## 3 Performance Evaluation and Analysis
### Property Analysis of WFCS  
Figure 2 demonstrates the original distribution of Iris dataset and the clustering results of the five algorithms. As shown in Figure 2a, Iris dataset contains three clusters of 50 data points each, where each cluster refers to a type of iris plant. It is obvious that Cluster1 is linearly separable from the other two while the latters are overlapped. Hence, it is more reasonable for data points in Cluster1 to be hard clustered than to be fuzzy clustered. (1) Clustering performance  
Figure 2 shows that clustering results of feature weighted clustering algorithms (WFCS, ESSC, WFCM and WSRFCM) are similar to the distribution of original data (shown in Figure 2 (a)). Data points in Cluster1 can be recognized very well by the five algorithms. Moreover, most data points in Cluster2 and Cluster3 can be recognized by the four feature weighted algorithms. In Figure 2 (f), it is obvious that some data in Cluster3 are misclassified into Cluster2 by FCS. The cluster centers of five algorithms are different from each other. Furthermore, the distance between Cluster1, Cluster2 and Cluster3 center obtained by the five algorithms are shown in Figure 3.  
With regard to WFCS, ESSC and FCS integrating the within-cluster compactness and between-cluster separation, the distances between the overlapped Cluster2 and Cluster3 center are larger than that of WSRFCM and WECM. However, FCS can't partition the data points belonging to Cluster2 or Cluster3 correctly for it has not included the feature weight though the biggest value of distance is obtained. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Datasets**} & \multicolumn{2}{c}{**WFCS**} & \multicolumn{2}{c}{**ESSC**} & \multicolumn{2}{c}{**WSRFCM**} & \multicolumn{2}{c}{**WFCM**} & \multicolumn{1}{c}{**FCS**} \\ \cline{2-7}  & \(\beta\) & \(\alpha\) & \(\gamma\) & \(\eta\) & \(\alpha\) & \(\alpha\) & \(\beta\) \\ \hline Australian & 1 & \(-\)7 & 1000 & 0.9 & \(-\)2 & \(-\)7 & 1 \\ Breast Cancer & 1 & \(-\)9 & 5 & 0.5 & \(-\)10 & \(-\)10 & 1 \\ Balance-scale & 0.01 & 4 & 100 & 0.7 & \(-\)6 & \(-\)5 & 0.05 \\ Heart & 0.005 & 2 & 100 & 0 & \(-\)5 & \(-\)10 & 0.5 \\ Iris & 1 & 2 & 1 & 0.01 & 2 & 2 & 1 \\ Pima & 0.5 & \(-\)9 & 100 & 0.2 & \(-\)6 & \(-\)9 & 1 \\ Vehicle & 1 & 2 & 50 & 0.01 & 4 & \(-\)10 & 1 \\ Wine & 1 & \(-\)1 & 50 & 0.01 & \(-\)1 & \(-\)1 & 1 \\ ENGINE & 1 & 2 & 1 & 0 & \(-\)10 & 2 & 1 \\ LTT & 1 & \(-\)2 & 1 & 0 & \(-\)1 & \(-\)10 & 1 \\ \hline \end{tabular}
\end{table} 
Table 3: Parameter values for 10 datasets. Figure 3: The distance between three cluster centers. Figure 2: **(a)** The original data distribution; **(b)** The clustering results of weighting fuzzy compactness and separation algorithm (WFCS); **(c)** The clustering results of enhanced soft subspace clustering algorithm (ESSC); **(d)** The clustering results of the feature weighting fuzzy clustering algorithm integrating rough sets and shadowed sets (WSRFCM); **(e)** The clustering results of the feature weighting fuzzy c-means algorithm (WFCM); **(f)** The clustering results of the fuzzy compactness and separation algorithm (FCS). Since different clustering algorithms have different objective functions, we introduce the iteration function \(F(t)=\dfrac{\sum\limits_{i=1}^{c}\sum\limits_{j=1}^{n}\mu_{j}\|x_{j}-a_{i}\|^{2}} {\sum\limits_{i=1}^{c}\sum\limits_{j=1}^{n}\mu_{j}\|a_{i}-\overline{X}\|^{2}}\) in order to evaluate the convergence of algorithm. Figure 4 shows the convergence curves of the five algorithms. As shown in Figure 4, the five convergence curves descend fast in the first two iterations, and the convergence curves vary slowly after three iterations. Furthermore, the smaller iteration number means the higher convergence speed. Overall, WFCS has a higher speed of convergence. The convergence speed of WFCM is lower than that of WFCS and ESSC, while the FCS has the lowest convergence speed.