Multiple Linear Regression Analysis of Interval Type-2 Fuzzy Observed Data in Full Compliance with the Extension Principle

Abstract

Modeling regression in fuzzy environment is far from a simple analogy of the classic regression. Numerous papers on fuzzy regression transform the fuzzy optimization problem into a sequence of crisp optimization problems that derive the components of the fuzzy solution. Such approaches generally mislead, since the fuzzy arithmetic and optimization are carried out separately, with no inter-connectivity. Searching for a binder, we learned that they can be unified via a solution concept based on the extension principle. We propose a regression analysis on data represented by interval Type-2 fuzzy numbers aiming to predict outputs in full accordance with the extension principle. The importance of our results can be perceived through: the robustness of the proposed extensions; the applicability of the extension principle to general regression analysis; and the improvement of the predictions expressed by thinner interval Type-2 fuzzy-set-valued outputs due to a “soft-and” aggregation operator.