A multiple linear regression model for imprecise information

Abstract

In standard regression analysis the relationship between the (response) variable and a set of (explanatory) variables is investigated. In the classical framework the response is affected by probabilistic uncertainty (randomness) and, thus, treated as a random variable. However, the data can also be subjected to other kinds of uncertainty such as imprecision. A possible way to manage all of these uncertainties is represented by the concept of fuzzy random variable (FRV). The most common class of FRVs is the L R family (L R FRV), which allows us to express every FRV in terms of three random variables, namely, the center, the left spread and the right spread. In this work, limiting our attention to the L R FRV class, we consider the linear regression problem in the presence of one or more imprecise random elements. The procedure for estimating the model parameters and the determination coefficient are discussed and the hypothesis testing problem is addressed following a bootstrap approach. Furthermore, in order to illustrate how the proposed model works in practice, the results of a real-life example are given together with a comparison with those obtained by applying classical regression analysis.

Central tendency for symmetric random fuzzy numbers

“…Several problems and techniques are being studied and developed along this century. For instance, testing about means (see Colubi et al [4], González-Rodríguez et al [14] or the recent review by BlancoFernández et al [1]), regression analysis (see, for instance, Ferraro et al [9], Ferraro and Giordani [10]), clustering (see, for instance, González-Rodríguez et al [12]), Bayesian analysis (see Stein et al [26]), actuarial developments, portfolio selection and mathematical programming (see, for instance, Shapiro [22], Li and Xu [16], Sakawa and Matsui [20]), and so on.…”

On the Generalization Performance of a Regression Model with Imprecise Elements

“…In Ferraro et al 18,19 and Ferraro and Giordani 20 we have fixed the transformation functions g and h and then we have estimated the regression parameters and the determination coefficient. In this paper the aim is considering a family of transforms, the Box-Cox transformation model (see, for more details, Box and Cox 6 ) and, by means of an algorithm, choosing the optimal parameters of the family.…”

“…are the parameters estimated on the basis of the b-th bootstrap sample (for more details, see Ferraro et al, 18 Ferraro and Giordani 20 ) and · is the Euclidean norm. The bootstrap gh-prediction error is expressed as…”

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]).…”