## 5 Simulation Results
Table 7 shows the results of the comparison for IDS7 with different approaches: (a) using IFCME and IIFPFCMC under different \(\eta\); and (b) using IFCMC and IIFPFCMC under different \(\eta\). From Table 7, the proposed IIFPFCME and IIFPFCMC appear better than the IFCME and IFCMC, respectively. Note that due to it being a real dataset, there is no information regarding the cluster center; the results only show the comparison on average times on ten (sec.) and average number of ten iterations (\(I\)) and no RMSE in the table. Figure 10.8 The results of IFCMC. (**a**) The results of the IIFPFCME parameter with (**b**) \(\eta\)\(=1\), (**c**) \(\eta\)\(=0.1\), and (**d**) \(\eta\)\(=0.01\) for IDS7. Figure 10.9 The results of the IIFPFCMC parameter with (**a**) \(\eta=0.01\) and (**b**) \(\eta=0.1\) for IDS6. The results of IFCMC with parameter \(m=2\) for IDS8 are shown in Figure 32a. The results of IIFPFCMC and IIFPFCME with parameters \(\eta=1\), \(\eta=0.1\), and \(\eta=0.01\) for IDS7 are shown in Figures 32b-d and 33, respectively. Table 8 shows the results of the comparison for IDS7 with different approaches: (a) using IFCME and IIFPFCME under different \(\eta\); and (b) using IFCMC and IIFPFCMC under different \(\eta\). From Table 8, the proposed IIFPFCME and IIFPFCMC appear better than the IFCME and IFCMC, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{(**a**)} \\ \hline
**Compared Index/Method** & \begin{tabular}{c} **IFCMC** \\ **(\(m=2\))** \\ \end{tabular} & \begin{tabular}{c} **IIFPFCME** \\ **(\(\eta=0.01\))** \\ \end{tabular} & \begin{tabular}{c} **IIFPFCME** \\ **(\(\eta=0.1\))** \\ \end{tabular} & \begin{tabular}{c} **IIFPFCME** \\ **(\(\eta=1\))** \\ \end{tabular} \\ \hline Average times on ten (sec.) & 0.011177 & 0.004754 & 0.005224 & 0.005234 \\ \hline Average number of ten iterations (\(l\)) & 39.6 & 13 & 16.4 & 16.5 \\ \hline \multirow{3}{*}{**Compared Index/Method**} & \begin{tabular}{c} **(b)** \\ **(\(m=2\))** \\ \end{tabular} & \begin{tabular}{c} **IIFPFCMC** \\ **(\(\eta=0.01\))** \\ \end{tabular} & \
\begin{tabular}{c} **IIFPFCMC** \\ **(\(\eta=0.1\))** \\ \end{tabular} &
\begin{tabular}{c} **IIFPFCMC** \\ **(\(\eta=1\))** \\ \end{tabular} \\ \hline Average times on ten (sec.) & 0.008484 & 0.008062 & 0.009802 & 0.009563 \\ \hline Average number of ten iterations (\(l\)) & 13.9 & 10.3 & 11.4 & 10.4 \\ \hline \end{tabular}
\end{table} 
Table 7: The results of the comparison for IDS7 with different approaches: (**a**) using IFCME and IIFPFCME under different \(\eta\); and (**b**) using IFCMC and IIFPFCMC under different \(\eta\). Figure 31: The results of the IIFPFCME parameter with (**a**) \(\eta\)\(=1\), (**b**) \(\eta=0.1\), and (**c**) \(\eta=0.01\) for IDS7. Figure 33: The results of the IIFPFCME parameter with (**a**) \(\eta\)\(=1\), (**b**) \(\eta\)\(=0.1\), and (**c**) \(\eta\)\(=0.01\) for IDS8. In order to assess the efficiency of the proposed methods, the silhouette index (SI) was used as a clustering validation index. The SI is a metric used to calculate the goodness of a clustering technique. Its value provides a measure of how similar an object is to its own cluster (cohesion) compared to other clusters (separation). The SI is particularly useful under scenarios where the true labels of the data are not known [38,39]. It provides a way to compare the effectiveness of different clustering methods on a given dataset:  
\[S_{\text{SI}}=\frac{b_{\text{SI}}-a_{\text{SI}}}{\max(a_{\text{SI}},b_{\text{ SI}})} \tag{30}\]  
where \(a_{\text{SI}}\) is the mean distance to the points in the same cluster, and \(b_{\text{SI}}\) is the mean distance to the points in the nearest other cluster. Regarding the interpretation of the SI, (a) the silhouette coefficient ranges from \(-1\) to \(+1\); (b) a high value indicates that the object is well matched to its own cluster and poorly matched to their neighboring clusters.