## 5 Results and Discussion
### HDBSCAN Clustering  
After doing an analysis on hard clustering algorithms, we evaluate the performance of the soft clustering approach (HDBSCAN) in this section. To evaluate HDBSCAN, we use the \(t\)-SNE approach to plot the original variants from our data and compare them with clusters we obtained after applying HDBSCAN. Since this is a soft clustering approach (overlapping allowed), there were a large number of clusters inferred for different feature selection methods (see Table 20 for the number of clusters). Therefore, we use \(t\)-SNE to plot the clusters to visually observe the patterns before and after clustering. Figure 5  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Variant**} & \multicolumn{4}{c}{**\(k\)-means (Cluster IDs)**} & \multicolumn{4}{c}{**\(k\)-modes (Cluster IDs)**} \\ \cline{2-10}  & **0** & **1** & **2** & **3** & **4** & **0** & **1** & **2** & **3** & **4** \\ \hline Alpha & 8762 & 86 & 2925 & 680 & 1513 & 11,403 & 7 & 184 & 1823 & 549 \\ Beta & 601 & 33 & 626 & 172 & 295 & 6 & 6 & 640 & 1060 & 15 \\ Epsilon & 7848 & 187 & 3155 & 638 & 956 & 1 & 0 & 11,170 & 947 & 666 \\ Delta & 2605 & 30 & 1342 & 868 & 2706 & 0 & 0 & 2894 & 690 & 3967 \\ Gamma & 22,140 & 50 & 3016 & 741 & 682 & 6 & 25,428 & 6 & 1128 & 61 \\ \hline \hline \end{tabular}
\end{table}
Table 16: Contingency tables of variants vs. clusters (Boruta feature selection, \(k\)-mers) with number of clusters \(=5\).  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Variant**} & \multicolumn{4}{c}{**Fuzzy (Cluster IDs)**} & \multicolumn{4}{c}{**Hierarchical (Cluster IDs)**} \\ \cline{2-10}  & **0** & **1** & **2** & **3** & **4** & **0** & **1** & **2** & **3** & **4** \\ \hline Alpha & 668 & 1513 & 78 & 2945 & 8762 & 9373 & 702 & 2641 & 1198 & 52 \\ Beta & 171 & 297 & 31 & 627 & 601 & 823 & 170 & 457 & 254 & 23 \\ Epsilon & 637 & 943 & 186 & 3170 & 7848 & 8419 & 644 & 2949 & 591 & 181 \\ Delta & 851 & 2713 & 28 & 1354 & 2605 & 2847 & 879 & 1563 & 2245 & 17 \\ Gamma & 739 & 669 & 47 & 3034 & 22,140 & 22,955 & 743 & 2330 & 560 & 41 \\ \hline \hline \end{tabular}
\end{table}
Table 18: Contingency tables of variants vs. clusters (Boruta feature selection, \(k\)-mers) with number of clusters \(=5\).  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Variant**} & \multicolumn{4}{c}{**Fuzzy (Cluster IDs)**} & \multicolumn{4}{c}{**Hierarchical (Cluster IDs)**} \\ \cline{2-10}  & **0** & **1** & **2** & **3** & **4** & **0** & **1** & **2** & **3** & **4** \\ \hline Alpha & 2172 & 11,083 & 438 & 268 & 5 & 1977 & 1585 & 543 & 9839 & 22 \\ Beta & 26 & 46 & 103 & 1552 & 0 & 898 & 7 & 198 & 623 & 1 \\ Epsilon & 3682 & 4 & 656 & 8442 & 0 & 546 & 2983 & 641 & 8613 & 1 \\ Delta & 1049 & 37 & 3269 & 3196 & 0 & 949 & 1235 & 2635 & 2731 & 1 \\ Gamma & 96 & 29 & 389 & 139 & 25,976 & 1703 & 26 & 2017 & 1192 & 21,691 \\ \hline \hline \end{tabular}
\end{table}
Table 19: Contingency tables of variants vs. clusters (Boruta feature selection, one-hot embedding) with number of clusters \(=5\).  
shows the comparison of the \(t\)-SNE plot on the original data versus \(t\)-SNE plots for the clustering results after applying HDBSCAN. Since overlapping is allowed in this setting, we cannot see any pure clusters as compared to the original \(t\)-SNE plot. An interesting finding from such a result is that not all sequences corresponding to a specific variant are similar. This means that a small cluster of sequences that initially belong to a certain variant can make another subgroup, which could eventually lead to the developing of a new variant. Therefore, using such overlapping clustering approach, we can visually observe if a group of sequences is diverging from its parent variant. Biologists and other decision-making authorities can then take relevant measures to deal with such scenarios. The \(t\)-SNE plots for different feature selection methods are given in Figure 6.