Interval-valued fuzzy regression: Philosophical and methodological issues

Abstract

This paper revisits interval-valued fuzzy regression and proposes a new unified framework to address interval-valued type-1 and type-2 fuzzy regression models. The paper focuses on two main objectives. First, some philosophical and methodological reflections about interval-valued type-1 fuzzy regression (IV-T1FR) and interval-valued type-2 fuzzy regression (IV-T2FR) are discussed and analyzed. These reflections aim at positioning fuzzy regression to avoid misinterpretations that may sometimes lead to erroneous or ambiguous considerations in practical applications. Consequently, the interest, relevance, representativeness and typology of interval-valued fuzzy regression are established. Therefore, IV-T1FR generalizes conventional interval regression (CIR) and increases its specificity. However, if the IV-T1FR can fit fuzzy data, then its formalism is not able to address the uncertainty phenomenon in the IV-T1FS representation. In this context, the IV-T2FR can be regarded as an uncertain IV-T1FR, i.e., a generalization of the IV-T1FR in an uncertain environment. Second, a new unified methodology to address fuzzy regression models using the concepts of gradual intervals (GIs) and thick gradual intervals (TGIs) is proposed. The proposed view allows handling regression models via an extension of the standard interval arithmetic (SIA) – initially proposed for conventional intervals (CIs) – to GIs and TGIs. The originality of the proposed approach resides in the fact that all the CIR methodologies can be extended to the IV-T1FR methods. Furthermore, all the IV-T1FR methodologies can be extended to the IV-T2FR framework. Our view does not depend on the model shape and preserves the flexibility and rigor of SIA computations in the propagation of fuzzy quantities through regression models. The proposed concepts are validated using illustrative examples.