Theoretical properties of angular halfspace depth

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 in 1987, ahD has never received ample recognition in the literature, mainly due to the lack of efficient algorithms for its computation. With the recent progress on the computational front, ahD however exhibits the potential for developing viable nonparametric statistics techniques for directional datasets. In this paper, we thoroughly treat the theoretical properties of ahD. 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 analogue of the quantiles for directional data.

Figures (3)

• Figure 1: Transformation $\xi$ that takes $\mathbb{S}^{d-1}\setminus\mathbb{S}^{d-1}_0$ to $\mathcal{G}$ for $d=2$ (left panel) and $d=3$ (right panel). The data from the northern hemisphere $\mathbb{S}^{d-1}_+$ are mapped into points of positive $P_\pm$-mass in $\mathcal{G}$ (red points), while data from the southern hemisphere $\mathbb{S}^{d-1}_-$ project to points of negative $P_\pm$-mass (green points). In the left panel we see also a halfspace $H_{0,u} \in \mathcal{H}_0$ (shaded halfplane). Only points $x_1 \in \mathbb{S}^{1}_+$ and $x_2 \in \mathbb{S}^{1}_-$ lie in $H_{0,u}$, which means that precisely $\xi(x_1)$ and $\xi(x_4)$ are contained in $G_u = H_{0,u} \cap \mathcal{G}$. This is in accordance with formula \ref{['eq: pm']}.
• Figure 2: Example \ref{['example: sets']}: A distribution $P \in \mathcal{P}\left({{\mathbb{S}^{1}}}\right)$ that does not satisfy the condition of strict monotonicity \ref{['eq: strict monotonicity']} at $\alpha = 3/8$. In the left panel, the outer red curve corresponds to the ahD of $x(\theta) = (\cos(\theta), \sin(\theta) )^\mathsf{T} \in \mathbb{S}^{1}$ as a function of the angle $\theta \in (-\pi,\pi]$; the depth ahD is maximized at the angle $\theta = 5\pi/8$ (solid straight line), and $ahD_{3/8}(P)$ corresponds to the angles $\theta \in [\pi/4, 3 \pi/4]$ (wedge with dashed boundary lines). The depth ahD is constant for $\theta \in [\pi/4, \pi/2]$, meaning that ahD is not strictly monotone for $P$. In the right panel, we have the function $\theta \mapsto ahD(x(\theta); P)$ (blue curve).
• Figure 3: The setup from Appendix \ref{['app: example']}: The five points on the sphere $\mathbb{S}^{2}$ (three black points in $\mathbb{S}^{2}_+$ and two orange points in $\mathbb{S}^{2}_-$; the origin is displayed in red. The depth ahD of the corresponding measure is equal to $1/5$ for all $x \notin B$ and $2/5$ for all $x \in B$, where $B$ is the arc displayed in brown. The plane on the left-hand side passing through the origin separates $\mathbb R^3$ into two open halfspaces, each containing only a single black point. In the right-hand panel, we see the same setup in the gnomonic projection in the plane $\mathbb R^2$. The pink halfplane $G_1$ contains $P_\pm$-mass $0$, while the orange one $G_2$ contains $P_\pm$-mass $-1/5$.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• proof
• Theorem 3
• Theorem 4
• Theorem 5
• Example 1
• Definition
• Theorem 6
• Theorem 7