B1624
Title: A new statistical 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 the 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, Tukey's depth cannot be extended to handle functional data because of its degeneracy. A new halfspace depth for functional data is proposed, 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 center, monotonicity relative to the deepest point, upper semi-continuity, and consistency, are established. Moreover, depending on the regularisation, the regularized halfspace depth can rank functional data with varying emphasis in shape or magnitude. A new outlier detection approach is also proposed, capable of detecting shape and magnitude outliers. It is applicable to trajectories in L2, a very general space of functions that include non-smooth trajectories. Based on extensive numerical studies, the methods are shown to perform well in detecting different types of outliers. Three real data examples showcase the proposed depth notion.
