Rough set theory is a well-known mathematical framework that can deal with inconsistent data by providing lower and upper approximations of concepts. A prominent property of these approximations is their granular representation: that is, they can be written as unions of simple sets, called granules. The latter can be identified with "if. . . , then. . . " rules, which form the backbone of rough set rule induction. It has been shown previously that this property can be maintained for various fuzzy rough set models, including those based on ordered weighted average (OWA) operators. In this paper, we will focus on some instances of the general class of fuzzy quantifier-based fuzzy rough sets (FQFRS). In these models, the lower and upper approximations are evaluated using binary and unary fuzzy quantifiers, respectively. One of the main targets of this study is to examine the granular representation of different models of FQFRS. The main findings reveal that Choquet-based fuzzy rough sets can be represented granularly under the same conditions as OWA-based fuzzy rough sets, whereas Sugeno-based FRS can always be represented granularly. This observation highlights the potential of these models for resolving data inconsistencies and managing noise.

翻译：粗糙集理论是一种著名的数学框架，通过提供概念的下近似和上近似来处理不一致数据。这些近似的一个显著特性是其粒化表示：即它们可以表示为简单集（称为粒）的并集。这些粒可被识别为“如果……则……”规则，构成粗糙集规则归纳的基础。先前研究表明，这一性质可推广至多种模糊粗糙集模型，包括基于有序加权平均（OWA）算子的模型。本文聚焦于模糊量词模糊粗糙集（FQFRS）广义类中的若干实例。在这些模型中，下近似和上近似分别通过二元和一元模糊量词进行评估。本研究的主要目标之一是考察不同FQFRS模型的粒化表示。主要发现表明：在OWA模糊粗糙集具备粒化表示的条件下，基于Choquet积分的模糊粗糙集同样可呈现粒化表示；而基于Sugeno积分的模糊粗糙集则始终可呈现粒化表示。这一发现揭示了这些模型在解决数据不一致性和处理噪声方面的潜力。