Regularized halfspace depth for functional data

Hyemin Yeon, Xiongtao Dai, Sara Lopez-Pintado

Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2025 · Volume 87 · Issue 5

Abstract

Data depth is a powerful tool originally proposed to rank multivariate data from centre outward. In this context, one of the most archetypical depth notions is Tukey’s halfspace depth. In the last few decades, notions of depth have also been proposed for functional data. However, a naive extension of Tukey’s depth cannot handle functional data because of its degeneracy. Here, we propose a new halfspace depth for functional data, which avoids degeneracy by regularization. The halfspace projection directions are constrained to have a small reproducing kernel Hilbert space norm. Desirable theoretical properties of the proposed depth, such as isometry invariance, maximality at centre, monotonicity relative to a deepest point, upper semi-continuity, and consistency are established. Moreover, the regularized halfspace depth can rank functional data with varying emphasis in shape or magnitude, depending on the regularization. A new outlier detection approach is also proposed, which is capable of detecting both shape and magnitude outliers. It is applicable to trajectories in the space of all square-integrable functions, a very general space of functions that include nonsmooth trajectories. Based on extensive numerical studies, our methods are shown to perform well in detecting outliers of different types. Real data examples showcase the proposed depth.

Keywords

functional data analysis, functional rankings, infinite dimension, outlier detection, robust statistics

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