The median of a random fuzzy number. The 1-norm distance approach

Abstract

In quantifying the central tendency of the distribution of a random fuzzy number (or fuzzy random variable in Puri and Ralescu's sense), the most usual measure is the Aumann-type mean, which extends the mean of a real-valued random variable and preserves its main properties and behavior. Although such a behavior has very valuable and convenient implications, 'extreme' values or changes of data entail too much influence on the Aumann-type mean of a random fuzzy number. This strong influence motivates the search for a more robust central tendency measure. In this respect, this paper aims to explore the extension of the median to random fuzzy numbers. This extension is based on the 1-norm distance and its adequacy will be shown by analyzing its properties and comparing its robustness with that of the mean both theoretically and empirically.

Bagged fuzzy clustering for fuzzy data: An application to a tourism market

“…Therefore two important drawbacks arise: first, this type of scale entails a source of vagueness and uncertainty in evaluation since it represents subjective knowledge [20,21]; second, respondent must automatically convert an opinion on a scale and this conversion can distort the original opinion that had to be captured [22]. One way to overcome these drawbacks is to transform Likert variables into fuzzy numbers [16,23]. In the tourism field there are relevant applications of this type of transformation [see, e.g., [20][21][22][24][25][26][27].…”

“…Fuzzy numbers are convex and normalized fuzzy sets with a piecewise continuous membership function defined in R that maps an interval to [0, 1]. The use of fuzzy sets and fuzzy numbers has gain attention in the literature mainly for the following reasons: 1) they are able to capture and measure the uncertainty of individual evaluations (20,23,74); 2) fuzzy numbers have a very intuitive meaning and it is more comprehensive than other methods [79]; 3) fuzzy sets can better describe complex processes of the real-life than traditional statistical methods [79]; 4) they can be adapted to a wide range of imprecise data due to the richness of the existing fuzzy scales [23,78,79]. As a consequence, when Likert-type scales, or any other linguistic variables, are used in a questionnaire it is useful to formalize them in terms of fuzzy numbers, in order to reduce the imprecision/vagueness of the observed data.…”

“…the fuzzy number), rather than only one score, per each individual answer. Note that, in order to avoid this problem, when designing the questionnaire a fuzzy rating scale could be adopted [23,80,81].…”

Bertoluzza et al.’s metric as a basis for analyzing fuzzy data

“…Other relevant summary measures of the distribution of a random fuzzy number are the Fréchet variance based on D φ W (see, for instance, Lubiano et al [23], Blanco-Fernández et al [2]), and the L 1 -medians by Sinova et al [30,31]. The covariance of two random fuzzy numbers can be also introduced (see González-Rodríguez et al [13], Blanco-Fernández et al [2]) in connection with the simple linear regression analysis between random fuzzy sets, although in this case it does not involve D Another statistical problem involving Bertoluzza et al's metric is that of testing about the population fuzzy-valued Aumann-type mean of one or more random fuzzy numbers on the basis of a sample of independent observations from it or them.…”

A parameterized metric between fuzzy numbers and its parameter interpretation

“…• to classify fuzzy data (see, for instance, Coppi et al [1], Ferraro and Giordani [2] and Guillaume et al [3]), • to obtain some limit and probabilistic results for random fuzzy numbers (see, for instance, Colubi et al [4], Molchanov [5], Terán [6,7], Quang and Thuan [8], Aletti and Bongiorno [9]), • in optimization problems (see, for instance, Abbasbandy and Asady [10], Abbasbandy and Amirfakhrian [11], Prochelvi et al [12], Báez-Sánchez et al [13], Bana and Coroianu [14], Bera et al [17], Coroianu [15], Coroianu et al [16]) • and especially in performing many statistical analyses (see, for instance, Näther [18,19], Körner and Näther [20], Körner [21], García et al [22], Montenegro et al [23,24], Gil et al [25], Coppi et al [26], González-Rodríguez et al [29,30,31], Ferraro et al [27], Ferraro and Giordani [28], Ramos-Guajardo and Lubiano [32], Sinova et al [39]). In the literature on fuzzy numbers and more general fuzzy sets, several metrics have been suggested (see, for instance, Puri and Ralescu [33], Klement et al [34]).…”