Incremental rule splitting in generalized evolving fuzzy regression models

Abstract

We propose an incremental rule splitting concept for generalized fuzzy rules in evolving fuzzy regression models in order to properly react on gradual drifts and to compensate inappropriate settings of rule evolution parameters; both occurrences may lead to oversized rules with untypically large local errors, which also usually affects the global model error. The generalized rules are directly defined in the multi-dimensional feature space through a kernel function, and thus allowing any rotated orientation of their shapes. Our splitting condition is based 1.) on the local error of rules measured in terms of a weighted contribution to the whole model error and 2.) on the size of the rules measured in terms of its volume. Thereby, we use the concept of statistical process control for automatic thresholding, in order to omit two extra parameters. The splitting technique relies on the eigendecompisition of the rule covariance matrix by adequately manipulating the largest eigenvector and eigenvalues in order to retrieve the new centers and contours of the two split rules. Thus, splitting is performed along the main principal component direction of a rule. The splitting concepts are integrated in the generalized smart evolving learning engine (Gen-Smart-EFS) and successfully tested on two real-world application scenarios, engine test benches and rolling mills, the latter including a real-occurring gradual drift (whose position in the data is known). Results show clearly improved error trend lines over time when splitting is applied: reduction of the error by about one third (rolling mills) and one half (engine test benches). In case of rolling mills, three rule splits right after the gradual drift starts were essential for this significant improvement.