## 3 Proposed Ensemble Clustering
### Problem Definition
#### 3.2.1 Production of Multiple Base Clustering Results
The proposed ensemble clustering method has the advantages of the kmedoids clustering algorithm, including its high speed. Meanwhile, it does not have its major weaknesses, i.e., the inability to detect non-spherical and non-uniform clusters. Indeed, the new ensemble clustering algorithm can improve stability and quality of kmedoids clustering algorithm and prove \"aggregating several weak clustering results is better than or equal to a strong clustering result.\" This study tries to solve all of the ensemble clustering problems by defining valid local clusters. In fact, this study calls data around a cluster center in the clustering of kmedoids as valid local data clusters. In order to generate a diverse clustering result, a strategy of sequential application of a weak clustering algorithm (i.e., using kmedoids clustering algorithm as base clustering algorithm) is used on non-appeared data in previously valid local clusters. Empirical analysis compares the proposed ensemble clustering algorithm with several existing ensemble clustering algorithms and three strong fundamental clustering algorithms running on a set \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Source** & **Dataset** & \(|\mathbf{D_{i1}}|\) & \(|\mathbf{D_{i1}}|\) & \(\mathbf{C}\) \\ \hline UCI dataset & Glass (Gl) & 214 & 9 & 6 \\ UCI dataset & Galaxy (Ga) & 323 & 4 & 7 \\ UCI dataset & Yeast (\(\Upsilon\)) & 1484 & 8 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Description of the benchmark datasets; the number of data objects (\(|D_{i1}|\)), the number of attributes (\(|D_{i1}|\)), the number of clusters (C). \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Evaluation** & \multicolumn{6}{c}{**Dataset Number**} & \multicolumn{3}{c}{**T-Test Results Wins**} \\ \cline{2-7}
**Measure** & **B** & **I** & **Gl** & **Ga** & **Y** & **W** & **Against-Draws with-Loses to** \\ \hline NMI [9] + & 95.73– & 82.89– & 41.38– & 21.71– & 34.45– & 91.83\(\pm\) & 5-1-0 \\ \hline MAX [44] & 96.39\(\pm\) & 83.21– & 42.63– & 20.57– & 33.89– & 91.29– & 5-1-0 \\ + ItoU & \multirow{2}{*}{95.16–} & \multirow{2}{*}{82.10–} & \multirow{2}{*}{41.98–} & \multirow{2}{*}{24.01–} & \multirow{2}{*}{34.12–} & \multirow{2}{*}{91.78\(\pm\)} & \multirow{2}{*}{5-1-0} \\ \cline{1-1} APMM [45] + ItoU & & & & & & \\ \hline ENMI [46] & 96.51\(\pm\) & 84.66– & 42.65– & 24.84– & 35.58– & 92.27+ & 4-1-1 \\ + ItoU & \multirow{2}{*}{97.28} & \multirow{2}{*}{86.05} & \multirow{2}{*}{44.79} & \multirow{2}{*}{29.44} & \multirow{2}{*}{38.20} & \multirow{2}{*}{92.13} \\ \cline{1-1} Proposed & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The NMI between the consensus partitioning results, produced by different ensemble clustering methods, and the ground truth labels validated by paired t-test [66] with 95% level of confidence. of artificial and real-world benchmark datasets. According to the empirical results, the performance of the proposed ensemble clustering algorithm is much more effective than the state-of-the-art ensemble clustering methods. In addition, we examined the efficiency of the proposed ensemble clustering algorithm, which is even suitable for dealing with large-scale datasets. This method works because it concentrates on finding local structures in valid small-cluster and then merging them. It also benefits from instability of kmedoids clustering algorithm to make a diverse ensemble. The main limitation of the paper is its usage of base kmedoids clustering algorithm. It seems that other base kmedoids clustering algorithms such as fuzzy cmeans clustering algorithm can be a better alternative and it will be discussed in future work. [PERSON] and [PERSON] designed the study. [PERSON], [PERSON] and [PERSON] wrote the paper. [PERSON] and [PERSON] revised the manuscript. [PERSON] and [PERSON] provided the data. [PERSON], [PERSON] and [PERSON] carried out tool implementation and all of the analyses. [PERSON] and [PERSON] generated all figures and tables. [PERSON], [PERSON] and [PERSON] performed the statistical analyses.