K-means clustering:: A half-century synthesis

This paper synthesizes the results, methodology, and research conducted concerning the K-means clustering method over the last fifty years. The K-means method is first introduced, various formulations of the minimum variance loss function and alternative loss functions within the same class are outlined, and different methods of choosing the number of clusters and initialization, variable preprocessing, and data reduction schemes are discussed. Theoretic statistical results are provided and various extensions of K-means using different metrics or modifications of the original algorithm are given, leading to a unifying treatment of K-means and some of its extensions. Finally, several future studies are outlined that could enhance the understanding of numerous subtleties affecting the performance of the K-means method.