Fuzzy Modeling using Vector Quantization and Local Linear Mapping

Fuzzy modeling has been extensively studied. It has been shown that fuzzy modeling using vector quantization (VQ) and steepest descent method (SDM) is effective in terms of the number of rules (parameters). In the methods, the initial parameters of fuzzy rules by using VQ with learning data firstly are determined, then the parameters are adjusted by using SDM. On the other hand, Neural Gas (NG) is known as a novel approach of VQ and NG with local linear mapping (LLM) has been applied to a time-series prediction problem. In the application, the predicted value in each of subregions is approximated by using a corresponding linear mapping. It has been demonstrated that, compared with RBF, NG with LLM is advantageous in terms of the accuracy and the number of rules. The idea of NG with LLM has been applied to fuzzy modeling with TS fuzzy model, and its effectiveness has been demonstrated. However, the effectiveness of this approach has not been been confirmed for simpler fuzzy model such as simplified fuzzy model. In this paper, the method using NG with LLM to fuzzy modeling with simplified fuzzy inference model is proposed and the similar method using k-means instead of NG is also proposed. In the proposed methods, the initial parameters of fuzzy rules including the weights in the consequent part by using NG (k-means) are firstly defined, and then the parameters by using SDM are adjusted. The effectiveness of the proposed methods with simplified fuzzy modeling is demonstrated in numerical simulations of function approximation and classification problems.