# A Novel Adaptive FCM with Cooperative Multi-Population Differential Evolution Optimization
## 4 Proposed Co-Evolutionary Framework
### Evaluation of Clustering Results

Table 9: Performance of FCM-AO and CDE-FCM for five bearing fault datasets. the same cluster and a true negative (\(TN\)) is a decision that assigns two dissimilar objects to different clusters. These are both desirable outcomes. The errors can either be a false-positive (\(FP\)) decision for an assignment of two dissimilar objects to the same cluster or a false-negative (\(FN\)) decision when two similar objects are assigned to different clusters. These metrics are evaluated using pairwise measurements--for a dataset of cardinality \(n\), there are \(n(n-1)/2\) pairs of objects. The precision (\(P\)) and recall (\(R\)) are defined as,

\[P=\frac{TP}{TP+FP},\ R=\frac{TP}{TP+FN} \tag{26}\]

The Rand index \(RI\) measures the percentage of decisions that are correct (also called the accuracy). The \(F\)-score is a measure of the harmonic mean of the method's precision and recall. The \(F\)-score is implemented as \(F_{1}\) in this paper (which gives equal weight to precision and recall) [67]. \[RI=\frac{TP+TN}{TP+FP+FN+TN},\ F_{1}=\frac{2 PR}{P+R} \tag{27}\]

Results of the sensitivity analysis comparing CDE-FCM with FCM-AO are provided in Table 10. As can be seen, the proposed method CDE-FCM achieves accuracies of approximately 75% with two of the five datasets. The best accuracy of FCM-AO approaches 75% for only one of the five datasets. The \(F_{1}\) score equally weighing precision and recall for CDE-FCM is also better than FCM-AO for all five datasets. This is a promising result that shows the superiority of the proposed method over the original FCM with randomized initialization. ## 7 Conclusions and Directions of Future Work

A novel method of initializing cluster prototypes for fuzzy c-means (FCM) is presented in this paper. The method also simultaneously finds the optimal number of clusters in the partition. These two constitute the biggest drawbacks of clustering techniques such as FCM. Many attempts have been made in addressing the two issues-we present a very detailed review of the existing literature in this paper. The concept presented in this paper not only complements the body of work in this field but is also a non-trivial improvement on present \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **TP** & **\(FP\)** & **TN** & **\(FN\)** & **\(P\)** & **\(R\)** & **\(RI\)** & **\(F_{1}\)** \\ \hline \multicolumn{8}{c}{**BearingData1, \(n=90\), \(c=3\)**} \\ \hline FCM-AO & 1295 & 721 & 1366 & 263 & 0.642 & 0.675 & 0.664 & 0.658 \\ \hline CDE-FCM & 1481 & 694 & 1404 & 426 & 0.681 & 0.777 & 0.720 & 0.725 \\ \hline \multicolumn{8}{c}{**BearingData2, \(n=90\), \(c=5\)**} \\ \hline FCM-AO & 1326 & 1926 & 1228 & 525 & 0.589 & 0.716 & 0.638 & 0.646 \\ \hline CDE-FCM & 1497 & 612 & 1478 & 418 & 0.710 & 0.782 & 0.742 & 0.744 \\ \hline \multicolumn{8}{c}{**BearingData3, \(n=90\), \(c=3\)**} \\ \hline FCM-AO & 1205 & 629 & 1516 & 655 & 0.657 & 0.648 & 0.679 & 0.652 \\ \hline CDE-FCM & 1422 & 598 & 1498 & 487 & 0.704 & 0.745 & 0.729 & 0.724 \\ \hline \multicolumn{8}{c}{**BearingData4, \(n=90\), \(c=6\)**} \\ \hline FCM-AO & 1375 & 826 & 1075 & 729 & 0.625 & 0.654 & 0.612 & 0.639 \\ \hline CDE-FCM & 1457 & 683 & 1190 & 675 & 0.680 & 0.683 & 0.661 & 0.682 \\ \hline \multicolumn{8}{c}{**BearingData5, \(n=150\), \(c=6\)**} \\ \hline FCM-AO & 4219 & 1622 & 4143 & 1191 & 0.772 & 0.780 & 0.748 & 0.750 \\ \hline CDE-FCM & 4522 & 1433 & 4291 & 929 & 0.760 & 0.830 & 0.789 & 0.793 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Comparative evaluation of FCM-AO and CDE-FCM for five bearing fault datasets. techniques. We propose a co-evolutionary multi-population differential evolution-based technique to evolve a candidate vector that would encode for the most optimal set of initial cluster prototypes to use and also for the optimal number of clusters to find. This is based on a smaller subset of the original data thus reducing effort during evolution. The best subset is chosen based on a sparse sampling statistic from literature.