## 5 The Proposed ECPSO-MTP for Data Clustering
### Comparison with Other Existing Approaches  
Five existing clustering approaches, including classic PSO, genetic algorithm (GA), differential evolution (DE), PSO with environmental factors (EPSO) [59], and PSO with an enhanced learning strategy and crossover operator (PSO-LC) [43], are compared in this computational experiment. And EPSO and PSO-LC employ different strategies to improve performance. EPSO integrates environmental factors based on a partitioning of the dataset into the generation process of particles. Three strategies, such as altering the exemplar particles, modifying adjusting parameters, including inertia weight and learning factors, and integrating PSO with GA, are applied in PSO-LC. The values of the employed parameters used in these comparative approaches are presented in Table 2.  
Specifically, mean squared error (MSE) [67], as the fitness function of comparative clustering approaches, is introduced in the comparison experiments, which is determined by (12) in the following:  
\[\min_{\mathbf{c}_{1},\mathbf{c}_{2},\cdots,\mathbf{c}_{K}}f=\min f(\mathbf{c} _{1},\mathbf{c}_{2},\cdots,\mathbf{c}_{K})=\min\frac{1}{M}\sum_{i=1}^{K}\sum_{ j=1}^{M}\rho_{ij}\|\mathbf{x}_{j}-\mathbf{c}_{i}\|^{2}, \tag{12}\]  
where \(\rho\) is the partition matrix of the dataset, if \(\rho_{ij}=1\), where indicates that data point \(\mathbf{x}_{j}\) belongs to cluster \(i\), otherwise, \(\rho_{ij}=0\). \(\mathbf{x}_{j}\) represents the \(j\)-th data point in the dataset, for \(1\leq j\leq M\). \(\mathbf{c}_{i}\) is the clustering center of cluster \(i\), for \(1\leq i\leq K\). The convergence results of \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Datasets** & **Data Instances** & **Features** & **Clusters** \\ \hline Data\_9\_2 & 900 & 2 & 9 \\ Square4 & 1000 & 2 & 4 \\ Iris & 150 & 4 & 3 \\ Newthyroid & 215 & 5 & 3 \\ Seeds & 210 & 7 & 3 \\ Yeast & 1484 & 8 & 10 \\ Glass & 214 & 9 & 6 \\ Wine & 178 & 13 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of eight test datasets in the comparison experiment. the six tested comparative approaches on eight datasets for typical runs are depicted in Figure 3.  
It can be clearly observed in Figure 3 that the proposed ECPSO-MTP achieves mostly the best MSE values on eight test datasets. To eliminate randomness, each test clustering method was independently implemented 50 times. Simple statistical results, including the worst values (Worst), best values (Best), mean values (Mean), and standard deviations (SD) of MSE, for these optimization approaches on the test datasets are presented in Table 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{6}{c}{**Comparative Approaches**} \\ \cline{2-7}  & **PSO** & **GA** & **DE** & **EPSO** & **PSO-LC** & **ECPSO-MTP** \\ \hline \(N\) & 200 & 200 & 200 & 200 & 200 & 200 \\ \(t_{\text{max}}\) & 200 & 200 & 200 & 200 & 200 & 200 \\ \(c_{1}\), \(c_{2}\) & 2,2 & N & N & 0.6,3 & (0,1) & 2,2 \\ \(c_{3}\) & N & N & N & (0,1) & N & N \\ \(r_{1}\), \(r_{2}\) & (0,1) & N & N & (0,1) & N & (0,1) \\ \(r_{3}\) & N & N & N & (0,1) & N & N \\ \(\omega_{\text{min}}\,\omega_{\text{max}}\) & 0.4,1.2 & N & N & 0.4,0.6 & N & 0.4,1.2 \\ \(\theta_{\omega}\) & N & N & N & N & 0.99 & N \\ \(P_{c}\) & N & 0.6 & 0.6 & N & N & N \\ \(P_{\text{m}}\) & N & 0.02 & N & N & N & N \\ \(\theta_{\varphi}\) & N & N & N & N & 0.994 & 0.994 \\ \(\theta_{\gamma}\) & N & N & N & N & 0.995 & 0.995 \\ \(F\) & N & N & (0.5,1) & N & N & N \\ \(m\) & N & N & N & N & N & 10 [66] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Preferences in the comparison experiments.