Improved Training of GMM-Based TSK Fuzzy System from Stability Perspective

With the help of the conditional mean, the mathematically equivalent bridge between a Gaussian mixture model (GMM) and a Takagi-Sugeno-Kang (TSK) fuzzy system has been built to form a GMM-based fuzzy system so as to share mutual support for their respective training from rich achievements about GMM and fuzzy system. However, only the conditional mean without the consideration of stability (i.e., conditional variance) may perhaps hinder the modeling performance of the GMM-based fuzzy system, especially for noisy data scenarios. In this study, with strict mathematical derivations, it is revealed that the stability of a GMM-based fuzzy system depends on the three factors, i.e., stability of each fuzzy rule, stability of each input/output subspace associated with each fuzzy rule, and weighted stability of all fuzzy rules and their input/output subspaces. Accordingly, based on the derived result about the above stabilities, the training method of the proposed EGMM-based (Enhanced GMM-based) TSK fuzzy system, which considers both accuracy and stability is developed to improve the modeling performance of the original GMM-based TSK fuzzy system by optimizing the GMM's parameters therein. The effectiveness of the proposed training method is manifested by the experimental results on 10 real regression datasets.