A New Wasserstein Based Distance for the Hierarchical Clustering of Histogram Symbolic Data

Abstract

Symbolic Data Analysis (SDA) aims to to describe and analyze complex and structured data extracted, for example, from large databases. Such data, which can be expressed as concepts, are modeled by symbolic objects described by multivalued variables. In the present paper we present a new distance, based on the Wasserstein metric, in order to cluster a set of data described by distributions with finite continue support, or, as called in SDA, by "histograms". The proposed distance permits us to define a measure of inertia of data with respect to a barycenter that satisfies the Huygens theorem of decomposition of inertia. We propose to use this measure for an agglomerative hierarchical clustering of histogram data based on the Ward criterion. An application to real data validates the procedure.