## 3 Proposed Ensemble Clustering
### Problem Definition
#### 3.2.1 Production of Multiple Base Clustering Results
\begin{table}
\begin{tabular}{c c c c c} \hline  & \multicolumn{2}{c}{**ARI**} & \multicolumn{2}{c}{**NMI**} \\ \cline{2-5}  & **Average \(\pm\) STD** & **L-D-W** & **Average \(\pm\) STD** & **L-D-W** \\ \hline EAC+SL & 66.87 \(\pm\) 3.39 & 0-2-6 & 64.43 \(\pm\) 2.43 & 1-0-7 \\ EAC+AL & 68.19 \(\pm\) 2.65 & 0-2-6 & 60.97 \(\pm\) 2.75 & 0-2-6 \\ WCT+SL & 60.11 \(\pm\) 3.23 & 0-1-7 & 55.78 \(\pm\) 2.69 & 1-1-6 \\ WCT+AL & 67.58 \(\pm\) 3.13 & 0-1-7 & 60.72 \(\pm\) 2.32 & 0-1-7 \\ WTQ+SL & 65.87 \(\pm\) 2.89 & 0-1-7 & 62.28 \(\pm\) 3.03 & 0-1-7 \\ WTQ+AL & 67.88 \(\pm\) 2.58 & 1-1-6 & 61.02 \(\pm\) 2.94 & 1-2-5 \\ CSM+AL & 58.67 \(\pm\) 3.68 & 1-0-7 & 48.90 \(\pm\) 2.72 & 1-0-7 \\ CSM+SL & 68.21 \(\pm\) 2.48 & 1-0-7 & 60.99 \(\pm\) 2.71 & 1-1-6 \\ CSPA & 59.97 \(\pm\) 2.55 & 1-0-7 & 54.55 \(\pm\) 2.43 & 0-0-8 \\ HGPA & 24.24 \(\pm\) 2.36 & 0-0-8 & 20.29 \(\pm\) 2.26 & 0-8 \\ MCLA & 66.21 \(\pm\) 3.29 & 0-2-6 & 58.26 \(\pm\) 2.54 & 0-1-7 \\ SUV & 48.82 \(\pm\) 3.08 & 1-0-7 & 40.93 \(\pm\) 2.31 & 1-1-6 \\ SWV & 52.76 \(\pm\) 2.83 & 1-0-7 & 47.43 \(\pm\) 3.25 & 1-1-6 \\ EM & 57.42 \(\pm\) 2.89 & 0-0-8 & 52.48 \(\pm\) 2.97 & 0-0-8 \\ IVC & 58.48 \(\pm\) 3.02 & 0-0-8 & 53.52 \(\pm\) 2.37 & 0-0-8 \\ PC+EAC+SL & 85.01 \(\pm\) 3.31 & 1-1-6 & 80.82 \(\pm\) 2.40 & 1-2-5 \\ PC+EEAC+AL & 89.51 \(\pm\) 2.07 & 1-0-7 & 83.29 \(\pm\) 1.42 & 1-2-5 \\ PC+CSPA & 84.63 \(\pm\) 2.75 & 0-1-7 & 78.02 \(\pm\) 2.76 & 0-0-8 \\ PC+HGPA & 55.77 \(\pm\) 2.98 & 0-0-8 & 55.26 \(\pm\) 3.52 & 0-0-8 \\ PC+MCLA & 91.42 \(\pm\) 2.09 & 1-0-7 & 83.03 \(\pm\) 2.25 & 1-2-5 \\ PC+EM & 86.37 \(\pm\) 2.44 & 1-1-6 & 79.80 \(\pm\) 2.35 & 0-0-8 \\ Proposed & 93.86 \(\pm\) 1.37 & & 89.58 \(\pm\) 2.15 & \\ \hline \end{tabular}
\end{table} 
Table 4: The summary of the results presented in Figure 5. The column _L-D-W_ indicates the number of datasets on which the proposed method _Loses to-Draws with-Wins against_ a rival validated by paired t-test [66] with the confidence level of 95%. \begin{table}
\begin{tabular}{c c c} \hline \hline \(|\mathbf{X}|\) & \(\sum_{i=1}^{B}c_{i}\) & **Time (in Sec.)** \\ \hline
10K & 91 & 11.23 \\
20K & 213 & 51.29 \\
30K & 225 & 80.11 \\
40K & 232 & 114.06 \\
50K & 233 & 138.91 \\
60K & 242 & 178.71 \\
70K & 245 & 197.62 \\
80K & 353 & 331.02 \\
90K & 461 & 516.96 \\
100K & 472 & 576.58 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The computational cost of the proposed ensemble clustering algorithm in terms of the number of data points. Figure 6: Experimental results of different strong clustering methods comparing with the results of the proposed ensemble clustering algorithm on different datasets in terms of (**a**) ARI, and (**b**) NMI. #### 4.5.3 Final Decisive Experimental Results

In this subsection, a set of six real-world datasets has been employed to evaluate the efficacy of the proposed method in comparison with some recently published papers. Three of these six datasets are the same Wine, Iris, and Breast datasets presented in Table 2. To make our final conclusion fairer and more general, three additional datasets, whose details are presented in Table 6, are used as benchmark in this subsection. Based on the NMI criterion, the comparison of the performances of different ensemble clustering algorithms on the real-world benchmark datasets is shown in Table 7. According to the experimental results, the proposed ensemble clustering algorithm can detect different clusters in an effective way and increase the performance of the state-of-the-art ensemble clustering algorithms. ## 5 Conclusions and Future Works

The kmedoids clustering algorithm, as a fundamental clustering algorithm, has been widely considered to be a low-computational one. However, it is also considered to be a weak clustering method, because its performances are affected by many factors, such as unsuitable selection of the initial cluster centers and dissimilar distribution of data. This study is carried out aiming to propose a new ensemble clustering algorithm using multiple kmedoids clustering algorithms.