Learning concurrently data and rule bases of Mamdani fuzzy rule-based systems by exploiting a novel interpretability index

Abstract

Interpretability of Mamdani fuzzy rule-based systems (MFRBSs) has been widely discussed in the last years, especially in the framework of multi-objective evolutionary fuzzy systems (MOEFSs). Here, multi-objective evolutionary algorithms (MOEAs) are applied to generate a set of MFRBSs with different trade-offs between interpretability and accuracy. In MOEFSs interpretability has often been measured in terms of complexity of the rule base and only recently partition integrity has also been considered. In this paper, we introduce a novel index for evaluating the interpretability of MFRBSs, which takes both the rule base complexity and the data base integrity into account. We discuss the use of this index in MOEFSs, which generate MFRBSs by concurrently learning the rule base, the linguistic partition granularities and the membership function parameters during the evolutionary process. The proposed approach has been experimented on six real world regression problems and the results have been compared with those obtained by applying the same MOEA, with only accuracy and complexity of the rule base as objectives. We show that our approach achieves the best trade-offs between interpretability and accuracy.

Appendix

In Tables 6, 7, we show the averages and the standard deviations of the MSEs on training and test sets $\left( {\overline{{{\text{MSE}}_{\text{TR}} }} (\sigma_{\text{TR}} )} \right.$ and $\overline{{{\text{MSE}}_{\text{TS}} }} (\sigma_{\text{TS}} ),$ respectively), the results of the Kolmogorov–Smirnov test (column k–s _TR and k–s _TS for the training and test sets, respectively), and the averages and the standard deviations of the interpretability index $\bar{I}\left( {\bar{I}(\sigma_{I} )} \right)$ for the MEDIAN and LAST solutions, respectively.

Table 6 Average MSEs on training and test sets and interpretability index for the MEDIAN solution

Table 7 Average MSEs on training and test sets and interpretability index for the LAST solution

In Tables 8 and 9, we show the averages and the standard deviations of the complexity $\left( {\overline{\text{COMP}} (\sigma_{\text{COMP}} )} \right),$ of the number of concrete rules $\left( {\overline{{M^{\text{c}} }} (\sigma_{{M^{\text{c}} }} )} \right),$ and of the concrete $\left( {\overline{{D^{\text{c}} }} (\sigma_{{D^{\text{c}} }} )} \right)$ and virtual $\left( {\overline{{D^{\text{v}} }} (\sigma_{{D^{\text{v}} }} )} \right)$ dissimilarities for the MEDIAN and LAST solutions, respectively.

Table 8 Average interpretability index I, complexity COMP, number M ^c of rules and average dissimilarities D ^c and D ^v for the MEDIAN solution

Table 9 Average interpretability index I, complexity COMP, number M ^c of rules and average dissimilarities D ^c and D ^v for the LAST solution