# A Novel Adaptive FCM with Cooperative Multi-Population Differential Evolution Optimization
## 4 Proposed Co-Evolutionary Framework
### Evaluation of Clustering Results
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & \(c\) & \(V_{B}\) & \(V_{T}\) & \(V_{K2}\) & \(V_{R}\) \\ \hline FCM-AO & 2 & 0.1233 & 0.0121 & 12.897 & 0.2868 \\  & 8 & 0.0129 & 0.6792 & 128.329 & 0.3862 \\ \hline ADEFC & 2 & 0.1234 & 0.0123 & 12.775 & 0.2854 \\ \hline PSO-V & 2 & 0.1293 & 0.0129 & 12.292 & 0.2841 \\  & 8 & 0.0137 & 0.6822 & 137.850 & 0.3922 \\ \hline GAKFCM & 2 & 0.1224 & 0.0174 & 12.325 & 0.2833 \\  & 8 & 0.0193 & 0.6891 & 129.825 & 0.3851 \\ \hline EwFCM & 2 & 0.1239 & 0.0132 & 12.457 & 0.2857 \\  & 8 & 0.0174 & 0.6891 & 135.839 & 0.3823 \\ \hline CDE-FCM & 2 & 0.1197 & 0.0121 & 12.775 & 0.2803 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Performance of algorithms on cancer data. **Wine** dataset has 178 samples of wine differentiated by 13 attributes--percent contents of alcohol, malic acid, ash, magnesium, total phenols, flavonoids, nonflavonoid phenols, proanthocyanins, alkalinity of ash, color intensity, OD280/OD315 of diluted wines (protein content), and proline content. The wines are categorized into three classes (red, white, and rose) with 59 instances in the first cluster, 71 in the second, and the rest in the third cluster. FCM-AO, PSO-V, GAKFCM, and EwFCM are run for \(c=2\) to \(c=13\). For ADEFC and CDE-FCM the vector representation is of length 169 with \(d=13\) and \(c_{max}=13\). The performances of different algorithms as measured by cluster validity criteria are listed in Table 7. The clusters are well-defined, and all the cluster validity measures reach their minimum at either \(c=3\) or \(c=4\). **Wine Quality** dataset has 1898 samples of wines divided into 11 classes (quality scores ranging from 0 to 10). The data are defined over 11 attributes--fixed acidity, volatile acidity, citric acid, residual sugar, chlorides, free SO2, total SO2, density, pH, sulphates, and alcohol content. This is the largest of the UCI datasets tested for this work both in terms of cardinality and dimensions. FCM-AO, PSO-V, GAKFCM and EwFCM are run for \(c=2\) to \(c=45\). For ADEFC and CDE-FCM the vector representation is of length 495 with \(d=11\) and \(c_{max}=45\). The performances of different algorithms as measured by cluster validity criteria are listed in Table 8. FCM-AO, PSO-V, GAKFCM, and EwFCM identify at least 8 of the classes with moderately high accuracy. Accuracy decreases for \(c=11\) as is evident from the cluster validity measures. ADEFC identifies \(c=10\) clusters, with an accuracy below 90% while the proposed CDE-FCM identifies 12 clusters with initial cluster centers almost optimal for 8 of the real classes. The FCM-AO implemented with cluster prototypes identified by CDE-FCM converges on an average of 12 iterations over 10 independent runs, while FCM-AO randomly initialized converges after an average of 33 iterations for \(c=8\) and 37 iterations for \(c=11\). ## 6 Case Study--Rolling Bearing Fault Analysis

Rolling element bearings are commonly used in supporting rotor components and assemblies in rotating machinery. Bearing defects can lead to undesirable vibrations, noise, \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & \(c\) & \(V_{B}\) & \(V_{T}\) & \(V_{K2}\) & \(V_{R}\) \\ \hline FCM-AO & 3 & 1.025 & 0.0997 & 8.726 & 2.970 \\ \hline ADEFC & 4 & 1.133 & 0.1029 & 12.339 & 3.892 \\ \hline PSO-V & 3 & 1.025 & 0.0987 & 8.697 & 2.835 \\ \hline GAKFCM & 3 & 1.025 & 0.0997 & 8.645 & 2.754 \\ \hline EwFCM & 3 & 1.175 & 0.0984 & 8.772 & 2.663 \\ \hline CDE-FCM & 3 & 1.025 & 0.0984 & 8.597 & 2.754 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Performance of algorithms on Wine data.