## 3 Performance Evaluation and Analysis
### Clustering Evaluation  
The best RI indexes of the five algorithms are presented in Table 4.  
Figure 4: Convergence of the five algorithms. It is evident in Table 4 that WFCS demonstrates the best performance except for Breast-cancer, Vehicle and ENGINE datasets. The performance of WFCM and WSRFCM are mostly comparable or better than that of ESSC and FCS. Even if FCS is not a feature weighted clustering algorithm, it is able to achieve the best clustering result performance for the dataset Wine. Tables 5 and 6 list the XB and WB index values of the five algorithms respectively. By comparing Tables 4-6, we found that the best clustering performance as indicated through RI is not always the smallest value as indicated through XB or WB index. Therefore, no single algorithm can always be superior to the others for all datasets. The average performances of the five algorithms are shown in Figure 6.  
Figure 5: Fuzzy membership value on the first cluster with different \(\beta\) (**a**) \(\beta\)=1 ; (**b**) \(\beta\)=0.5 ; (**c**) \(\beta\)=0.05 ; (**d**) \(\beta\)=0.005. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Dataset** & **WFCS** & **ESSC** & **WSRFCM** & **WFCM** & **FCS** \\ \hline \multicolumn{5}{c}{Australian} \\ \hline mean & **0.7336** & 0.7162 & 0.7302 & 0.7265 & 0.6995 \\ std & 0.0000 & 0.1134 & 0.0033 & 0.0569 & 0.0125 \\ \hline \multicolumn{5}{c}{Breast Cancer} \\ \hline mean & 0.8721 & **0.8779** & 0.8630 & 0.8600 & 0.8627 \\ std & 0.0009 & 0.0057 & 0 & 0.0000 & 0.0000 \\ \hline \multicolumn{5}{c}{Balance-scale} \\ \hline mean & **0.6427** & 0.6389 & 0.6101 & 0.6099 & 0.6201 \\ std & 0.0758 & 0.0287 & 0.0586 & 0.0578 & 0.0662 \\ \hline \multicolumn{5}{c}{Heart} \\ \hline mean & **0.7163** & 0.7114 & 0.7120 & 0.6939 & 0.6833 \\ std & 0.0000 & 0.0019 & 0.0023 & 0.0000 & 0.0000 \\ \hline \multicolumn{5}{c}{Iris} \\ \hline mean & **0.9495** & 0.9195 & **0.9495** & **0.9495** & 0.8679 \\ std & 0.0000 & 0.0000 & 0.0081 & 0.0000 & 0.0000 \\ \hline \multicolumn{5}{c}{Pima} \\ \hline mean & **0.5841** & 0.5564 & 0.5698 & 0.5837 & 0.5576 \\ std & 0.0000 & 0.0005 & 0.0044 & 0.0009 & 0.0000 \\ \hline \multicolumn{5}{c}{Vehicle} \\ \hline mean & 0.6654 & 0.6539 & **0.6778** & 0.6581 & 0.6528 \\ std & 0.0025 & 0.0028 & 0.0006 & 0.0038 & 0.0000 \\ \hline \multicolumn{5}{c}{Wine} \\ \hline mean & **0.9551** & 0.9398 & 0.9324 & 0.9398 & **0.9551** \\ std & 0.0000 & 0.0095 & 0.0000 & 0.0000 & 0.0000 \\ \hline \multicolumn{5}{c}{ENGINE} \\ \hline mean & **0.8600** & 0.7823 & 0.7693 & **0.8600** & 0.7903 \\ std & 0.0000 & 0.0005 & 0.0067 & 0.0000 & 0.0000 \\ \multicolumn{5}{c}{LTT} \\ \hline mean & **0.9671** & 0.96 & 0.9543 & **0.9671** & 0.9607 \\ std & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ \hline \end{tabular}
\end{table} 
Table 4: The best clustering results obtained for the 10 datasets with rand index (RI). Figure 6: The average performances of the five algorithms. In Figure 6, we can see that WFCS obtained the best mean values of RI (0.7946) and XB (0.1976) with the least standard variation (0.0079, 0.0021 respectively) for the 10 datasets. WFCM, WSRFCM and ESSC perform similarly in terms of RI. It can be seen that the feature weighting clustering algorithms are superior to the un-weighted clustering one. The average XB and WB index values of WFCS, ESSC and FCS are smaller than those of WFCM and WSRFCM, which demonstrate that these three algorithms integrating between-cluster separation and within-cluster compactness can partition data points more reasonably.