New estimation methods in fuzzy regression analysis, based on projection theorem and decoupling principle

In this paper we introduce a new formal criterion, named the decoupling principle, which allows us to naturally extend the projection theorem into a fuzzy regression framework. This criterion justifies a radical revision of the usual estimation methods. It should be mentioned here that actually dominant in the fuzzy regression analysis is another approach, which resorts to linear programming and fuzzy arithmetic in order to minimize the fuzziness of the model, subject to some possibilistic constraints. Our criticism relating to this approach will be concluded by the proposal of a new strategy, where we give up using fuzzy arithmetic (identified as a source of estimation distortions) and we insist on the reestablishment of the natural framework for solving a minimum norm problem: that one in which we benefit from the projection theorem, valid – as it is well known – not in arbitrary normed spaces, but only in Hilbert spaces. Since the orthogonality concept is related to the choice of a norm induced by an inner product, our fuzzy estimation procedure inevitably leads to a quadratic programming problem and not to a linear one. The decoupling principle consists of a set of rules, which allow us to express the fuzzy regression model as a system of two classical equations and to choose the corresponding projection subspaces. We explore the algorithmic consequences, which follow from such theoretical considerations for various situations and concrete types of problems, and we propose some MATLAB implementations.