# Majority Voting Based Multi-Task Clustering of Air Quality Monitoring Network in Turkey
## 4 Multi-task Clustering
The SSE results of different pollutants under different algorithms are shown as \(\text{C}_{\text{pollutantName}}\) where \"pollutantName\" is one of the pollutants (PM\({}_{10}\), SO\({}_{2}\), NO\({}_{2}\), NO or O\({}_{3}\)) in Table 8. \(\text{C}_{\text{ALL}}\) is the average SSE value coming from all pollutants. KM is applied with two different versions in terms of initialization method used. KM with the random initialization is denoted as KM and KM initialized with K-Means++ is displayed as KM++. Hierarchical clustering is implemented with different link types among clusters. \(\text{HIER}_{\text{Sing}}\), \(\text{HIER}_{\text{Comp}}\), \(\text{HIER}_{\text{Avg}}\), \(\text{HIER}_{\text{Mean}}\) and \(\text{HIER}_{\text{Centro}}\) represent the hierarchical clustering types with single link, complete link, average link, mean link and centroid link, respectively. The bold notations in Table 9 show the best results in the respective rows. We can conclude that the proposed MV-MTC method outperforms all single clustering algorithms that similar AQMSs are assigned to the same cluster group more accurately when multi-task clustering is applied. Besides, the most promising output of MV-MTC is obtained by KM++. In the case of single clustering algorithms, EM performs the best among the other applied techniques. Final cluster assignments after performing MV-MTC with KM++ are shown in Figure 2. It points out the geographical locations of AQMSs in the map of Turkey with different colored markers where each color represents a cluster. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Cluster 1**} & \multicolumn{2}{c}{**Cluster 2**} & \multicolumn{2}{c}{**Cluster 3**} & \multicolumn{2}{c}{**Cluster 4**} & \multicolumn{2}{c}{**Cluster 5**} \\ \cline{2-10}  & **Weight** & **Label** & **Weight** & **Label** & **Weight** & **Label** & **Weight** & **Label** & **Weight** & **Label** \\ \hline PM\({}_{10}\) & 0.2002 & \(L_{2}\) & 0.3142 & \(L_{3}\) & 0.1309 & \(L_{1}\) & 0.3982 & \(L_{4}\) & 0.5977 & \(L_{5}\) \\ SO\({}_{2}\) & 0.0713 & \(L_{1}\) & 0.2328 & \(L_{3}\) & 0.1497 & \(L_{2}\) & 0.5799 & \(L_{5}\) & 0.4996 & \(L_{4}\) \\ NO\({}_{2}\) & 0.0902 & \(L_{1}\) & 0.2177 & \(L_{2}\) & 0.3322 & \(L_{3}\) & 0.4396 & \(L_{4}\) & 0.6593 & \(L_{5}\) \\ NO & 0.0726 & \(L_{1}\) & 0.1382 & \(L_{2}\) & 0.5186 & \(L_{5}\) & 0.1781 & \(L_{3}\) & 0.1783 & \(L_{4}\) \\ O\({}_{3}\) & 0.4208 & \(L_{3}\) & 0.3237 & \(L_{2}\) & 0.7482 & \(L_{5}\) & 0.4873 & \(L_{4}\) & 0.2082 & \(L_{1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The mean weight values of each cluster group and their intra-cluster labels obtained by KM++. In MV-MTC approach, a clustering algorithm is performed for \(t\) tasks and a merging operation is done in the final step (\"majority voting\"). In this study, the best results were obtained using the KMeans++ algorithm. The time complexity of K-Means++ is O (\(n\times k+n\times k\times I\)), where \(n\) is the number of instances, \(k\) is the number of clusters, and \(I\) is the number of iterations needed for convergence [56]. After single-task clustering step, the merging process takes the runtime cost of O (\(t\times n\)), where \(t\) is the number of tasks. Considering this, the total time complexity of MV-MTC algorithm is O (\(n\times k+n\times k\times I+t\times n\)). This time complexity indicates that the proposed MV-MTC algorithm requires computation time that grows linearly with the number of instances, clusters and tasks. Thus, the execution time of the algorithm will still be reasonable even if we process a large volume of data. Table 10 shows the execution time (in seconds) to perform MV-MTC algorithm in terms of different clustering methods. Single task clustering results are also shown as \(\text{C}_{\text{PolluntName}}\), and \(\text{C}_{\text{ALL}}\) represents the sum of the running time of all single task clustering results. Experiments were performed on a desktop computer with Intel Core i7-6700 3.40 GHz processor and 8 GB memory.