Possibilistic and fuzzy clustering methods for robust analysis of non-precise data

This work focuses on robust clustering of data affected by imprecision. The imprecision is managed in terms of fuzzy sets. The clustering process is based on the fuzzy and possibilistic approaches. In both approaches the observations are assigned to the clusters by means of membership degrees. In fuzzy clustering the membership degrees express the degrees of sharing of the observations to the clusters. In contrast, in possibilistic clustering the membership degrees are degrees of typicality. These two sources of information are complementary because the former helps to discover the best fuzzy partition of the observations while the latter reflects how well the observations are described by the centroids and, therefore, is helpful to identify outliers. First, a fully possibilistic k-means clustering procedure is suggested. Then, in order to exploit the benefits of both the approaches, a joint possibilistic and fuzzy clustering method for fuzzy data is proposed. A selection procedure for choosing the parameters of the new clustering method is introduced. The effectiveness of the proposal is investigated by means of simulated and real-life data

Fitting parametric link functions in a regression model with imprecise random variables

A regression model for imprecise random variables has been introduced in our previous works. The imprecision of a random element has been formalized by means of the fuzzy random variable (FRV). In detail, a particular case of FRVs characterized by a center, a left and a right spread, the LR family (LR FRV), has been considered. The idea is to jointly consider three regression models in which the response variables are the center, and two transformations of the left and the right spreads in order to overcome the non-negativity conditions of the spreads. Response transformations could be fixed, as we have done so far, but all inferential procedures, such as estimation, hypothesis tests on the regression parameters, linearity test etc., are affected by this choice. For this reason we consider a family of parametric link functions, the Box-Cox transformation model, and by means of a computational procedure we will look for the transformation parameters that maximize the goodness of fit of the model

A linear regression model with LR fuzzy random variables

In standard regression analysis the relationship between one (response) variable and a set of (explanatory) variables is investigated. In a classical framework the response is affected by probabilistic uncertainty (randomness) and, thus, treated as a random variable. However, the data can be also subjected to other kinds of uncertainty, such as imprecision, vagueness, etc. 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 LR family, which allows us to express every FRV in terms of three random variables, namely, the center, the left and the right spread. In this work, limiting our attention to the LR FRVs, we address the linear regression problem in presence of one or more imprecise random elements. The procedure for estimating the model parameters is discussed, and the statistical properties of the estimates are analyzed. Furthermore, in order to illustrate how the proposed model works in practice, the results of some case-studies are given

Cluster analysis for networks using a fuzzy approach

As the network representation is widely used to describe problems in an increasing number of disciplines, novel methodologies are needed to handle such complexity. In particular, cluster analysis is an interesting and challenging task in the network framework. In this work, we focus on how to represent networks for fuzzy clustering and how to apply standard fuzzy algorithms for clustering multiple networks on synthetic data

Testing linearity for a regression model with imprecise elements: a power analysis

A linearity test for a simple regression model with imprecise random elements is analyzed. The concept of LR fuzzy random variable is used to formalize imprecise random elements. The proposed linearity test is based on the comparison of the simple linear regression model and the nonparametric regression. In details, based on the variability explained by the above two models, the test statistic is constructed. The asymptotic significance level and the power under local alternatives are established. Since large samples are required to obtain suitable asymptotic results a bootstrap approach is investigated. Furthermore, in order to illustrate how the proposed test works in practice, some simulation and real-life examples are given

Different linearity tests for a regression model with an imprecise response

Recently a new linear regression model with fuzzy response and scalar explanatory variables has been introduced and deeply analyzed. Since the inferences developed for such a model are meaningful only if the relationship is indeed linear, it is important to check the linearity for the regression model. Two different linearity tests have been introduced. The first one is based on the comparison of the simple linear regression model and the nonparametric regression. In details, the test statistic is constructed based on the variability explained by the two models. The second one consists in using the empirical process of the regressors marked by the residuals. Both tests have been analyzed by means of a bootstrap approach. In particular, a wild bootstrap and a residual bootstrap have been investigated

A linear regression model for imprecise response

A linear regression model with imprecise response and p real explanatory variables is analyzed. The imprecision of the response variable is functionally described by means of certain kinds of fuzzy sets, the LR fuzzy sets. The LR fuzzy random variables are introduced to model usual random experiments when the characteristic observed on each result can be described with fuzzy numbers of a particular class, determined by 3 random values: the center, the left spread and the right spread. In fact, these constitute a natural generalization of the interval data. To deal with the estimation problem the space of the LR fuzzy numbers is proved to be isometric to a closed and convex cone of R3 with respect to a generalization of the most used metric for LR fuzzy numbers. The expression of the estimators in terms of moments is established, their limit distribution and asymptotic properties are analyzed and applied to the determination of confidence regions and hypothesis testing procedures. The results are illustrated by means of some case-studies. © 2010 Elsevier Inc. All rights reserved