The Kantorovich-Wasserstein distance for spatial statistics: The Spatial-KWD library

Abstract

In this paper we present Spatial-KWD, a free open-source tool for efficient computation of the Kantorovich-Wasserstein Distance (KWD), also known as Earth Mover Distance, between pairs of binned spatial distributions (histograms) of a non-negative variable. KWD can be used in spatial statistics as a measure of (dis)similarity between spatial distributions of physical or social quantities. KWD represents the minimum total cost of moving the “mass” from one distribution to the other when the “cost” of moving a unit of mass is proportional to the euclidean distance between the source and destination bins. As such, KWD captures the degree of “horizontal displacement” between the two input distributions. Despite its mathematical properties and intuitive physical interpretation, KWD has found little application in spatial statistics until now, mainly due to the high computational complexity of previous implementations that did not allow its application to large problem instances of practical interest. Building upon recent advances in Optimal Transport theory, the Spatial-KWD library allows to compute KWD values for very large instances with hundreds of thousands or even millions of bins. Furthermore, the tool offers a rich set of options and features to enable the flexible use of KWD in diverse practical applications.