## 5 The Proposed ECPSO-MTP for Data Clustering
### Friedman Test Statistics  
To investigate the statistical significance of the proposed ECPSO-MTP, the Friedman test is employed in this analysis, and the average of MSE obtained by test clustering approaches in comparison experiments is also introduced as the reference object [70]. In this case, the null hypothesis assumes that all test comparative approaches have equal values of MSE in the experiment. More details about the Friedman test in mathematical description are given as follows [71]:  
The average MSE obtained by comparative approaches on eight datasets is ranked from smallest to largest. Let \(r_{ij}\) is the rank associated with approach \(j\) on dataset \(i\), for \(1\leq i\leq 8,1\leq j\leq 6\). Noted that if \(r_{ij}=1\), which represents the lowest value of MSE among these approaches. \(\frac{1}{2}(p+1)\) represents the average of these ranks, and its value is equal to 3 in this case, where \(p\) is the totality of test clustering approaches, and, \(p=6\). The Friedman test statistic is defined by (14) in the following:  
\[\chi^{2}_{r}=\frac{12}{np(p+1)}\sum_{j=1}^{p}\left(\sum_{i=1}^{n}r_{ij}\right) ^{2}-3n(p+1), \tag{14}\]  
where \(n\) is the totality of datasets, and, \(n=8\). The ranks of the average obtained by the test clustering approaches are presented in Table 5. The value of the [PERSON] test statistic \(\chi^{2}_{r}\) is 38.393, as shown in Table 5. At the significance level of \(\alpha=5\%\), the null hypothesis is rejected with \(p-1\) degrees of freedom, and the critical value of \(\chi^{2}\) is 11.070. Therefore, the proposed ECPSO-MTP is statistically superior to other clustering approaches in terms of the MSE measure.  
\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{2}{*}{Data\_9\_2} & Mean & 0.8413 & 0.9128 & 0.9179 & 0.9212 & 0.9144 & 0.9214 \\  & S.D. & 0.0382 & 0.0078 & 0.0304 & 0.0022 & 0.0201 & 0.0015 \\ \multirow{2}{*}{Square4} & Mean & 0.9313 & 0.9333 & 0.9346 & 0.9350 & 0.9350 & 0.9350 \\  & S.D. & 0.0033 & 0.0027 & 0.0018 & \(3.36\times 10^{-16}\) & 0.0001 & \(3.36\times 10^{-16}\) \\ \multirow{2}{*}{Iris} & Mean & 0.8771 & 0.8923 & 0.8857 & 0.8907 & 0.8932 & 0.8933 \\  & S.D. & 0.0551 & 0.0106 & 0.0319 & 0.0056 & 0.0107 & 0.0026 \\ \multirow{2}{*}{Newthyroid} & Mean & 0.8007 & 0.8361 & 0.8536 & 0.8541 & 0.8502 & 0.8605 \\  & S.D. & 0.0448 & 0.0144 & 0.0273 & 0.0392 & 0.0298 & \(7.85\times 10^{-16}\) \\ \multirow{2}{*}{Seeds} & Mean & 0.8510 & 0.8879 & 0.8929 & 0.8952 & 0.8947 & 0.8976 \\  & S.D. & 0.0906 & 0.0077 & 0.0065 & 0.0119 & 0.0084 & \(7.85\times 10^{-16}\) \\ \multirow{2}{*}{Yeast} & Mean & 0.3494 & 0.3559 & 0.3925 & 0.4385 & 0.3785 & 0.5044 \\  & S.D. & 0.0260 & 0.0266 & 0.0287 & 0.0235 & 0.0179 & 0.0215 \\ \multirow{2}{*}{Glass} & Mean & 0.4911 & 0.4967 & 0.5185 & 0.5355 & 0.4965 & 0.5877 \\  & S.D. & 0.0450 & 0.0199 & 0.0154 & 0.0222 & 0.0152 & 0.0062 \\ \multirow{2}{*}{Wine} & Mean & 0.7013 & 0.7016 & 0.7018 & 0.7022 & 0.7022 & 0.7022 \\  & S.D. & 0.0058 & 0.0022 & 0.0041 & \(2.24\times 10^{-16}\) & \(2.24\times 10^{-16}\) & \(2.24\times 10^{-16}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Performance of six clustering approaches on eight datasets (Purity).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Datasets**} & \multicolumn{6}{c}{**Comparative Approaches**} \\ \cline{2-7}  & **PSO** & **GA** & **DE** & **EPSO** & **PSO-LC** & **ECPSO-MTP** \\ \hline Data\_9\_2 & 6 & 5 & 4 & 2 & 3 & 1 \\ Square4 & 6 & 5 & 4 & 1 & 3 & 1 \\ Iris & 6 & 5 & 4 & 2 & 3 & 1 \\ Newthyroid & 6 & 5 & 4 & 2 & 3 & 1 \\ Seeds & 6 & 5 & 4 & 2 & 3 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Computation results obtained by Friedman test statistics (Mean).