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 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 linear-time method (i.e., running in time O(np)) computing with endpoints of the intervals. (Here, n is the number of observations and p is 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(n2p)-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.