Incremental Rule Splitting in Generalized Evolving Fuzzy Systems for Autonomous Drift Compensation

Gradual drifts in data streams are usually hard to detect and often do not necessarily trigger the evolution of new fuzzy rules during model adaptation steps in order to represent the new, drifted data distribution(s) appropriately in the fuzzy model. Over time, they thus lead to oversized rules with untypically large local errors (typically also worsening the global model error), as representing joint local data distributions before and after a drift happened likewise. We therefore propose an incremental rule splitting concept for generalized fuzzy rules in order to autonomously compensate these negative effects of gradual drifts. Our splitting condition is based on the local error of rules measured in terms of a weighted contribution to the whole model error and on the size of the rules measured in terms of the volume of the associated clusters. We use the concept of statistical process control in order to omit an extra threshold parameter in our splitting condition. The splitting technique relies on the eigendecomposition of the rule covariance matrix to adequately manipulate the largest eigenvector and eigenvalues in order to retrieve the new centers and contours of the two split rules. Furthermore, we guarantee sufficient flexibility in adapting the shapes and consequents of the split rules to the new drifted situation in the stream by integrating a specific dynamic and smooth forgetting concept of older samples, which formed the original (nonsplit) rules. Robustness against outliers is guaranteed by the realization of a two-layer model building process, where one layer represents the cluster partition and the other layer the rule partition: Only clusters becoming significant over time are accepted as rules in the fuzzy model. The splitting concepts are integrated in the generalized smart evolving learning engine for fuzzy systems (termed as 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, compared to the case when it is not applied: reduction of the mean absolute model error by about one third (rolling mills) and about one half (engine test benches).