# \(K\)-Means Cloning: Adaptive Spherical \(K\)-Means Clustering
###### Abstract
Their algorithm can automatically find data clusters by examining the data without need to set a fixed number of clusters. Specifically, the number of clusters is treated as a variable to be optimized within the clustering method. In [28], the authors proposed an intelligent \(K\)-means method, called \(iK\)-means, that locates the number of clusters by extracting odd examples from the data one-by-one. They used the Gaussian distribution to generate cluster models. Therefore, the clustering process can be analyzed through the usage of Gaussians in the data generation and specifics \(K\)-means clusters. Then, they could compare the performance of methods in terms of their capabilities in recovering cluster centroids as well as their abilities to recover of the number of clusters or clusters themselves. Their final version of \(iK\)-means utilizes the cluster number tuning, the cluster recovery, and the centroid recovery. [PERSON] and [PERSON] [29] proposed the \(G\)-means algorithm, which is based on assuming a statistical test checking if a subset of data follows a Gaussian distribution or not. \(G\)-means implements \(K\)-means with increasing values of \(K\) in a hierarchical structure until the hypothesis test is accepted. The authors of [30] presented a projected Gaussian method for data clustering called \(PG\)-means. Their method tries to discover an appropriate number of Gaussian clusters, as well as their positions and orientations. Therefore, their clustering strategy inherits the features of the well-known Gaussian mixture model. The main contribution of that paper is to use projections and statistical tests to determine if a generated mixture model fits its data or not. [PERSON] et al. [31] introduced a new measurement to estimate the number of clusters based on comparing the change within-cluster dispersion through some statistical tests. That measure can be used to the output of any clustering method in order to adapt the number of clusters. [PERSON] and [PERSON] [32] presented a Bayesian \(K\)-means based on the mixture model learning using the expectation-maximization method. Their method can be implement as a top-down approach \"Bayesian k-means\" or as a bottom-up one \"agglomerative clustering\". In [33], [PERSON] et al. introduced a new algorithm, \(X\)-means, to find the best configuration of the cluster locations and the number of clusters to optimize the Bayesian or Akaike information criteria. In each \(K\)-means run, the current centroids should be checked by the splitting conditions using Bayesian information criterion. A modified version of the \(X\)-means algorithm is presented in [34]. The main deviation is to apply a new cluster merging procedure to avoid unsuitable splitting caused by the cluster splitting order. [PERSON] et al. [23] presented an extensive review of the existing methods for selecting the number of clusters in \(K\)-means algorithms. In addition, the authors introduced a new measure to evaluate the selection of the number \(K\). In [35], the authors proposed a new clustering technique based on a modified version of \(K\)-means and a union-split algorithm to find a better value of the number of clusters. Those clustering methods are applied to undirected graphs to cluster similar graph nodes into clusters with a minimum size constraint to the cluster number. Other studies followed dynamic clustering strategies, e.g., in [36], cluster combination is performed using an evidence accumulation approach. Also [PERSON] et al. [37] have developed \(Y\)-means as an alternative clustering method of \(K\)-means. In \(Y\)-means, the number of clusters can be dynamically determined via assigning or removing data points to or from generated clusters depending on an Eacludian metric. [PERSON] et al. [38] proposed a dynamic data clustering method based on the combinatorial particle swarm optimization technique. Their method treats the data clustering problem as a single problem, including the sub-problem of finding the best number of clusters.