# Majority Voting Based Multi-Task Clustering of Air Quality Monitoring Network in Turkey
## 4 Multi-task Clustering
In each experiment, the algorithms were executed 10 times and then the average values were reported. The empirical results show that the running time of the proposed K-means++ algorithm under MV-MTC framework is better than EM and hierarchical clustering algorithms. Besides, The MV-MTC algorithm has comparable speed with the traditional clustering algorithms when we compare \(\text{C}_{\text{ALL}}\) and MV-MTC results on the datasets. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & **KM** & **KM++** & **EM** & **Canopy** & **FFint** & **HERSign** & **HERComp** & **HERReg** & **HERReg** & **HERReg** & **HERCento** \\ \hline \(\text{C}_{\text{PMLO}}\) & 396.40 & 406.53 & 392.42 & 476.39 & 415.03 & 532.51 & 411.91 & 516.16 & 456.02 & 526.03 \\ \(\text{C}_{\text{SG2}}\) & 260.31 & 194.96 & 226.03 & 415.04 & 214.27 & 216.13 & 204.56 & 214.27 & 214.27 & 214.27 \\ \(\text{C}_{\text{NO2}}\) & 367.95 & 315.79 & 319.25 & 406.52 & 356.39 & 437.90 & 317.32 & 437.90 & 373.54 & 437.90 \\ \(\text{C}_{\text{NO}}\) & 335.17 & 351.29 & 294.79 & 426.31 & 305.73 & 319.82 & 303.40 & 303.40 & 319.82 & 319.82 \\ \(\text{C}_{\text{O3}}\) & 398.83 & 394.42 & 371.31 & 478.08 & 418.25 & 645.29 & 402.39 & 410.27 & 514.38 & 446.52 \\ \(\text{C}_{\text{ALL}}\) & 350.73 & 332.60 & 320.76 & 440.47 & 341.93 & 430.33 & 327.92 & 376.40 & 375.61 & 388.91 \\ MV-MTC & 113.80 & 108.12 & 116.16 & 122.84 & 134.41 & 161.96 & 124.92 & 161.96 & 144.16 & 161.96 \\ \hline \end{tabular}
\end{table} 
Table 9: The results of the performance evaluation in terms of SSE values. Figure 2: Results from the implementation of multi-task clustering obtained by KM++. MV-MTC algorithm was compared with one of the recently proposed MTC methods, MTCMRL [45], in terms of time complexity. In [45], multi-task clustering is combined with model relation learning (MTCMRL) method to automatically learn the model parameter relatedness between each pair of tasks by providing a solution to a non-convex optimization problem. Even though the proposed algorithm has a better clustering performance compared to other multi-task clustering methods, it still does not offer the expected performance in terms of time complexity, which is O (\(n^{2}*m\)), where \(m\) is the number of features and \(n\) is the number of instances per task, thus it increases exponentially when \(n\) is increased to larger volumes. On the other hand, MV-MTC is still reasonable to be preferred because of its linearly changing time complexity. With this study, it was aimed to identify similar regions in terms of air quality. It enables flexible decision-making at the cluster level. Thus, decision makers on the control of air quality can take actions similarly for the members of the same cluster. Since many air quality monitoring station data are summarized in several clusters, it provides richer but compacted information for control and modeling. It finds structure in air quality data and is therefore exploratory in nature. Representing the whole environmental data by few clusters may offer the great advantage of simplification in analyzing the data. Identification of the monitoring station groups can be used to understand why these stations in a same cluster are similar. Clustering monitoring stations minimizes the overload of information. Grouping similar information and summarizing common characteristics help the environmental scientists understand the current situation more clearly. In addition, it is also possible to classify a new station by assigning it to the cluster with the closest center. The potential contributions of this study to the prediction of air quality can be listed as follows:

* The multi-task clustering can also be used to label all the observed elements before air quality prediction, by calculating the distance between each centroid and each element in the data, and then selecting the cluster label (or level) with minimum distance.