## 5 The Proposed ECPSO-MTP for Data Clustering
### Comparison with Other Existing Approaches
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Datasets**} & \multirow{2}{*}{**Statistics**} & \multicolumn{6}{c}{**Comparative Approaches**} \\ \cline{3-8}  & & **PSO** & **GA** & **DE** & **EPSO** & **PSO-LC** & **ECPSO-MTP** \\ \hline \multirow{4}{*}{Data\_9\_2} & Worst & 0.7260 & 0.5597 & 0.6505 & 0.5240 & 0.6223 & 0.5248 \\  & Best & 0.6476 & 0.5240 & 0.5230 & 0.5215 & 0.5218 & 0.5212 \\  & Mean & 0.6854 & 0.5407 & 0.5361 & 0.5222 & 0.5343 & 0.5214 \\  & S.D. & 0.0184 & 0.0079 & 0.0275 & 0.0006 & 0.0252 & 0.0007 \\  & Worst & 7.3164 & 7.0736 & 6.9807 & 6.9520 & 6.9525 & 6.9520 \\  & Best & 6.9780 & 6.9733 & 6.9520 & 6.9519 & 6.9519 & 6.9519 \\  & Mean & 7.1100 & 7.0195 & 6.9569 & 6.9519 & 6.9520 & 6.9519 \\  & S.D. & 0.0862 & 0.0274 & 0.0051 & \(6.81\times 10^{-6}\) & \(8.59\times 10^{-5}\) & \(5.49\times 10^{-6}\) \\  & Worst & 1.0158 & 0.5873 & 0.9578 & 0.6619 & 0.5729 & 0.5263 \\ Iris & Best & 0.5268 & 0.5450 & 0.5266 & 0.5263 & 0.5287 & 0.5263 \\  & Mean & 0.5663 & 0.5621 & 0.5433 & 0.5357 & 0.5418 & 0.5263 \\  & S.D. & 0.1178 & 0.0099 & 0.0614 & 0.0236 & 0.0103 & \(1.09\times 10^{-5}\) \\  & Worst & 152.0870 & 157.1877 & 175.1488 & 136.4225 & 138.1924 & 132.9317 \\  & Best & 135.9901 & 133.9973 & 133.1129 & 132.8665 & 135.0995 & 132.8379 \\  & Mean & 143.5047 & 139.2553 & 137.6728 & 133.9052 & 136.5018 & 132.8426 \\  & S.D. & 4.3043 & 4.6420 & 7.2724 & 1.2267 & 0.7346 & 0.0187 \\  & Worst & 4.8188 & 3.0923 & 3.0838 & 3.0387 & 3.2346 & 2.7968 \\  & Best & 2.8261 & 2.9253 & 2.8306 & 2.8119 & 2.7975 & 2.7968 \\  & Mean & 3.2956 & 3.0059 & 2.9215 & 2.8862 & 2.9042 & 2.7968 \\  & S.D. & 0.7528 & 0.0382 & 0.0467 & 0.0530 & 0.1070 & \(2.03\times 10^{-6}\) \\  & Worst & 0.0822 & 0.0797 & 0.0661 & 0.0465 & 0.0668 & 0.0341 \\  & Best & 0.0521 & 0.0578 & 0.0586 & 0.0388 & 0.0494 & 0.0306 \\ Yeast & Mean & 0.0721 & 0.0674 & 0.0623 & 0.0427 & 0.0585 & 0.0315 \\  & S.D. & 0.0082 & 0.0076 & 0.0016 & 0.0019 & 0.0043 & 0.0008 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance of Compared to other clustering approaches, the proposed ECPSO-MTP mostly achieves the best performance, as shown in Table 3. The external index, i.e., Purity [68], is introduced as an evaluation criterion for the clustering results obtained by these approaches, which is defined by (13) in the following [69]:  
\[Purity=\frac{1}{M}\sum_{i=1}^{K}\max_{j}\bigl{|}D_{i}\cap C_{j}\bigr{|}, \tag{13}\]  
where \(D_{i}\) is the \(i\)-th cluster obtained by a clustering approach. \(C_{j}\) is the \(j\)-th label of the real cluster. \(\bigl{|}D_{i}\cap C_{j}\bigr{|}\) is the totality of data points that belong to both \(D_{i}\) and \(C_{j}\). The clustering results of Purity obtained by these clustering approaches from eight datasets are given in Table 4, and overall, ECPSO-MTP has the best performance among the six clustering approaches. Therefore, all these comparison results validate the clustering efficiency of the proposed ECPSO-MTP. Figure 3: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Datasets**} & \multirow{2}{*}{**Statistics**} & \multicolumn{6}{c}{**Comparative Approaches**} \\ \cline{3-8}  & & **PSO** & **GA** & **DE** & **EPSO** & **PSO-LC** & **ECPSO-MTP** \\ \hline \multirow{4}{*}{Glass} & Worst & 5.0427 & 3.3902 & 2.7652 & 2.2345 & 3.8383 & 1.6684 \\  & Best & 2.7025 & 2.1930 & 2.4661 & 1.7229 & 2.3312 & 1.5705 \\  & Mean & 3.5114 & 2.6536 & 2.6326 & 2.0258 & 2.6359 & 1.5795 \\  & S.D. & 0.5915 & 0.3020 & 0.0713 & 0.1198 & 0.2870 & 0.0241 \\  & Worst & 13,616.9408 & 13,415.0092 & 14,752.6728 & 13,362.1693 & 13,396.0147 & 13,318.5101 \\  & Best & 13,318.6675 & 13,327.0278 & 13,320.7178 & 13,325.9217 & 13,327.5156 & 13,318.4817 \\  & Mean & 13,407.7736 & 13,361.6433 & 13,355.5274 & 13,340.4952 & 13,353.1583 & 13,318.4858 \\  & S.D. & 61.9835 & 18.9552 & 201.6508 & 8.3583 & 18.6450 & 0.0057 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._Figure 3: Comparison of convergence results from six test clustering approaches on eight datasets.