A Multiobjective Evolutionary Approach to Concurrently Learn Rule and Data Bases of Linguistic Fuzzy-Rule-Based Systems

In this paper, we propose the use of a multiobjective evolutionary approach to generate a set of linguistic fuzzy-rule-based systems with different tradeoffs between accuracy and interpretability in regression problems. Accuracy and interpretability are measured in terms of approximation error and rule base (RB) complexity, respectively. The proposed approach is based on concurrently learning RBs and parameters of the membership functions of the associated linguistic labels. To manage the size of the search space, we have integrated the linguistic two-tuple representation model, which allows the symbolic translation of a label by only considering one parameter, with an efficient modification of the well known (2 + 2) Pareto archived evolution strategy (PAES). We tested our approach on nine real world datasets of different sizes and with different numbers of variables. Besides the (2 + 2)PAES, we have also used the well known nondominated sorting genetic algorithm (NSGA-II) and an accuracy-driven single-objective evolutionary algorithm (EA). We employed these optimization techniques both to concurrently learn rules and parameters and to learn only rules. We compared the different approaches by applying a nonparametric statistical test for pairwise comparisons, thus taking into consideration three representative points from the obtained Pareto fronts in the case of the multiobjective EAs. Finally, a data complexity measure, which is typically used in pattern recognition to evaluate the data density in terms of average number of patterns per variable, has been introduced to characterize regression problems. Results confirm the effectiveness of our approach, particularly for (possibly high dimensional) datasets with high values of the complexity metric.