Fuzzy clustering of distribution-valued data using adaptive L2 Wasserstein distances

TL;DR

This paper introduces adaptive fuzzy c-means algorithms for clustering distribution-valued data using Wasserstein distances, incorporating variable relevance weights for improved cluster detection.

Contribution

The paper develops new fuzzy clustering algorithms that adaptively weight variables and components based on Wasserstein distances for distributional data.

Findings

Algorithms outperform standard fuzzy c-means in cluster detection

Variable relevance weights improve clustering accuracy

Effective on both artificial and real-world data

Abstract

Distributional (or distribution-valued) data are a new type of data arising from several sources and are considered as realizations of distributional variables. A new set of fuzzy c-means algorithms for data described by distributional variables is proposed. The algorithms use the $L2$ Wasserstein distance between distributions as dissimilarity measures. Beside the extension of the fuzzy c-means algorithm for distributional data, and considering a decomposition of the squared $L2$ Wasserstein distance, we propose a set of algorithms using different automatic way to compute the weights associated with the variables as well as with their components, globally or cluster-wise. The relevance weights are computed in the clustering process introducing product-to-one constraints. The relevance weights induce adaptive distances expressing the importance of each variable or of each component in the clustering process, acting also as a variable selection method in clustering. We have tested the proposed algorithms on artificial and real-world data. Results confirm that the proposed methods are able to better take into account the cluster structure of the data with respect to the standard fuzzy c-means, with non-adaptive distances.