# 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 & 0.2302 & 0.0682 & 8.9245 & 0.4789 \\  & 3 & 0.1349 & 0.1371 & 19.235 & 0.5529 \\  & 9 & 0.0284 & 0.3672 & 67.224 & 0.5849 \\ \hline ADEFC & 9 & 0.0296 & 0.3619 & 71.344 & 0.5849 \\ \hline PSO-V & 2 & 0.2311 & 0.0671 & 8.9382 & 0.4762 \\  & 3 & 0.1245 & 0.1371 & 19.235 & 0.5529 \\  & 9 & 0.0278 & 0.374 & 67.334 & 0.5872 \\ \hline GAKFCM & 2 & 0.2372 & 0.0680 & 8.9263 & 0.4821 \\  & 3 & 0.1262 & 0.1371 & 18.936 & 0.5587 \\ \hline EwFCM & 3 & 0.1263 & 0.1362 & 19.292 & 0.5529 \\  & 4 & 0.1021 & 0.1942 & 32.039 & 0.6232 \\  & 9 & 0.0287 & 0.3672 & 71.348 & 0.5869 \\ \hline CDE-FCM & 2 & 0.2235 & 0.0590 & 8.4385 & 0.4741 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance of algorithms on Iris data. Figure 10: Partition by CDE-FCM, \(c=3\) with initial cluster prototypes shown in Figure 7.

**Cancer** data has 683 datapoints in two classes (malignant and benign). The data has 9 features (attributes)--clump thickness, cell size uniformity, cell shape uniformity, single epithelial cell size, bare nuclei, bland chromatin, normal nucleoli, and mitoses. There are 440 instances belonging to the benign cluster and 243 instances in the malignant cluster. The cancer dataset is known to be linearly inseparable and it is often difficult for clustering algorithms to achieve high levels of accuracy with this dataset. FCM-AO, PSO-V, GAKFCM and EwFCM are run for \(c=2\) to \(c=26\). For ADEFC and CDE-FCM the vector representation is of length 234 with \(d=9\) and \(c_{max}=26\). The performances of different algorithms as measured by the cluster validity criteria are listed in Table 5. Little more than 18% of the candidate vectors in the terminating elite population of CDE-FCM encoded for \(c=2\) while approximately 15% of candidate vectors encoded for \(c=8\). The cluster prototypes identified by CDE-FCM in 72% of the candidates in the terminating elite population are almost optimal, and a further application of FCM-AO converges on average in 8.5 iterations over 10 independent runs of the algorithm. On the other hand, the best-performing instance of the naive FCM-AO with random initializations took 31 iterations. **Glass** data has 214 different glass samples with 9 features (refractive index, weight percent of corresponding oxide of Na, Mg, Al, Si, K, Ca, Ba, and Fe). There are 6 different types of glasses--building window (float), building window (non-float), vehicle window (float), container glass, tableware glass and headlamp glass. FCM-AO, PSO-V, GAKFCM and EwFCM are run for \(c=2\) to \(c=15\). For ADEFC and CDE-FCM the vector representation is of length 135 with \(d=9\) and \(c_{max}=15\). The performances of different algorithms as measured by cluster validity criteria are listed in Table 6. Almost all cluster validity indices agreed, and all algorithms produced very similar partitions at either \(c=5\), \(c=6\), and \(c=8\). \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & \(c\) & \(V_{B}\) & \(V_{T}\) & \(V_{K2}\) & \(V_{R}\) \\ \hline FCM-AO & 5 & 0.8251 & 0.1925 & 12.854 & 3.332 \\  & 6 & 0.8149 & 0.1786 & 12.779 & 3.297 \\ \hline ADEFC & 5 & 0.8135 & 0.1883 & 12.775 & 3.472 \\ \hline PSO-V & 5 & 0.8238 & 0.1937 & 12.854 & 3.974 \\  & 8 & 0.9244 & 0.2215 & 32.975 & 8.773 \\ \hline GAKFCM & 5 & 0.8188 & 0.1875 & 12.695 & 3.224 \\  & 6 & 0.8275 & 0.1786 & 12.702 & 3.133 \\ \hline EwFCM & 6 & 0.8175 & 0.1775 & 12.972 & 3.297 \\  & 8 & 0.8924 & 0.2232 & 37.395 & 9.725 \\ \hline CDE-FCM & 6 & 0.8077 & 0.1760 & 12.715 & 3.109 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance of algorithms on glass data.