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.

本文回顾了区间值模糊回归，并提出了一个新的统一框架来处理区间值的1型和2型模糊回归模型。本文着重于两个主要目标。首先，讨论并分析了有关区间值1型模糊回归（IV-T1FR）和区间值2型模糊回归（IV-T2FR）的一些哲学和方法学思考。这些思考旨在定位模糊回归，以避免在实际应用中有时可能导致错误或歧义的误解。因此，建立了区间值模糊回归的兴趣，相关性，代表性和类型。因此，IV-T1FR推广了常规间隔回归（CIR）并提高了其特异性。但是，如果IV-T1FR可以拟合模糊数据，则其形式主义无法解决IV-T1FS表示形式中的不确定性现象。在这种情况下，IV-T2FR可被视为不确定的IV-T1FR，即，IV-T1FR在不确定环境中的推广。其次，提出了一种新的统一方法，该方法使用渐进间隔（GI）和粗渐进间隔（TGI）的概念来处理模糊回归模型。提议的视图允许处理回归模型通过将标准间隔算法（SIA）扩展到GI和TGI，SIA是最初针对常规间隔（CI）提出的。所提出方法的独创性在于所有CIR方法都可以扩展到IV-T1FR方法。此外，所有IV-T1FR方法都可以扩展到IV-T2FR框架。我们的观点不依赖于模型的形状，而是保留了通过回归模型传播模糊量时SIA计算的灵活性和严格性。所提出的概念使用说明性示例进行了验证。