Multi-Objective Evolutionary Learning of Mamdani Fuzzy Rule-Based Systems

In the last years, the numerous successful applications of Mamdani Fuzzy Rule-Based Systems (MFRBSs) to several different domains have produced a considerable interest in methods to generate MFRBSs from data. Most of the methods proposed in the literature, however, focus on performance maximization and omit to consider MFRBS comprehensibility. Only recently, the problem of finding the right trade-off between performance and interpretability, in spite of the original nature of fuzzy logic, has arisen a growing interest in methods which take both the aspects into account. In this Ph.D. thesis, we propose the use of multi-objective evolutionary algorithms to design the structure of MFRBSs with good trade-off between accuracy and complexity. Complexity is always measured as sum of the conditions which compose the antecedents of the rules included in the MFRBS while the accuracy depends on the specific application of the system. As regards the application to regression problems, we first introduce a variant of the well-known (2+2) Pareto Archived Evolutionary Strategy ((2+2)PAES), which adopts the one-point crossover and two appropriately defined mutation operators, to generate a set of non-dominated rule bases (RBs) in the error-complexity space. Then, we extend this approach to learn concurrently both the RB and the data base (DB) in the multi-objective evolutionary framework. In particular, we introduce two approaches that allow to learn concurrently the RB and the partition granularity or the membership function parameters, respectively. In the first case, we introduce the concept of virtual RB, in order to handle RBs defined on different variable partitions, while in the latter case we exploit the linguistic 2-tuple representation for the fuzzy sets, which allows the symbolic translation of a linguistic label by only considering one parameter. Regarding classification with imbalanced datasets, we exploit a three-objective evolutionary algorithm, namely NSGA-II, to generate a set of RBs for Fuzzy Rule Based Classifiers with different trade-offs among sensitivity, specificity and complexity. Then, we use the ROC convex hull method to select the potentially optimal classifiers in the projection of the Pareto front approximation onto the ROC plane. Intensive experimentations have been performed and the results obtained with the proposed approaches and with comparison techniques have been extensively discussed.