Fuzzy Ridge Regression with Non Symmetric Membership Functions and Quadratic Models
Fuzzy regression models has been traditionally considered as a problem of linear programming. The use of quadratic programming allows to overcome the limitations of linear programming as well as to obtain highly adaptable regression approaches. However, w
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Abstract. Fuzzy regression models has been traditionally considered as a problem of linear programming. The use of quadratic programming allows to overcome the limitations of linear programming as well as to obtain highly adaptable regression approaches. However, we verify the existence of multicollinearity in fuzzy regression and we propose a model based on Ridge regression in order to address this problem.
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Introduction
Regression analysis tries to model the relationship among one dependent variable and one or more independent variables. During the regression analysis, an estimate is computed from the available data though, in general, it is very difficult to obtain an exact relation. Probabilistic regression assumes the existence of a crisp aleatory term in order to compute the relation. In contrast, fuzzy regression (first proposed by Tanaka et al. [15]) considers the use of fuzzy numbers. The use of fuzzy numbers improves the modeling of problems where the output variable (numerical and continuous) is affected by imprecision. Even in absence of imprecision, if the amount of available data is small, we have to be cautious in the use of probabilistic regression. Fuzzy regression is also a practical alternative if our problem does not fulfill the suppositions of probabilistic regression (as, for example, that the coefficient of the regression relation must be constant). Fuzzy regression analysis (with crisp input variables and fuzzy output variable) can be categorized in two alternative groups: – Proposals based on the use of possibility concepts [10,11,12,13,16,17]. – Proposals based on the minimization of central values, mainly through the use of the least squares method[7,9]. Possibilistic regression is frequently carried out by means of the use of linear programming. Nevertheless, implemented in such a way, this method does not consider the optimization of the central tendency and usually derives a high number of crisp estimates. H. Yin et al. (Eds.): IDEAL 2007, LNCS 4881, pp. 135–144, 2007. c Springer-Verlag Berlin Heidelberg 2007
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S. Donoso, N. Mar´ın, and M.A. Vila
In this work we introduce a proposal where both approaches of fuzzy regression analysis are integrated. We also show that the use of quadratic programming can improve the management of multicollinearity among input variables. To address this problem, we propose a new version of Fuzzy Ridge Regression. The paper is organized as follows: next section presents new regression models based on the use of quadratic programming, section 3 describes a new version of Fuzzy Ridge Regression based on the methods of section 2, section 4 is devoted to presents and example, and, finally, section 5 concludes the paper.
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Fuzzy Linear Regression
Let X be a data matrix of m variables X1 , ..., Xm , with n observations each one (all of them real numbers), and Yi (i = 1, .., n) be a fuzzy set characterized by a LR membership function μYi (x), with center yi , left spread pi , and right spread qi (Yi = (yi , pi , qi )). The problem of fuzzy regression is to fi
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