Online Active Learning for Evolving Error Feedback Fuzzy Models within a Multi-Innovation Context

In data stream modeling problems, online active learning plays an important role for reducing model update times and costs (or efforts) associated with measuring and collecting target values, which are typically required for supervised updates in evolving fuzzy models. We propose an online active learning (oAL) approach for evolving error feedback fuzzy models (EEF-FM), which integrate an auto-regressive (AR) based noise correction component on consequent hyper-planes to gain higher robustness of predictions, especially in the case of measurement noise. Thereby, we first propose a multi-innovation based recursive update scheme for the consequent parameters, substituting conventional recursive fuzzily weighted least squares (RFWLS) approach in order to reduce the sub-optimality of consequent parameters in the case of structural changes in a fully single-pass manner $\rightarrow$ EEF-FM-MI. Our oAL technique applies a sample selection strategy in a fully single-pass manner to elicit those samples which are expected to be most important for improving the robustness of model parameters and enriching the model structure when being used for model updates. It is based on three criteria: i) maximization of the D-optimality criterion in the consequent space for addressing the problem to best reduce the parameter uncertainty; it relies on the determinant of the Hessian matrix, which is updated and represented in a multi-innovation context, where a threshold for selection is automatically derived by kernel density estimation; ii) overlap degree in the antecedent space for the purpose to ‘sharpen’ the local trends in the transition regions between the hyper-planes; iii) novelty content in the antecedent space for indicating required knowledge expansion through rule evolution. The results on noisy real-world data sets showed i) that multi-innovation RFWLS could significantly outperform conventional RFWLS in all cases, with achieving an even more significant out-performance of classical EFS (not using any AR-based correction component), ii.) that our oAL strategy was able to reduce the measurement effort by up to 90% with a small increase in the error trend lines, and this with about quartering the computation times for model updates.