Wasserstein spatial depth

Abstract

Modeling observations as random distributions in Wasserstein spaces has gained significant attention due to its ability to capture the variability and geometric structure of data. However, the non-linear geometry of Wasserstein space presents challenges for conventional statistical tools de- signed for Euclidean spaces. This necessitates the development of tailored methodologies for analysis within these spaces. In this presentation, we introduce Wasserstein spatial depth (WSD), a novel extension of statistical depth tailored to distribution-valued data. WSD ranks and orders distributions, enabling order-based clustering and inference tools. We demonstrate that WSD retains essential properties of classical depths, such as invariance under transformations, continuity, vanishing at infinity, and maximization at the geometric median. Additionally, we propose a simple plug-in estimator for WSD, based on empirical distributions, and establish its consistency and asymptotic normal- ity. Finally, we highlight the practical utility of WSD through extensive sim- ulations and real-data applications, showcasing its potential in a variety of scientific domains.

Keywords: Depth – Optimal transport – Quantiles – Robustness