A topologically valid definition of depth for functional data

TL;DR

This paper proposes a formal, topologically valid definition of statistical depth for functional data, addressing partial observability and robustness, and evaluates six existing depth measures against these criteria.

Contribution

It introduces a rigorous, topologically grounded definition of depth for functional data and systematically assesses six existing depth proposals based on this framework.

Findings

Six functional depth proposals are evaluated against the new properties.

The definition ensures robustness and performance guarantees under partial observability.

Provides a systematic basis for selecting appropriate depth functions.

Abstract

The main focus of this work is on providing a formal definition of statistical depth for functional data on the basis of six properties, recognising topological features such as continuity, smoothness and contiguity. Amongst our depth defining properties is one that addresses the delicate challenge of inherent partial observability of functional data, with fulfilment giving rise to a minimal guarantee on the performance of the empirical depth beyond the idealised and practically infeasible case of full observability. As an incidental product, functional depths satisfying our definition achieve a robustness that is commonly ascribed to depth, despite the absence of a formal guarantee in the multivariate definition of depth. We demonstrate the fulfilment or otherwise of our properties for six widely used functional depth proposals, thereby providing a systematic basis for selection of a depth function.