Monotone Fuzzy Rule Interpolation for TSK-FIS-Like n-ary Aggregation Functions

Fuzzy Rule Interpolation (FRI) is important for fuzzy inference systems modeling pertaining to a sparse fuzzy rule base system. The focus of this paper is on a specific class of FRI, i.e., monotone FRI (MFRI), for modeling monotone Takagi-Sugeno-Kang Fuzzy Inference System (TSK-FIS) in the presence of a monotone sparse fuzzy rule base. On the other hand, a function is denoted as an n-ary aggregation function for a given n-dimensional input space and an output space when both the monotone and boundary properties are satisfied. In this paper, a set of sufficient conditions derived from the principles of Ordered Weighted Averaging (OWA) and the concept of orness for TSK-FIS to obey the monotone property is firstly formulated. We show that it is necessary to have a dense fuzzy rule base, which can be obtained by interpolation of fuzzy rules in a sparse fuzzy rule base, for constructing a monotone TSK-FIS. We then devise a two-stage MFRI for establishing monotone TSK-FIS. The first stage comprises a sufficient condition, inspired from the orness concept, to generate intermediate fuzzy membership functions (FMFs). The second stage deduces the monotone consequent of each intermediate rule from the available sparse fuzzy rules. We further extend our MFRI formulation to form TSK-FIS-like n-ary aggregation functions.