Another look at halfspace depth:
Flag halfspaces with applications

Denote by $\mathcal{M}\left(\mathbb{R}^{d}\right)$ the set of all finite Borel measures on the Euclidean space $\mathbb{R}^{d}$. The halfspace (or Tukey) depth of $x\in\mathbb{R}^{d}$ with respect to (w.r.t.) $\mu\in\mathcal{M}\left(\mathbb{R}^{d}\right)$ is defined as¹¹1We consider the halfspace depth w.r.t. finite measures $\mu$, that is when $\mu(\mathbb{R}^{d})<\infty$. Compared to the usual setup of probability measures, this extension is minor, and made only for notational convenience. All our results could be considered also for probability measures only, with obvious modifications.

where $\mathcal{H}(x)$ is the collection of closed halfspaces in $\mathbb{R}^{d}$ that contain $x$ on their boundary. The halfspace depth quantifies the centrality of $x$ w.r.t. the mass of $\mu$. That is quite useful in nonparametric statistics, as it allows us to rank sample points according to their depth, from the central to the peripheral ones. As such, the depth enables the introduction of rankings, orderings, and quantile-like inference to multivariate datasets [3, 20, 21]. The upper level sets of the halfspace depth of $\mu$, given for $\alpha\geq 0$ by

play in nonparametric statistics the role of the inner quantile regions of $\mu$. They are often called the (halfspace) central regions of $\mu$. The sets (2) are nested, closed and convex; they are compact for $\alpha>0$, and non-empty for $\alpha\leq\alpha^{*}(\mu)$, where $\alpha^{*}(\mu)=\sup_{x\in\mathbb{R}^{d}}0pt{x}{\mu}$ is the maximum halfspace depth of $\mu$. Of special importance is the set $D^{*}\left(\mu\right)=D_{\alpha^{*}(\mu)}$, which contains points that are the most centrally positioned w.r.t. $\mu$. It is called the set of the halfspace medians of $\mu$ and, as its name suggests, it generalises the median to $\mathbb{R}^{d}$. The halfspace depth has many applications in multivariate statistics, and is already for 30 years a subject of active research [11, 12, 13, 15, 16]. Although many other statistical depth functions have been developed [2, 14, 21], in this paper we focus on the halfspace depth, and sometimes write simply depth instead of halfspace depth. We call a measure $\mu\in\mathcal{M}\left(\mathbb{R}^{d}\right)$ smooth if the $\mu$-mass of every hyperplane in $\mathbb{R}^{d}$ is zero. A measure with a density is smooth; examples of non-smooth measures are those with an atom. The infimum in (1) is attained for smooth measures. That is why theoretical results on the halfspace depth are often formulated only for smooth measures, and why the analysis of the sample halfspace depth (that is, the halfspace depth evaluated w.r.t. empirical measures of random samples) is performed using different techniques [10, 12, 13]. In this paper we introduce flag halfspaces — symmetrised variants of closed halfspaces that may be considered in (1) instead of $\mathcal{H}(x)$ without altering the depth, with the property that a flag halfspace attaining the depth always exists. We will see that our restatement of formula (1) simplifies many theoretical derivations about the halfspace depth, as it is no longer needed to distinguish whether the infimum in (1) is attained.

Flag halfspaces are introduced in Section 2. Two applications to the computation of the depth are given in Section 3. In Section 3.1, we investigate the dimensionality and the structure of the median set $D^{*}\left(\mu\right)$ for $\mu$ an empirical measure. We show that for datasets sampled from absolutely continuous probability measures in $\mathbb{R}^{d}$, the halfspace median set cannot be of dimension $d-1$, almost surely. In a series of examples in $\mathbb{R}^{3}$ we demonstrate that already for random samples of size $n=8$ from the standard Gaussian distribution, halfspace median sets of dimensions $0$, $1$, and $3$ occur with positive probability. In Section 3.2 we deal with the special situation of data of dimension $d=2$. We show that if the dataset satisfies a mild condition of general position, then the halfspace median set must be either a full-dimensional polygon, or a data point. Both these advances find applications in the computation of the halfspace median and the central regions (2), where the dimensionality of $D^{*}\left(\mu\right)$ plays a crucial role [6, 11]. The paper is complemented by online Supplementary Material containing R and Mathematica scripts with visualisations and computations completing examples from Section 3.

For $\mu\in\mathcal{M}\left(\mathbb{R}^{d}\right)$ and $x\in\mathbb{R}^{d}$ we call $H\in\mathcal{H}(x)$ a minimising halfspace of $\mu$ at $x$ if $\mu(H)=0pt{x}{\mu}$. For $d=1$ minimising halfspaces always trivially exist. They also exist if $\mu$ is smooth, or if $\mu$ is supported in a finite number of points. In general, however, the infimum in (1) does not have to be attained. We give a simple example.

The problem with measures not attaining the infimum in (1) is elegantly resolved by considering flag halfspaces instead of the usual closed halfspaces.

The name flag comes from geometry [17], where an analogous recursive construction is considered, involving nested faces of convex polytopes. The formal definition of flag halfspaces is somewhat convoluted, but these sets appear naturally. In $\mathbb{R}^{2}$, a flag halfspace at $x$ is the union of an open halfplane $G_{2}$ whose boundary passes through $x$, a relatively open halfline $G_{1}$ originating at $x$ contained in the one-dimensional affine space (line) $\operatorname{bd}\left(G_{2}\right)$, and the $0$-dimensional point $x$ itself. For an example see Figure 1. A flag halfspace is neither an open nor a closed set. In contrast to a usual closed halfspace, a complement of a flag halfspace $F\in\mathcal{F}(x)$ is, except for its central point $x$, again a flag halfspace from $\mathcal{F}(x)$, i.e. $(\mathbb{R}^{d}\setminus F)\cup\{x\}\in\mathcal{F}(x)$. Several more interesting properties and characterisations of flag halfspaces can be found in [9].

We define a minimising flag halfspace of $\mu$ at $x$ to be any $F\in\mathcal{F}(x)$ that satisfies $\mu(F)=D(x;\mu)$. In the following Theorem 1 we show that the halfspace depth (1) of any measure can be expressed in terms of the $\mu$-mass of flag halfspaces, and a minimising flag halfspace always exists. The intuition behind this result is as follows: Even if the minimising closed halfspace of $x\in\mathbb{R}^{d}$ does not exist, there is a sequence of closed halfspaces $\{H_{n}\}_{n=1}^{\infty}\subset\mathcal{H}(x)$ that satisfies

Because the unit normals $\{v_{n}\}_{n=1}^{\infty}$ of these halfspaces come from the compact set $\mathbb{S}^{d-1}$, we can also assume that the sequence of halfspaces is convergent and $\lim_{n\to\infty}v_{n}=v\in\mathbb{S}^{d-1}$ (otherwise, we extract a convergent subsequence). For $n$ large enough, $\mu\left(H_{n}\right)$ is arbitrarily close to $0pt{x}{\mu}$, but this fact alone, of course, does not imply that the $\mu$-mass of the limit $H\equiv H_{x,v}$ defined as $H=\lim_{n\to\infty}H_{n}$ is equal to $0pt{x}{\mu}$. It turns out that for general measures, it is not possible to find any useful upper bound on the mass $\mu\left(H\right)$, but it is possible to bound the mass of its interior by $\mu\left(\operatorname{int}\left(H\right)\right)\leq 0pt{x}{\mu}$. The interior of $H$ is the first open halfspace $G_{d}$ in the construction of the minimising flag halfspace (3). The remaining relatively open halfspaces $G_{k}$ are found by iterating the same process inside the relative boundary of the previous $G_{k+1}$, $k=1,\dots,d-1$. In the right hand panel of Figure 2 we see a visualisation of our setup, with $d=2$. In the situation displayed, as $n\to\infty$, the halfspaces $H_{n}$ do not intersect the halfline $G_{1}^{-}\subset\operatorname{bd}\left(H\right)$ originating at $x$, so the $\mu$-mass of $G_{1}^{-}$ does not contribute to the depth of $x$, and $G_{1}^{-}$ should not be contained in $F$. The formal statement of our theorem follows.

In Example 1, the single minimising flag halfspace of $\mu$ at $x$ is

In formula (12) in the proof of Theorem 1 we unveiled the recursive nature of the halfspace depth. The following result formalises that observation. In the special situation of an empirical measure $\mu\in\mathcal{M}\left(\mathbb{R}^{d}\right)$, a related result has been observed in [5, Theorems 1 and 2] and successfully applied in the task of exact computation of the halfspace depth.

We now use flag halfspaces to derive several properties of the sample halfspace median that are of interest in the practice of the depth; additional applications of flag halfspaces to the theory of the halfspace depth can be found in [8, 9]. Write $\mathcal{A}\left(\mathbb{R}^{d}\right)$ for the set of all empirical measures $\mu\in\mathcal{M}\left(\mathbb{R}^{d}\right)$, that is all purely atomic probability measures with a finite number $n$ of atoms, each atom having $\mu$-mass $1/n$, for some $n=1,2,\dots$. These measures are typically obtained observing a random sample $X_{1},\dots,X_{n}\in\mathbb{R}^{d}$ from a probability distribution $\nu\in\mathcal{M}\left(\mathbb{R}^{d}\right)$, each sample point corresponding to an atom. To approximate the halfspace depth of $\nu$, the depth of $\mu$ is computed. The latter depth function is standardly used for inference about the unknown distribution $\nu$. Naturally, it is therefore crucial to understand the behaviour of the halfspace depth w.r.t. empirical measures. We provide results on the dimensionality of the median set, assuming that the atoms of $\mu\in\mathcal{A}\left(\mathbb{R}^{d}\right)$ lie in a sufficiently general position. The last assumption is not restrictive; it is satisfied if, for instance, the measure $\nu$ from which we sample is smooth. The proof of the following lemma is standard and omitted.