Wasserstein $k$-Centers Clustering for Distributional Data

We develop a novel clustering method for distributional data, where each data point is regarded as a probability distribution on the real line. For distributional data, it has been challenging to develop a clustering method that utilizes modes of variation of the data because the space of probability distributions lacks a vector space structure, preventing the application of existing methods devised for functional data. Our clustering method for distributional data takes account of the differences in both means and modes of variation of clusters, in the spirit of the $k$-centers clustering approach proposed for functional data. Specifically, we consider the space of distributions equipped with the Wasserstein metric and define geodesic modes of variation of distributional data using the notion of geodesic principal component analysis. Then, we utilize geodesic modes of clusters to predict the cluster membership of each distribution. We theoretically show the validity of the proposed clustering criterion by studying the probability of correct membership. Through a simulation study and real data application, we demonstrate that the proposed distributional clustering method can improve the quality of the cluster compared to conventional clustering algorithms.