# An Optimal and Stable Algorithm for Clustering Numerical Data
## 3 Proposed Clustering Algorithm with a Constant Seeding Selection  
### Proposed Seeding Selection Method  
Traditionally, seeding selection for clustering algorithms, such as the k-means, k-modes, and fuzzy c-means algorithms, including the k-AMH algorithm, involves randomly generated seeds by default. Unlike the k-means framework, the k-AMH algorithm exploits the k-objects (usually known as medoids) from the seeding process until they are clustered. Thus, each object needs to be replaced individually to find the final k-objects if the objective function is optimized. For object replacement, the objective function minimization is mainly based on the minimization process of each set of k-objects forming a k-combination. Therefore, an appropriate combination of k-objects may lead to better clustering results and vice versa.  
Figure 1 depicts two scenarios that may happen in the random seeding method. Consider three clusters \(C_{1},C_{2}\), and \(C_{3}\) with objects colored in green, black, and blue, respectively, and the seeding of k-objects (\(k=3\)) that are taken from the objects randomly, represented in red. It is a well-known problem that many cases result in the worst-case scenario shown in Figure 1a, in which the three objects are seeded from the same cluster (the green cluster). However, it may also result in the best-case scenario shown in Figure 1b, in which the three objects are seeded and represented by each cluster. These scenarios may lead toeither poorer clustering results (the worst-case scenario) or better clustering results (the best-case scenario). Therefore, for every run, the clustering results may vary, ranging from the worst-case to the best-case clustering performance.  
Roughly speaking, the proposed seeding concept is derived from a special point in a Cartesian coordinate system called the origin. Figure 2 depicts a conceptual idea of a new seeding mechanism that seeds the k-AMH algorithm with zero-point multidimensional spaces for each k-cluster.  
The idea is to offer a globally optimal solution for each object by pointing to the same data point, known as the origin shown in Figure 1(a). In this process, selecting the k-objects seem to be more appropriate in setting up the best possible k-object combination shown in Figure 1(d). Using the same k-AMH (numerical) framework, the proposed approach begins with the first k-object and testing of the objects individually to determine the object that will serve as the center of the first cluster, whereas the other clusters continue pointing to the origin, as shown in Figure 1(b). This process is repeated for the second k-object shown in Figure 1(c), then for the third k-object shown in Figure 1(d). Under these circumstances, it is believed that the k-AMH algorithm converges in a globally optimal fashion for the best possible combination of k-objects. In fact, using the fixed reference point for every seeding process gives the new seeding method and k-AMH numeric algorithm the advantage of producing constant results. In such convergence spaces, the clustering results may reach stability and the optimal clustering results may be achieved in every run.  
Figure 1: Current seeding approach with random seeds for three clusters: \(C_{1},C_{2}\), and \(C_{3}\). (**a**) Random seeding approach, which may result in three seeds from the same cluster. (**b**) Random seeding approach, which may result in seeding objects that represent the three clusters equally.  
Figure 2: Cont.