Ana Belén Ramos-Guajardo

A distance-based statistical analysis of fuzzy number-valued data

Abstract Real-life data associated with experimental outcomes are not always real-valued. In particular, opinions, perceptions, ratings, etc., are often assumed to be vague in nature, especially when they come from human valuations. Fuzzy numbers have extensively been considered to provide us with a convenient tool to express these vague data. In analyzing fuzzy data from a statistical perspective one finds two key obstacles, namely, the nonlinearity associated with the usual arithmetic with fuzzy data and the lack of suitable models and limit results for the distribution of fuzzy-valued statistics. These obstacles can be frequently bypassed by using an appropriate metric between fuzzy data, the notion of random fuzzy set and a bootstrapped central limit theorem for general space-valued random elements. This paper aims to review these ideas and a methodology for the statistical analysis of fuzzy number data which has been developed along the last years.

K-sample tests for equality of variances of random fuzzy sets

The problem of testing equality of variances often arises when distributions of random variables are compared or linear models between them are considered. The usual tests for variances given normality of the underlying populations are highly non-robust to non-normality and are strongly dependent on the kurtosis. Some alternative formulations of Levenes test statistic for testing the homoscedasticity have been shown to be powerful and robust under non-normality. On the basis of Levenes classical procedure, a test for the equality of variances of k fuzzy-valued random elements is developed. Accordingly, consistent asymptotic and bootstrap tests are established and their empirical behaviour is analyzed by means of extensive simulation studies. In addition, the proposed test is compared with a Bartlett-type test. A case-study illustrating the applicability of the procedure is presented.

Inclusion degree tests for the Aumann expectation of a random interval

An extension of the inclusion test of the expected value of a real random variable in an interval to the case of general random intervals is introduced. The hypothesis of strict inclusion is relaxed by considering a measure of the degree of inclusion. Thus, partial inclusions are also tested. Asymptotic and bootstrap techniques are established. The performance of the bootstrap test is also analyzed by means of some simulations. A case-study regarding the blood pressure classification in adults is considered.

Distance-based linear discriminant analysis for interval-valued data

Interval-valued observations arise in many real-life situations as either the precise representation of the objective entity or the representation of incomplete knowledge. Thus given p features observed over a sample of objects belonging to one of two possible classes, each observation can be perceived as a non-empty closed and bounded hyperrectangle on R p . The aim of the paper is to suggest a p-dimensional classification method for random intervals when two or more classes are considered, by the generalization of Fishers procedure for linear discriminant analysis. The idea consists of finding a directional vector which maximizes the ratio of the dispersion between the classes and within the classes of the observed hyperrectangles. A classification rule for new observations is also provided and some simulations are carried out to compare the behavior of the proposed classification procedure with respect to other methods known from the literature. Finally, the suggested methodology are applied on a real-life situation example.

Similarity Test for the Expectation of a Random Interval and a Fixed Interval

A hypothesis test for analyzing the degree of similarity between the expected value of a random interval and a fixed interval is introduced. It is based on a measure of the similarity between classical convex sets proposed in the literature. Asymptotic techniques are firstly applied to analyze the limit distribution of the proposed test statistic. Afterwards, a bootstrap approach is presented to better approximate the sampling distribution. Finally, the performance of the test is investigated by means of simulation studies.

Testing the Variability of Interval Data: An Application to Tidal Fluctuation

A methodology for analyzing the variability of the tidal fluctuation in a specific area is proposed in this work. The idea is to consider intervals determined by the minimum and maximum height reached by the tides in a day. Thus, the theoretical aim is to develop hypothesis tests about the variance of one or more interval-valued random elements (i.e., random intervals). Some simulations showing the empirical behavior and consistency of the proposed tests are carried out by considering different models. Finally, the procedure is applied to a real-life study concerning the fluctuation of tides in the port of Gijon (Asturias).

A fuzzy clustering procedure for random fuzzy sets

A fuzzy clustering method for random fuzzy sets is proposed. The starting point is a p-value matrix with elements obtained by comparing the expected values of random fuzzy sets by means of a bootstrap test. As such, the p-value matrix can be viewed as a relational data matrix since the p-values represent a kind of similarity between random fuzzy sets. For this reason, in order to cluster random fuzzy sets, fuzzy clustering techniques for relational data can be applied. In this context, the so-called NE-FRC algorithm is considered. One of the most important advantages of the NE-FRC is that the relational data could not be derived from Euclidean distances. Some simulations are presented to show the behavior of the proposed procedure and two applications to real-life situations are also included.

Inclusion and exclusion hypothesis tests for the fuzzy mean

Abstract A hypothesis test for the inclusion of the expected value of a fuzzy-valued random element in a given fuzzy set is provided. The exclusion, or the empty intersection of the expected value of a random fuzzy set and a fuzzy set, is also tested, that allows us to define one-sided tests for this expected value without the need of considering any restrictive ordering. Both tests are developed by using a measure of the intersection between fuzzy sets. Asymptotic and bootstrap techniques are established. Some simulations are included to show the performance of the bootstrap approaches. Finally, the methodology proposed is applied on a real-life situation related to the field of sensorial analysis.

Power Analysis of the Homoscedasticity Test for Random Fuzzy Sets

Some tools for testing hypotheses about the variance of random fuzzy sets are already available. Asymptotically correct procedures for the k-sample homoscedasticity tests have been recently developed. However, the power of such procedures has not been analyzed yet. In this paper, some studies about the power function of the asymptotic procedure for the homoscedasticity test are presented. The theoretical analysis is carried out by considering the capability of the test under local alternatives. Finally, the behavior of the power function is illustrated by means of simulation studies.

A new framework for the statistical analysis of set-valued random elements

Abstract 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.