Projection depth for functional data: Theoretical properties

TL;DR

This paper introduces a regularized projection depth for functional data in Hilbert spaces, addressing degeneracy issues and providing robust, theoretically sound depth measures without moment assumptions.

Contribution

It proposes a novel regularized projection depth for functional data that overcomes degeneracy and satisfies key properties, with strong theoretical backing.

Findings

Requires no moment assumptions on data

Satisfies invariance, monotonicity, and vanishing at infinity

Generates a robust median and detects shape outliers

Abstract

We introduce a novel projection depth for data lying in a general Hilbert space, called the regularized projection depth, with a focus on functional data. By regularizing projection directions, the proposed depth does not suffer from the degeneracy issue that may arise when the classical projection depth is naively defined on an infinite-dimensional space. Compared to existing functional depth notions, the regularized projection depth has several advantages: (i) it requires no moment assumptions on the underlying distribution, (ii) it satisfies many desirable depth properties including invariance, monotonicity, and vanishing at infinity, (iii) its sample version uniformly converges under mild conditions, and (iv) it generates a highly robust median. Furthermore, the proposed depth is statistically useful as it (v) does not produce ties in the induced ranks and (vi) effectively detects shape outlying functions. This paper focuses mainly on the theoretical properties of the regularized projection depth.