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Some hypothesis tests for analyzing the degree of overlap between the expected value of random intervals are provided. For this purpose, a suitable measure to quantify the overlapping grade between intervals is considered on the basis of the Szymkiewicz-Simpson coefficient defined for general sets. It can be seen as a kind of likeness index to measure the mutual information between two intervals. On one hand, an estimator for the proposed degree of overlap between intervals is provided and its strong consistency is analyzed. On the other hand, two tests are also proposed in this framework: a one-sample test to examine the degree of overlap between the expected value of a random interval and a given interval, and a two-sample test to check the degree of overlap between the expected value of two random intervals. To solve such hypothesis tests, two statistics are suggested and their limit distributions are studied by considering both asymptotic and bootstrap techniques. Their power has been also explored by means of local alternatives. In addition, some simulation studies are carried out to investigate the behavior of the proposed approaches. Finally, the performance of the tests is also reported in a real-life application. [ABSTRACT FROM AUTHOR]

The space of nonempty convex and compact (fuzzy) subsets of R p , K c ( R p ) , has been traditionally used to handle imprecise data. Its elements can be characterized via the support function, which agrees with the usual Minkowski addition, and naturally embeds K c ( R p ) into a cone of a separable Hilbert space. The support function embedding holds interesting properties, but it lacks of an intuitive interpretation for imprecise data. As a consequence, it is not easy to identify the elements of the image space that correspond to sets in K c ( R p ) . Moreover, although the Minkowski addition is very natural when p = 1 , if p > 1 the shapes which are obtained when two sets are aggregated are apparently unrelated to the original sets, because it tends to convexify. An alternative and more intuitive functional representation will be introduced in order to circumvent these difficulties. The imprecise data will be modeled by using star-shaped sets on R p . These sets will be characterized through a center and the corresponding polar coordinates, which have a clear interpretation in terms of location and imprecision , and lead to a natural directionally extension of the Minkowski addition. The structures required for a meaningful statistical analysis from the so-called ontic perspective are introduced, and how to determine the representation in practice is discussed. [ABSTRACT FROM AUTHOR]

Abstract: The supervised classification of fuzzy data obtained from a random experiment is discussed. The data generation process is modelled through random fuzzy sets which, from a formal point of view, can be identified with certain function-valued random elements. First, one of the most versatile discriminant approaches in the context of functional data analysis is adapted to the specific case of interest. In this way, discriminant analysis based on nonparametric kernel density estimation is discussed. In general, this criterion is shown not to be optimal and to require large sample sizes. To avoid such inconveniences, a simpler approach which eludes the density estimation by considering conditional probabilities on certain balls is introduced. The approaches are applied to two experiments; one concerning fuzzy perceptions and linguistic labels and another one concerning flood analysis. The methods are tested against linear discriminant analysis and random K-fold cross validation. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: A clustering method to group independent fuzzy random variables observed on a sample by focusing on their expected values is developed. The procedure is iterative and based on the p-value of a multi-sample bootstrap test. Thus, it simultaneously takes into account fuzziness and stochastic variability. Moreover, an objective stopping criterion leading to statistically equal groups different from each other is provided. Some simulations to show the performance of this inferential approach are included. The results are illustrated by means of a case study. [Copyright &y& Elsevier]

Abstract: This work deals with the simulation of fuzzy random variables, which can be used to model various realistic situations, where uncertainty is not only present in form of randomness but also in form of imprecision, described by means of fuzzy sets. Utilizing the common arithmetics in the space of all fuzzy sets only induces a conical structure. As a consequence, it is difficult to directly apply the usual simulation techniques for functional data. In order to overcome this difficulty two different approaches based on the concept of support functions are presented. The first one makes use of techniques for simulating Hilbert space-valued random elements and afterwards projects on the cone of all fuzzy sets. It is shown by empirical results that the practicability of this approach is limited. The second approach imitates the representation of every element of a separable Hilbert space in terms of an orthonormal basis directly on the space of fuzzy sets. In this way, a new approximation of fuzzy sets useful to approximate and simulate fuzzy random variables is developed. This second approach is adequate to model various realistic situations. [Copyright &y& Elsevier]

Abstract: Interpretability is acknowledged as the main advantage of fuzzy systems and it should be given a main role in fuzzy modeling. Classical systems are viewed as black boxes because mathematical formulas set the mapping between inputs and outputs. On the contrary, fuzzy systems (if they are built regarding some constraints) can be seen as gray boxes in the sense that every element of the whole system can be checked and understood by a human being. Interpretability is essential for those applications with high human interaction, for instance decision support systems in fields like medicine, economics, etc. Since interpretability is not guaranteed by definition, a huge effort has been done to find out the basic constraints to be superimposed during the fuzzy modeling process. People talk a lot about interpretability but the real meaning is not clear. Understanding of fuzzy systems is a subjective task which strongly depends on the background (experience, preferences, and knowledge) of the person who makes the assessment. As a consequence, although there have been a few attempts to define interpretability indices, there is still not a universal index widely accepted. As part of this work, with the aim of evaluating the most used indices, an experimental analysis (in the form of a web poll) was carried out yielding some useful clues to keep in mind regarding interpretability assessment. Results extracted from the poll show the inherent subjectivity of the measure because we collected a huge diversity of answers completely different at first glance. However, it was possible to find out some interesting user profiles after comparing carefully all the answers. It can be concluded that defining a numerical index is not enough to get a widely accepted index. Moreover, it is necessary to define a fuzzy index easily adaptable to the context of each problem as well as to the user quality criteria. [Copyright &y& Elsevier]