A1723
Title: Halfspace depth for directional data
Authors: Stanislav Nagy - Charles University (Czech Republic) [presenting]
Rainer Dyckerhoff - University of Cologne (Germany)
Abstract: The angular halfspace depth (ahD) is a natural modification of the celebrated halfspace (or Tukey) depth to the setup of directional data. It allows us to define elements of nonparametric inference, such as the median, the inter-quantile regions, or the rank statistics, for datasets supported in the unit sphere. Despite being introduced previously, ahD has never received ample recognition in the literature, mainly due to the lack of efficient algorithms for its computation. We address both the computation and the theory of ahD. First, we express the angular depth ahD in the unit sphere in the d-dimensional space as a generalized (Euclidean) halfspace depth in dimension d-1, using a projection approach. That allows us to develop fast exact computational algorithms for ahD in any dimension d. Second, we show that similarly to the classical halfspace depth for multivariate data, also ahD satisfies many desirable properties of a statistical depth function. Further, we derive uniform continuity/consistency results for the associated set of directional medians, and the central regions of ahD, the latter representing a depth-based analog of the quantiles for directional data.
