# An Optimal and Stable Algorithm for Clustering Numerical Data
## 1 Introduction
The issue related to seeding selection continues to exist and has attracted a significant amount of research attention, and various ideas have been proposed over the past few years. For example, a new seeding approach called the k-centroid initialization algorithm (PkCIA) based on eigenvalues/eigenvectors for computing initial cluster centroids was recently proposed [24]. Another seeding selection mechanism was proposed based on a small subset of non-degenerate observation points extracted from an original dataset [25]. In addition, [PERSON] and [PERSON] [26] conducted an intensive study on the seeding selection issue and split it into six categories: random points, farthest point heuristic, sorting heuristic, density-based, projection-based, and splitting technique; however, no clear conclusion was made as to which seeding method or category works better [26]. Therefore, further improvement of the seeding mechanism is needed, and attempts to achieve the required optimality, particularly for k-means is an ongoing research problem. In addition, to tackling seeding selection issues, a recent idea, aimed at optimal clustering results, is the repeated k-means. This method must be performed multiple times with different seeding selection mechanisms while maintaining the lowest SSE value [26]. At present, attempts are ongoing in this regard. The k-AMH algorithm, however, seems to present a comparably less significant seeding selection issue. For example, the original k-AMH algorithm for categorical data was found to be superior to other extended k-means approaches such as the k-modes and fuzzy k-modes methods [17]. A key advantage of the k-AMH algorithm is that it provides a higher minimum accuracy score compared with the other algorithms, as determined by performing 100 experimental runs. Apparently, the main factor that contributes to its superiority over other algorithms is that the k-AMH framework is not heavily dependent on the seeding selection mechanism, even though it uses random seeds. This advantage also exists for the k-AMH numerical extensions, called k-AMH numeric I and II [27], and these approaches have been found to be comparable to the fuzzy c-means technique. An earlier attempt to improve the random seeding mechanism for the original k-AMH algorithm, in particular for Y-STR clustering, yielded very promising results [28]. The aforementioned points discussed above present a basis to employ the k-AMH framework in order to yield highly constant results for each seeding while still maintainingtheir optimality. As discussed above, numerous attempts at seeding, such as the popular k-means++ approach, seem to be primarily focused on obtaining optimal clustering results. This is because optimality is prioritized when solving clustering problems. Benchmarking the optimality is subject to certain comparisons with existing solutions or algorithms, particularly for k-means algorithm. Notably, very few studies on cluster stability have been conducted thus far. This can be attributed to the fact that cluster stability is required only after desirable optimality by the appropriate seeding selection mechanism. According to [PERSON] and [PERSON] (2016), k-means can be improved from two aspects: better initialization (Seeding selection) and repeated k-means. At present, no concrete solution has been obtained. For k-AMH algorithm, a recent attempt for the repeated k-AMH has been proposed ([PERSON] and [PERSON], 2017). This finding is quite promising; however, it must be noted that an optimal objective function is not necessarily produced optimal clustering results. Clustering should be a structure on a dataset that is stable. When applied to several datasets from the same underlying model or of the same data-generating process, the clustering algorithm should yield similar results ([PERSON] and [PERSON], 2017).