B0587
Title: Regularized halfspace depth for functional Data
Authors: Hyemin Yeon - Kent State University (United States) [presenting]
Xiongtao Dai - University of California Berkeley (United States)
Sara Lopez Pintado - Northeastern University (United States)
Abstract: Data depth is a powerful nonparametric tool originally proposed to rank multivariate data from center outward. For multivariate data, one of the most archetypical depth notions is the halfspace depth by Tukey. In the last few decades, notions of depth have been proposed for functional data. However, the halfspace depth by Tukey cannot be extended to handle functional data because of a degeneracy issue. In our work, we propose a new halfspace depth for functional data and avoid 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 center, monotonicity relative to a deepest point, and upper semi-continuity, are established. Moreover, the proposed regularized halfspace depth can rank functional data with a 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 $L^2$, a highly general space of functions including non-smooth trajectories. Based on an extensive numerical study, our methods are shown to perform well in terms of detecting outliers of different types. Three real data examples showcase the proposed depth notion.
