## 5 Simulation Results
If most objects have a high value, then the clustering configuration is appropriate; (c) if many points have a low or negative value, then the clustering configuration may have too many or too few clusters. Due to IDS7 and IDS8 being real datasets, there is no information regarding their cluster centers. Hence, the SI was used as a clustering validation index to assess the efficiency of the proposed methods. The compared results used the SI of IDS7 with IFCME and the proposed IIFPFCME with different \(\eta\) in Table 9. The compared results used the SI of IDS8 with IFCME and the proposed IIFPFCME with different \(\eta\) in Table 10. Tables 9 and 10 show that the proposed methods have good results in terms of efficiency with the SI. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Silhouette** & **IFCMC (\(m=2\))** & **IIFPFCMC** & **IIFPFCMC** & **IIFPFCMC** \\
**Index/Method** & **IFCMC (\(m=2\))** & **(\(\eta=0.01\))** & **(\(\eta=0.1\))** & **(\(\eta=1\))** \\ \hline SI of IDS7 & 0.612450 & 0.655276 & 0.655276 & 0.655276 \\ \hline SI of IDS8 & 0.733026 & 0.760424 & 0.760424 & 0.760424 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: The compared results used the SIs of IDS7–8 with IFCMC and the proposed IIFPFCMC with different \(\eta\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{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.) & 0.007042 & 0.003014 & 0.003111 & 0.003673 \\ \hline Average number of ten iterations (\(l\)) & 23.2 & 8.2 & 8.1 & 9.7 \\ \hline \hline \multicolumn{4}{c}{(b)} \\ \hline
**Compared Index/Method** & \begin{tabular}{c} **IFCMC** \\ **(\(m=2\))** \\ \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.) & 0.007925 & 0.007317 & 0.006542 & 0.007304 \\ \hline Average number of ten iterations (\(l\)) & 13.5 & 8.8 & 7.8 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The results of the comparison for IDS8 with different approaches: (**a**) using IFCME and IIFPFCME under different \(\eta\); and (**b**) using IFCMC and IIFPFCMC under different \(\eta\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Silhouette** & **IFFCME (\(m=2\))** & \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} &
\begin{tabular}{c} **IIFPFCME** \\ **(\(\eta=1\))** \\ \end{tabular} \\ \hline SI of IDS7 & 0.613169 & 0.670552 & 0.669509 & 0.669509 \\ \hline SI of IDS8 & 0.678284 & 0.764616 & 0.764616 & 0.764616 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The compared results used the SIs of IDS7–8 with IFCME and the proposed IIFPFCME with different \(\eta\). In the above tests, we used three types of data points and added outlier data to these three types for testing. The algorithm of IIFPFCMC exhibits better clustering results in the diamond-shaped and square-shaped distribution data points, and the outlier data were added to these three types of data. The performance and average of ten iterations of the IIFPFCMC algorithm was also found to be better. Although the number of iterations of the proposed IIFPFCMC was reduced, it takes more computing time to obtain the median. The results displayed in all tables were based on converging to the best group center point to count the time and the number of iterations.