Connecting pairwise spheres by depth: DCOPS

TL;DR

This paper extends the concept of spherical depth to data on Riemannian manifolds, establishing its properties, consistency, and asymptotic behavior, with applications demonstrated through examples.

Contribution

It introduces a new depth measure for manifold data, proving its theoretical properties and consistency, and illustrating its practical utility.

Findings

Depth satisfies desirable properties on manifolds

Empirical depth is uniformly consistent

Asymptotic distribution is derived

Abstract

We extend the classical notion of the spherical depth in \mathbb{R}^k, to the important setup of data on a Riemannian manifold. We show that this notion of depth satisfies a set of desirable properties. For the empirical version of this depth function both uniform consistency and the asymptotic distribution are studied. Consistency is also shown for functional data. The behaviour of the depth is illustrated through several examples for Riemannian manifold data.