Catálogo de publicaciones - libros

Fuzzy Set Theory has been developed during the second half of last century, with a starting point in L.A. Zadeh seminal paper [14]. From that moment on, there has been a harsh debate between scientifics supporting it and others believing that it was an unnecesary mathematical construct, generally opposing it to Probability Theory. Those researchers usually complained about the alleged lack of mathematical soundness of Fuzzy Logic and its applications. For a succint review on this debate, see Section 1 of [2].

A new measure of skewness for real random variables is proposed in this paper. The measure is based on a fuzzy representation of real-valued random variables which can be used to characterize the distribution of the original variable through the of the ‘fuzzified’ random variable. Inferential studies concerning the expected value of fuzzy random variables provide us with a tool to analyze the asymmetry degree from random samples. As a first step, we propose an asymptotic test of symmetry. We present some examples and simulations to illustrate the behaviour of the proposed test.

In this paper we propose a new way of representing the distribution of a real random variable by means of the expected value of certain kinds of fuzzifications of the original variable. We will analyze the usefulness of this representation from a descriptive point of view. We will show that the graphical representation of the fuzzy expected value displays in a visible way relevant features of the original distribution, like the central tendency, the dispersion and the symmetry. The fuzzy representation is valuable for representing continuous or discrete distributions, thus, it can be employed both for representing population distributions and for exploratory data analysis.

In this paper a method is introduced to simulate fuzzy random variables by using the support function. On the basis of the support function, the class of values of a fuzzy random variable can be ‘identified’ with a closed convex cone of a Hilbert space, and we now suggest to simulate Hilbert space-valued random elements and to project later into such a cone. To make easier the projection above we will consider isotonic regression. The procedure will be illustrated by means of several examples.

Friedman’s test is traditionally applied for testing independence between orderings ( > 2). In the paper we show how to generalize Friedman’s test for situations with missing information or non-comparable outputs. This contribution is a corrected version of our previous paper [8].

The aim of this work is to give a summary of some of the known properties of sets of measure-free martingales in vector lattices and Banach spaces. In particular, we consider the relationship between such sets of martingales and the ranges of the underlying filtration of conditional expectation operators.

This note aims at presenting the most general framework for a class of random upper semicontinuous functions, namely random elements whose sample paths are upper semicontinuous (u.s.c.) functions, defined on some locally compact, Hausdorff and second countable base space, extending Matheron’s framework for random closed sets. It is shown that while the natural embedding process does not provide compactness for , the topology does.

In this paper, we shall prove some properties of set-valued asymptotic martingale (amart for short) and provide an optional sampling theorem. We also prove a quasi Risez decomposition theorem for set-valued amarts. Then we shall discuss the existence of selections of set-valued amarts and give a representation theorem.

We extend some topologies on the space of upper semicontinuous functions with compact support to those on that of general upper semicontinuous functions and see that graphical topology and modified topology are the same. We then define random upper semicontinuous functions using their topological Borel field and finally give a Choquet theorem for random upper semicontinuous functions.

We present a general law of large numbers in a (separable complete) metric space endowed with an abstract convex combination operation. Spaces of fuzzy sets are shown to be particular cases of that framework. We discuss the compatibility of the usual definition of expectation with the abstract one. We close the paper with two applications to the theory of fuzzy random variables (fuzzy random variables and level-2 fuzzy random variables in a space).