Parametric conditions for a Monotone TSK Fuzzy Inference System to be an n-ary Aggregation Function

Abstract

Despite the popularity and practical importance of the Fuzzy Inference System (FIS), the use of an FIS model as an n -ary aggregation function, which is characterized by both the monotonicity and boundary properties, is yet to be established. This is because research on ensuring that FIS models satisfy the monotonicity property, i.e., monotone FIS, is relatively new, not to mention the additional requirement of satisfying the boundary property. The aim of this paper, therefore, is to establish the parametric conditions for the Takagi-Sugeno-Kang (TSK) FIS model to operate as an n -ary aggregation function (hereafter denoted as n -TSK-FIS) via the specifications of fuzzy membership functions (FMFs) and fuzzy rules. An absorption property with fuzzy rules interpretation is outlined, and the use of n -TSK-FIS as a uni-norm is explained. Exploiting the established parametric conditions, a framework for which an n -TSK-FIS model can be constructed from data samples is formulated and analyzed, along with a number of remarks. Synthetic data sets and a benchmark example on education assessment are presented and discussed. To be best of the authors' knowledge, this study serves as the first use of the TSK-FIS model as an n -ary aggregation function.