Conditional regularized halfspace depth for sparse functional data and its applications

TL;DR

This paper introduces a new depth measure called CRHD for sparse functional data, enabling direct analysis without reconstructing trajectories, and demonstrates its theoretical properties and practical applications.

Contribution

The paper proposes the conditional regularized halfspace depth (CRHD), a novel method for analyzing sparse functional data directly at observations, overcoming limitations of existing methods.

Findings

CRHD provides meaningful rankings for complex functional data.

CRHD performs well in rank-based tests on real datasets.

Theoretical properties of CRHD clarify its behavior as a depth measure.

Abstract

Many functional datasets are observed sparsely and irregularly. Ordering such data is challenging because only limited information is available from each observation, while the underlying trajectories remain infinite-dimensional. This paper develops a novel depth notion for sparse functional data, called the conditional regularized halfspace depth (CRHD). CRHD is defined as the infimum of conditional halfspace probabilities of the underlying trajectory given the observed sparse measurements, thereby enabling depth evaluation directly at sparse observations without requiring trajectory reconstruction. We study several basic theoretical properties of CRHD that clarify its behavior as a depth measure. The proposed depth is applicable even to extremely sparsely observed functional data, overcoming key limitations of existing sparse functional depths that often rely on reconstructed curves. In addition, CRHD induces meaningful rankings for complex functional data. Its numerical performance is demonstrated through rank-based tests, and its practical utility is illustrated using an infant growth dataset.