دانلود رایگان مقاله انگلیسی رگرسیون خطی امکانی با داده های فازی: رویکرد تحمل با اطلاعات قبلی - الزویر 2018

Abstract

We introduce the tolerance approach to the construction of fuzzy regression coefficients of a possibilistic linear regression model with fuzzy data (both input and output). The method is very general: the only assumption is that α-cuts of the fuzzy data are efficiently computable. We take into account possible prior restrictions of the parameters space: we assume that the restrictions are given by linear and quadratic constraints. The method for construction of the possibilistic regression coefficients is in a reduction of the fuzzy-valued model to an interval-valued model on a given α-cut, which is further reduced to a lineartime method (i.e., running in O(np)) computing with endpoints of the intervals. The speed of computation makes the method applicable for huge datasets. Unlike various approaches based on mathematical programming formulations, the tolerance-based construction preserves central tendency of the resulting regression coefficients. In addition, we prove further properties: if inputs are crisp and outputs are fuzzy, then the construction preserves piecewise linearity and convex shape of fuzzy numbers. We derive an O(n2p)-algorithm for enumeration of breakpoints of the membership function of the estimated coefficients. (Here, n is the number of observations and p is the number of regression parameters). Similar results are also derived for the fuzzy input-and-output model. We illustrate the theory for the case of triangular and asymmetric Gaussian fuzzy inputs and outputs of an inflation-consumption model.

6.4. Conclusions.

We have adapted the tolerance approach for possibilistic linear regression with fuzzy-valued inputs and/or outputs. The method is applicable to any class of unimodal fuzzy numbers, not necessarily with a bounded support: in illustrative examples we used both triangular fuzzy data (which are bounded) and asymmetric Gaussian fuzzy data (which are unbounded). The method constructs fuzzy regression coefficients b respecting the central tendency of a crisp-data estimator applied to defuzzified data, and is minimal with respect to a user-given tolerance vector c. If the data are piecewise linear fuzzy numbers, then the resulting coefficients are piecewise-linear (in the crisp-input-fuzzyoutput model) ot piecewise-hyperbolic (in the fuzzy-input-fuzzy-output model). Moreover, the method is computationally very “cheap”, and thus can be used for large datasets.