## 3 Proposed Ensemble Clustering
### Problem Definition
#### 3.2.1 Production of Multiple Base Clustering Results
In each of the state-of-the-art clustering algorithms, the number of clusters per each base clustering result in the ensemble is set according to the method they used; and the kmeans clustering algorithm is also used to produce their basic clustering results. The parameter \(B\) is always set to 40. For these compared methods, we determine their parameters based on their authors' suggestions. The quality of each clustering algorithm is reported as an average over 50 independent runs. A Gaussian kernel has been employed for the NSC algorithm and a value in a range greater than or equal to 0.1 and less than or equal to 2 with step size 0.1 is chosen as the kernel parameter \(\sigma^{2}\). In these parameters, the best clustering result has been selected for comparison. DBSCAN and CFSFDP algorithms also require the input parameter \(\varepsilon\). We estimated the value of \(\varepsilon\) using the average distance between all data points and their average point, denoted by \(\overline{ASE}\). However, each of these algorithms may require different values of \(\varepsilon\). Therefore, we have evaluated each of these algorithms with ten different values from the following set \(\left\{\frac{\overline{ASE}}{\overline{1}},\ \frac{\overline{ASE}}{\overline{2}},\ \frac{ \overline{ASE}}{\overline{3}},\ \frac{\overline{ASE}}{\overline{4}},\ \frac{ \overline{ASE}}{\overline{5}},\ \frac{\overline{ASE}}{\overline{6}},\ \frac{ \overline{ASE}}{\overline{7}},\ \frac{\overline{ASE}}{\overline{8}},\ \frac{ \overline{ASE}}{\overline{9}},\ \frac{\overline{ASE}}{\overline{10}}\right\}\) and the best clustering result is used for comparison. ### Experimental Results

#### 4.5.1 Comparison with the State of the Art Ensemble Methods

Different consensus functions have been first used to extract the final clustering result out of the output ensemble of the Algorithm 1. As it is clear, the clustering results in the output ensemble of the Algorithm 1 are not complete; therefore, to apply EAC on the output ensemble of the Algorithm 1, edited EAC (EEAC) is needed [64; 65]. CSPA, HGPA, and MCLA are also applied on the output ensemble of the Algorithm 1. EM has been also used to extract the final clustering result out of the output ensemble of the Algorithm 1. Therefore, seven methods including PC + EEAC + SL, PC + EEAC + AL, PC + CSPA, PC + HGPA, PC + MCLA, PC + EM, and the proposed mechanism presented in Section 3.4 have been used as different consensus functions to extract the final clustering result out of the output ensemble of the Algorithm



Based on the ARI and NMI criteria, the comparison of the performances of different ensemble clustering algorithms on the artificial and real-world benchmark datasets is shown respectively in Figures 4 and 5, and they are summarized in Table 4. As shown in Figures 4 and 5, we observe that the proposed ensemble clustering algorithm has a high clustering accuracy in the benchmark synthetic and real-world datasets compared to other existing ensemble clustering algorithms. 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. Also, as shown in Table 4, the efficiency of the proposed ensemble clustering algorithm is significantly better than other ensemble clustering algorithms in the artificial datasets. However, accuracy improvement of the proposed ensemble clustering algorithm in the real-world datasets is marginally better than other ensemble clustering algorithms. The main reason for this is that the complexity of the real-world datasets is larger than the complexity of the artificial datasets. Considering the performance of each method on different datasets to be a variable, a Friedman test is performed. It is proved (with a \(p\)-value about 0.006) that there is a significant difference between our variables.