The inductive learning of fuzzy rule-based classification systems suffers from exponential growth of the fuzzy rule search space when the number of patterns and/or variables becomes high. This growth makes the learning process more difficult and, in most cases, it leads to problems of scalability (in terms of the time and memory consumed) and/or complexity (with respect to the number of rules obtained and the number of variables included in each rule). In this paper, we propose a fuzzy association rule-based classification method for high-dimensional problems, which is based on three stages to obtain an accurate and compact fuzzy rule-based classifier with a low computational cost. This method limits the order of the associations in the association rule extraction and considers the use of subgroup discovery, which is based on an improved weighted relative accuracy measure to preselect the most interesting rules before a genetic postprocessing process for rule selection and parameter tuning. The results that are obtained more than 26 real-world datasets of different sizes and with different numbers of variables demonstrate the effectiveness of the proposed approach.

Over the past few decades, fuzzy systems have been widely used in several application fields, thanks to their ability to model complex systems. The design of fuzzy systems has been successfully performed by applying evolutionary and, in particular, genetic algorithms, and recently, this approach has been extended by using multiobjective evolutionary algorithms, which can consider multiple conflicting objectives, instead of a single one. The hybridization between multiobjective evolutionary algorithms and fuzzy systems is currently known as multiobjective evolutionary fuzzy systems. This paper presents an overview of multiobjective evolutionary fuzzy systems, describing the main contributions on this field and providing a two-level taxonomy of the existing proposals, in order to outline a well-established framework that could help researchers who work on significant further developments. Finally, some considerations of recent trends and potential research directions are presented.

Linguistic fuzzy modeling allows us to deal with the modeling of systems by building a linguistic model which is clearly interpretable by human beings. However, since the accuracy and the interpretability of the obtained model are contradictory properties, the necessity of improving the accuracy of the linguistic model arises when complex systems are modeled. To solve this problem, one of the research lines in recent years has led to the objective of giving more accuracy to linguistic fuzzy modeling without losing the interpretability to a high level. In this paper, a new postprocessing approach is proposed to perform an evolutionary lateral tuning of membership functions, with the main aim of obtaining linguistic models with higher levels of accuracy while maintaining good interpretability. To do so, we consider a new rule representation scheme base on the linguistic 2-tuples representation model which allows the lateral variation of the involved labels. Furthermore, the cooperation of the lateral tuning together with fuzzy rule reduction mechanisms is studied in this paper, presenting results on different real applications. The obtained results show the good performance of the proposed approach in high-dimensional problems and its ability to cooperate with methods to remove unnecessary rules.

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.

Linguistic fuzzy modeling in high-dimensional regression problems poses the challenge of exponential-rule explosion when the number of variables and/or instances becomes high. One way to address this problem is by determining the used variables, the linguistic partitioning and the rule set together, in order to only evolve very simple, but still accurate models. However, evolving these components together is a difficult task, which involves a complex search space. In this study, we propose an effective multiobjective evolutionary algorithm that, based on embedded genetic database (DB) learning (involved variables, granularities, and slight fuzzy-partition displacements), allows the fast learning of simple and quite-accurate linguistic models. Some efficient mechanisms have been designed to ensure a very fast, but not premature, convergence in problems with a high number of variables. Further, since additional problems could arise for datasets with a large number of instances, we also propose a general mechanism for the estimation of the model error when using evolutionary algorithms, by only considering a reduced subset of the examples. By doing so, we can also apply a fast postprocessing stage for further refining the learned solutions. We tested our approach on 17 real-world datasets with different numbers of variables and instances. Three well-known methods based on embedded genetic DB learning have been executed as references. We compared the different approaches by applying nonparametric statistical tests for multiple comparisons. The results confirm the effectiveness of the proposed method not only in terms of scalability but in terms of the simplicity and generalizability of the obtained models as well.