On Fractional Tikhonov Regularization: Application to the Adaptive Network-Based Fuzzy Inference System for Regression Problems

In this article, we introduce a variant of the adaptive network-based fuzzy inference system (ANFIS). The proposed variant does not use backpropagation and grid partitioning, but the least-squares method with fractional Tikhonov regularization. The fractional regularization is a generalization of the standard regularization and is applied here to the learning process of the ANFIS scheme for the first time. This results in a simpler rule base, with a low number of rules, allowing to handle problems with many input variables with relatively low computational time while keeping high accuracy. We present new theoretical results on the fractional Tikhonov regularization. Such results are the basis for a formal discussion on how much the choice of a different architecture, resulting in a different matrix in the least-squares minimization, could affect the accuracy. We perform several numerical experiments on benchmark examples, first to assess the impact of the fractional regularization on the accuracy and then to compare our results against the most recent ones reported in the literature by other ANFIS-like or neuro-fuzzy systems. The numerical results show the good performance of the proposed approach.

I. Introduction

Over the last three decades, there have been several variants of adaptive network-based fuzzy inference system (ANFIS) [1]. ANFIS is a five-layered network architecture representing the Takagi–Sugeno–Kang (TSK) fuzzy inference system (FIS). It is based on two sets of parameters, namely premise parameters and consequent parameters. The relationship between them is set by fuzzy “if-then” rules.