Learning knowledge bases of multi-objective evolutionary fuzzy systems by simultaneously optimizing accuracy, complexity and partition integrity

Abstract

In the last few years, several papers have exploited multi-objective evolutionary algorithms (MOEAs) to generate Mamdani fuzzy rule-based systems (MFRBSs) with different trade-offs between interpretability and accuracy. In this framework, a common approach is to distinguish between interpretability of the rule base (RB), also known as complexity, and interpretability of fuzzy partitions, also known as integrity of the database (DB). Typically, complexity has been used as one of the objectives of the MOEAs, while partition integrity has been ensured by enforcing constraints on the membership function (MF) parameters. In this paper, we propose to adopt partition integrity as an objective of the evolutionary process. To this aim, we first discuss how partition integrity can be measured by using a purposely defined index based on the similarity between the partitions learned during the evolutionary process and the initial interpretable partitions defined by an expert. Then, we introduce a three-objective evolutionary algorithm which generates a set of MFRBSs with different trade-offs between complexity, accuracy and partition integrity by concurrently learning the RB and the MF parameters of the linguistic variables. Accuracy is assessed in terms of mean squared error between the actual and the predicted values, complexity is calculated as the total number of conditions in the antecedents of the rules and integrity is measured by using the purposely defined index. The proposed approach has been experimented on six real-world regression problems. The results have been compared with those obtained by applying the same MOEA, but with only accuracy and complexity as objectives, both to learn only RBs, and to concurrently learn RBs and MF parameters, with and without constraints on the parameter tuning. We show that our approach achieves the best trade-offs between interpretability and accuracy. Finally, we compare our approach with a similar MOEA recently proposed in the literature.

Multi-objective genetic learning of serial hierarchical fuzzy systems for large-scale problems

“…However, in large-scale problems (from 18 to 40 variables) the approach is the most accurate but with a higher number of rules. However, notice that the number of variables per rule is lower in GSHFS-Tuning than the different MOEAs implemented by Antonelli et al (2011). It means that although the number of rules obtained is high, each individual rule is simpler.…”

“…PAES-SF and PAES-SFC learn concurrently the RB and the MF parameters by means of the piecewise linear transformation. These algorithms have been taken fromAntonelli et al (2011). • A three-objective evolutionary algorithm: PAES-SF3(Antonelli et al 2011) seeks for different balances among complexity, accuracy, and partition integrity by concurrently learning the RB and MF parameters of the linguistic variables.…”

OFP_CLASS: a hybrid method to generate optimized fuzzy partitions for classification

“…In this study we will focus on this initial phase, with the aim of discretizing continuous attributes domains. This is a crucial step since there are classifiers that cannot deal with continuous attributes, and there are other classifiers that exhibit better performance when these attributes are discretized, since discretization reduces the number of continuous attribute values, enabling faster and more accurate learning (Antonelli et al 2010;Marzuki and Ahmad 2007).…”

Finding Pareto-front Membership Functions in Fuzzy Data Mining

“…In the fuzzy-control field, many approaches have been proposed for tuning parameters and learning membership functions [2,3,7,17]. As to fuzzy mining, Kaya et al proposed an approach that integrated the multi-objective genetic algorithm into clustering for fuzzy mining [5].…”