Possibilistic linear regression with fuzzy data: Tolerance approach with prior information

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 alpha-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 alpha-cut, which is further reduced to a linear-time method (i.e., running in time O(np)) computing with endpoints of the intervals. (Here, n is the number of observations and pis the number of regression parameters.) 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(n(2)p)-algorithm for enumeration of breakpoints of the membership function of the estimated coefficients. 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. (C) 2017 Elsevier B.V. All rights reserved.