision problems with fuzzy-valued consequences is presented. The model is based on the notion of fuzzy random variable, as defined by Puri and Ralescu, and also on a crisp ranking method for fuzzy numbers introduced by Campos and Gonzalez. Fundamentals of the fuzzy utility function representing the preference pattern of the decision maker are established to guarantee the existence of this function by means of an axiomatic development. Bayesian analyses of these statistical decision problems in normal and extensive forms are formalized, and conditions for the equivalence of these analyses are given. Finally, an example illustrating the Bayesian analysis is considered.

Reversing the order of integration in iterated expectations of fuzzy random variables, and statistical applications

In this paper conditions are given to compute iterated expectations of fuzzy random variables irrespectively of the order of integration. To this purpose, some studies about the measurability and the integrable boundedness of the fuzzy expected value of a fuzzy random variable with respect to a regular conditional probability are first developed. The conclusions obtained are applied later to statistical problems involving fuzzy random variables, like those concerning some hierarchical models and mixture distributions (more precisely, some techniques to obtain fuzzy unbiased estimators and the Bayesian analysis of statistical problems).

Tools for fuzzy random variables: Embeddings and measurabilities

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.

A method to derive strong laws of large numbers for random upper semicontinuous functions

In this paper, we suggest a new method which is useful to study convergences of certain random upper semicontinuous elements by taking advantages of some well-known results concerning random cadlag functions. The suggested method is applied in the paper to developed strong laws of large numbers.

Influence diagrams with super value nodes involving imprecise information

The concept of super value nodes was established to allow dynamic programming to be performed within the theory of influence diagrams and to reduce the computational complexity in solving problems by means of influence diagrams. This paper is focused on how influence diagrams with super value nodes are affected by the presence of imprecise information. We analyze how to reduce the complexity when evaluating an influence diagram in this framework by modelling these kinds of nodes and random magnitudes in terms of fuzzy random variables. Finally, an applied example of the theoretical results is developed.