Evolutionary parallel and gradually distributed lateral tuning of fuzzy rule-based systems

Abstract

The tuning of Fuzzy Rule-Based Systems is often applied to improve their performance as a post-processing stage once an initial set of fuzzy rules has been extracted. This optimization problem can become a hard one when the size of the considered system in terms of the number of variables, rules and, particularly, data samples is big. Distributed Genetic Algorithms are excellent optimization algorithms which exploit the nowadays available parallel hardware (multicore microprocessors and clusters) and could help to alleviate this growth in complexity. In this work, we present a study on the use of the Distributed Genetic Algorithms for the tuning of Fuzzy Rule-Based Systems. To this end, we analyze the application of a specific Gradual Distributed Real-Coded Genetic Algorithm which employs eight subpopulations in a hypercube topology and local parallelization at each subpopulation. We tested our approach on nine real-world datasets of different sizes and with different numbers of variables. The empirical performance in solution quality and computing time is assessed by comparing its results with those from a highly effective sequential tuning algorithm. The results show that the distributed approach achieves better results in terms of quality and execution time as the complexity of the problem grows.

Appendix: Wilcoxon’s 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 [52, 53]. Let d _i be the difference between the performance scores of the two algorithms on the i-th 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:

$$ \begin{aligned} R^+ & = \sum_{d_i > 0} {\hbox{rank}(d_i)} + \frac{1}{2} \sum_{d_i = 0} {\hbox{rank}(d_i)},\\ R^-& = \sum_{d_i < 0} {\hbox{rank}(d_i)} + \frac{1}{2} \sum_{d_i = 0} {\hbox{rank}(d_i)}. \end{aligned} $$

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 [54]), 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.