Multiobjective genetic fuzzy rule selection of single granularity-based fuzzy classification rules and its interaction with the lateral tuning of membership functions

Abstract

Multiobjective genetic fuzzy rule selection is based on the generation of a set of candidate fuzzy classification rules using a preestablished granularity or multiple fuzzy partitions with different granularities for each attribute. Then, a multiobjective evolutionary algorithm is applied to perform fuzzy rule selection. Since using multiple granularities for the same attribute has been sometimes pointed out as to involve a potential interpretability loss, a mechanism to specify appropriate single granularities at the rule extraction stage has been proposed to avoid it but maintaining or even improving the classification performance. In this work, we perform a statistical study on this proposal and we extend it by combining the single granularity-based approach with a lateral tuning of the membership functions, i.e., complete contexts learning. In this way, we analyze in depth the importance of determining the appropriate contexts for learning fuzzy classifiers. To this end, we will compare the single granularity-based approach with the use of multiple granularities with and without tuning. The results show that the performance of the obtained classifiers can be even improved by obtaining the appropriate variable contexts, i.e., appropriate granularities and membership function parameters.

Appendix: Wilcoxon signed-rank test

The Wilcoxon signed-rank test is a pair-wise test that aims to detect significant differences between two sample means: it is the analogous to the paired t test in non-parametric statistical procedures. If these means refer to the outputs of two algorithms, then the test practically assesses the reciprocal behavior of the two algorithms (Sheskin 2003; Wilcoxon 1945). Let $d_i$ be the difference between the performance scores of the two algorithms on the ith out of $N_{ds}$ datasets. The differences are ranked according to their absolute values; average ranks are assigned in case of ties. Let $R^+$ be the sum of ranks for the datasets on which the first algorithm outperformed the second, and $R^-$ the sum of ranks for the contrary outcome. Ranks of $d_i = 0$ are split evenly among the sums; if there is an odd number of them, one is ignored:

$$ R^+ = \sum_{d_i > 0} rank(d_i) + {\frac{1}{2}} \sum_{d_i = 0} rank(d_i), $$

$$ R^- = \sum_{d_i < 0} rank(d_i) + {\frac{1}{2}} \sum_{d_i = 0} rank(d_i). $$

Let T be the smaller of the sums, $T = \min(R^+,R^-).$ If T is less than, or equal to, the value of the distribution of Wilcoxon for $N_{ds}$ degrees of freedom [Table B.12 in Zar (1999)], the null hypothesis of equality of means is rejected.

The Wilcoxon signed-rank test is more sensible than the t test. It assumes commensurability of differences, but only qualitatively: greater differences still count for more, which is probably desired, but the absolute magnitudes are ignored. From the statistical point of view, the test is safer since it does not assume normal distributions. Also, the outliers (exceptionally good/bad performances on a few datasets) have less effect on the Wilcoxon test than on the t test. The Wilcoxon test assumes continuous differences $d_i;$ therefore, they should not be rounded to one or two decimals, since this would decrease the test power due to a high number of ties.

When the assumptions of the paired t test are met, the Wilcoxon signed-rank test is less powerful than the paired t test. On the other hand, when the assumptions are violated, the Wilcoxon test can be even more powerful than the t test. This allows us to apply it to the means obtained by the algorithms in each dataset, without any assumption about the distribution of the obtained results.