Fuzzy linear least squares for the identification of possibilistic regression models

Abstract

The stochastic approach to regression, e.g. the least-squares method, is a commonly used technique in a wide range of disciplines as it offers an intuitive and well-founded solution to the problem of fitting a set of parameters to existing data. Specifically, in the case where the parameters of meta models are identified, the linear least-squares formulation receives wide appreciation among engineers due to its simple yet powerful formulation and solution. Usually, this approach provides a reasonable fit, depending largely on the quality of the model that is employed. However, it has two main drawbacks. From a statistical point of view, the premises justifying its use cannot always be assumed in good faith, and most importantly, the resulting model will almost certainly fail at correctly predicting the output without falling victim to the effect of over-fitting.The presented fuzzy regression algorithm originates from a set of different axioms. For the solution of the linear least-squares problem, it is able to provide the worst-case optimal parameter variation necessary to cover all of the crisp data points while encoding meaningful information in the membership function. This is accomplished by means of a sensible fuzzification procedure of the crisp data points and the application of an exact inverse fuzzy arithmetic for estimating the fuzzy-valued parameters of the model in question.