Top–Down Sparse Fuzzy Regression Modeling from Data with Improved Coverage

Abstract

We propose a new fuzzy modeling algorithm from data for regression problems. It acts in a top–down manner by allowing the user to specify an upper number of allowed rules in the rule base which is sparsed out with the usage of an iterative constrained numerical optimization procedure. It is based on the combination of the least squares error and the sum of rule weights over all rules to achieve minimal error with lowest possible number of significantly active rules. Two major novel concepts are integrated into the optimization process: the first respects a minimal coverage degree of the sample space in order to approach \(\epsilon \)-completeness of the rule base (an important interpretability criterion) and the second optimizes the positioning and ranges of influence of the rules, which is done synchronously to the optimization of the rule weights within an intervened, homogeneous procedure. Based on empirical results achieved for several high-dimensional (partially noisy) data sets, it can be shown that our advanced, intervened optimization yields fuzzy systems with a better coverage and a higher degree of \(\epsilon \)-completeness compared to the fuzzy models achieved by related data-driven fuzzy modeling methods. This is even achieved with a significantly lower or at least equal number of rules and with a similar model error on separate validation data.