## 5 Results and Discussion  
In this section, we report the results for all clustering approaches without and with feature selection methods. We use the weighted \(F_{1}\) score to assess the performance of clustering. Since we do not have labels available for clusters, we label each cluster using the variant that has most of its sequences in that cluster (e.g., we give the label \"Alpha\" to that cluster if most of the sequences belong to the Alpha variant). Now, we calculate the \(F_{1}\) score (weighted) for each cluster individually using these given labels. For different methods, the weighted \(F_{1}\) scores are provided in Table 2. Note that we did not mention the \(F_{1}\) scores for HDBSCAN since it is an overlapping clustering approach. From the results, we can observe that LASSO regression is more consistent as compared to Boruta to efficiently cluster all variants. One interesting pattern we can observe is the pure clusters of some variants in the case of RFF. It shows that RFF is able to cluster some variants very efficiently. However, it fails to generalize over all variants. In terms of different clustering methods, \(k\)-means and \(k\)-modes are performing better and able to generalize more on all variants as compared to the other clustering methods. In addition, the results of other clustering validation metrics of Table 3 (silhouette coefficient, Calinski-Harabasz index, and Davies-Bouldin index) signify that for \(k\)-mersfeature vectors and without using any feature selection methods, the fuzzy clustering method has acceptable results, which is a high silhouette coefficient score, high Calinski-Harabasz index and low Davies-Bouldin index, but higher running time (bold rows in Table 3). Interestingly, using \(k\)-mers and applying the fuzzy clustering method with the Boruta feature selection method give even better results in terms of the three different metrics and running time. Other acceptable performances that validate the effectiveness of using \(k\)-mers and feature selection methods regarding clustering quality and running time are related to the combination of \(k\)-means clustering and LASSO feature selection method and fuzzy clustering and RFF feature selection method. In addition, we can see that there is a trade-off between the quality of clustering and runtime between \(k\)-means and Fuzzy clustering methods. Moreover, comparing the results of using \(k\)-mers with the results of using one-hot embedding feature vectors shows that \(k\)-means and LASSO feature selection method has a high silhouette coefficient, high Calinski-Harabasz index score and low Davies-Bouldin index, and the lowest runtime. However, in general, the improvement in quality of clustering and runtimes are observable when we compare using one-hot encoding and \(k\)-mers, the same metrics for \(k\)-means clustering and the LASSO feature selection method, and other different clustering methods given in Table 3.  
In terms of studying the behavior of different known variants, we are interested to see if there is any similarity between variants, that is, if two different variants can form a single cluster. Since we can observe from Table 2 that almost all variants apart from the Beta variant form separate clusters (for some experimental settings), it is evident that these variants are not directly related to each other. From this analysis, we can conclude that, for example, designing a single vaccination for all variants may not be appropriate. Another point was to see if a single variant could make more than one cluster. The idea of this analysis was to observe if a new variant could be developed from an existing variant. For this purpose, we used the soft clustering approach HDBSCAN. We can observe in Figures 5 and 6 that there are certain small clusters appearing in the data. They may be potential candidates for a developing new coronavirus variant.