Wasserstein Spatial Depth
Fran\c{c}ois Bachoc, Alberto Gonz\'alez-Sanz, Jean-Michel Loubes, Yisha Yao

TL;DR
This paper introduces Wasserstein spatial depth, a new statistical measure for ranking distributions in Wasserstein spaces, enabling robust analysis and inference for distribution-valued data.
Contribution
The paper extends statistical depth concepts to Wasserstein spaces, developing Wasserstein spatial depth with properties, estimators, and applications for distribution data.
Findings
WSD ranges within [0,1] and is geodesically invariant.
The empirical WSD estimator is consistent and asymptotically normal.
WSD-based two-sample test performs well in simulations and real data.
Abstract
Modeling observations as random distributions embedded within Wasserstein spaces is becoming increasingly popular across scientific fields, as it captures the variability and geometric structure of the data more effectively. However, the distinct geometry and unique properties of Wasserstein spaces pose challenges to the application of conventional statistical tools, which are primarily designed for Euclidean spaces. Consequently, adapting and developing new methodologies for analysis within Wasserstein spaces has become essential. The space of distributions on with is not linear, and "mimic" the geometry of a Riemannian manifold. In this paper, we extend the concept of statistical depth to distribution-valued data, introducing the notion of Wasserstein spatial depth. This new measure provides a way to rank and order distributions, enabling the development of…
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