Theory of angular depth for classification of directional data

Abstract

Depth functions offer an array of tools that enable the introduction of quantile- and ranking-like approaches to multivariate and non-Euclidean datasets. We investigate the potential of using depths in the problem of nonparametric supervised classification of directional data, that is classification of data that naturally live on the unit sphere of a Euclidean space. In this paper, we address the problem mainly from a theoretical side, with the final goal of offering guidelines on which angular depth function should be adopted in classifying directional data. A set of desirable properties of an angular depth is put forward. With respect to these properties, we compare and contrast the most widely used angular depth functions. Simulated and real data are eventually exploited to showcase the main implications of the discussed theoretical results, with an emphasis on potentials and limits of the often disregarded angular halfspace depth.

Appendices

Appendix A: Gnomonic projection of the sphere and angular depths

1.1 A.1. Gnomonic projection and its basic properties

A crucial mathematical tool that allows us to connect the angular depths in the unit sphere $\mathbb {S}^{d-1}$ with usual depths in the linear space ${\mathbb {R}}^{d-1}$ is the gnomonic projection and its inverse. The gnomonic projection takes points from the unit sphere $\mathbb {S}^{d-1}$, and maps them into the hyperplane $H = \left\{ (x_1, \dots , x_d) \in {\mathbb {R}}^d :x_d = 1 \right\}$ that is tangent to $\mathbb {S}^{d-1}$ at its “northern pole" $e_d = \left( 0, \dots , 0, 1 \right)$. Formally speaking, writing $\mathbb {S}^{d-1}_0 = \left\{ \left( x_1, \dots , x_d\right) \in \mathbb {S}^{d-1}:x_d = 0 \right\}$ for the “equator" of the unit sphere, the gnomonic projection is defined by

$$\begin{aligned} \Pi :\mathbb {S}^{d-1}\setminus \mathbb {S}^{d-1}_0 \rightarrow H :x = \left( x_1, \dots , x_d\right) \mapsto x/x_d. \end{aligned}$$

The map $\Pi$ is not defined for the points on the equator $\mathbb {S}^{d-1}_0$ of $\mathbb {S}^{d-1}$. If necessary, one could identify those points with their unit vectors, and expand the map** $\Pi$ by projecting $x \in \mathbb {S}^{d-1}_0$ into a “point in infinity” of H in the direction of the vector x. Mathematically, that construction corresponds to extending the map** $\Pi$ from $\mathbb {S}^{d-1}$ to the projective space associated with H (Berger (2010), Sect. I.5). For our purposes, this however turns out not to be necessary.

The gnomonic projection is a bijection, that is an invertible, one-to-one map**, between the “northern" hemisphere $\mathbb {S}^{d-1}_+ = \left\{ \left( x_1, \dots , x_d \right) \in \mathbb {S}^{d-1}:x_d > 0 \right\}$ and H. Also, it is a bijection between the “southern" hemisphere $\mathbb {S}^{d-1}_- = - \mathbb {S}^{d-1}_+$ and H, meaning that altogether it is a double cover of H. When restricted to $\mathbb {S}^{d-1}_+$ only, the inverse of this map taking any point in H back into $\mathbb {S}^{d-1}_+$ is denoted by $\Pi ^{-1}$. Of course, it is defined by

$$\begin{aligned} \Pi ^{-1}:H \rightarrow \mathbb {S}^{d-1}_+ :x \mapsto x/\left\| x \right\| . \end{aligned}$$

An interesting feature of the map $\Pi$ is that

$$\begin{aligned} \Pi \text{ maps } \text{ great } \text{ circles } \text{ in } \mathbb {S}^{d-1}\setminus \mathbb {S}^{d-1}_0 \text{ into } \text{ straight } \text{ lines } \text{ in } \text{ H }. \end{aligned}$$

(19)

This can be seen directly by considering a great circle as an intersection of a (two-dimensional) plane L passing through the origin in ${\mathbb {R}}^d$ with the sphere $\mathbb {S}^{d-1}$. Since L contains the origin, its gnomonic projection is exactly the intersection $L \cap H$, which is a straight line in H. This setup is visualized in Fig. 9. Another important consequence of (19) is that halfspaces in ${\mathbb {R}}^d$ whose boundary passes through the origin in ${\mathbb {R}}^d$ are mapped via $\Pi$ into halfspaces in H, when identified with the linear space ${\mathbb {R}}^{d-1}$, and that spherical convex sets in $\mathbb {S}^{d-1}_+$ map into convex subsets of H. All these results are direct consequences of (19).

Fig. 9

The gnomonic projection of the sphere $\mathbb {S}^{2}$. Points from the northern hemisphere $x \in \mathbb {S}^{2}_+$ (black point) map to H along the direction of the unit vector x from the origin (orange point); points from the southern hemisphere $y \in \mathbb {S}^{2}_-$ (red point) project in the direction of the antipodal point $-y \in \mathbb {S}^{2}_+$ (color figure online)

1.2 A.2. Gnomonic projection and angular depths

In Dyckerhoff and Nagy (2023), the gnomonic projection is used to devise a fast computation algorithm for the angular halfspace depth. It turns out that most angular depths relate very closely with usual depths of points in the linear space H (identified with ${\mathbb {R}}^{d-1}$), when the measure $P \in {\mathcal {P}}\left( {\mathbb {S}^{d-1}}\right)$ is properly mapped into H via the gnomonic projection. For instance, if P is concentrated only in the open hemisphere $\mathbb {S}^{d-1}_+$ (where $\Pi$ is invertible), the angular halfspace (or simplicial) depth of any $x \in \mathbb {S}^{d-1}_+$ is equivalent with the usual halfspace (or simplicial) depth of $\Pi (x)$ w.r.t. the measure of P projected into H via $\Pi$. This can be seen directly from the property (19), since halfspaces passing through the origin in ${\mathbb {R}}^d$ map via $\Pi$ into halfspaces in H, and spherical convex hulls on $\mathbb {S}^{d-1}_+$ map via $\Pi$ into (linear) convex hulls in H. Thus, for e.g. the angular simplicial depth of any $x \in \mathbb {S}^{d-1}_+$ w.r.t. $P \in {\mathcal {P}}\left( {\mathbb {S}^{d-1}_+}\right)$ concentrated in the northern hemisphere we can write

$$\begin{aligned} asD(x;P) = sD\left( \Pi (x);P_{\Pi }\right) , \end{aligned}$$

where $P_{\Pi }$ stands for the distribution P projected into H via $\Pi$, and $sD(y;Q)$ is the usual simplicial depth of $y \in {\mathbb {R}}^{d-1}$ w.r.t. $Q \in {\mathcal {P}}\left( {{\mathbb {R}}^{d-1}}\right)$ (Liu 1990).

More generally, observe that thanks to the property (19), it is very useful to visualize angular depths $aD$ for distributions $P \in {\mathcal {P}}\left( {\mathbb {S}^{2}_+}\right)$ supported in the northern hemisphere of $\mathbb {S}^{2}$ by displaying

$$\begin{aligned} {\mathbb {R}}^2 \rightarrow [0,\infty ) :y \mapsto aD\left( y;P_{\Pi }\right) \end{aligned}$$

(20)

instead of the more complicated function $x \mapsto aD(x;P)$. In (20), of course, we interpret y as $\Pi (x)$ for $x \in \mathbb {S}^{2}_+$. Using the contour plot of (20) we can, for example, detect departures of the depth $aD\left( \cdot ; P\right)$ from (star) convexity (in $\mathbb {S}^{d-1}$), observe multimodal features of the angular depth surface, or assess rotational symmetry around the direction given by the northern pole. This tool is used in Sect. 3 to infer the contours of angular depths in $\mathbb {S}^{2}$.

Appendix B: Proofs of theoretical results

1.1 B.1. Proof of Theorem 1

Take any directional median $\mu$ that satisfies (3); the existence of such a point $\mu$ follows from ($\textrm{D}_5$). By ($\textrm{D}_4$) we then have $aD(-\mu ;P) = \inf _{x \in \mathbb {S}^{d-1}} aD(x;P)$. Consider the orthogonal transform given by the matrix $O = -I_d \in {\mathbb {R}}^{d \times d}$, for $I_d$ the identity matrix, and apply the induced rotation in ($\textrm{D}_1$). We obtain

$$\begin{aligned} aD(x;P) = aD(-x; P_{-X}) = aD(-x;P) \quad \text{ for } \text{ all } x \in \mathbb {S}^{d-1}, \end{aligned}$$

where we used the fact that $P_{-X} = P$ by the assumption of central symmetry of P. In particular, applying the previous formula to $x = \mu$, we get

$$\begin{aligned} \sup _{x \in \mathbb {S}^{d-1}} aD(x;P) = aD(\mu ;P) = aD(-\mu ;P) = \inf _{x \in \mathbb {S}^{d-1}} aD(x;P), \end{aligned}$$

and necessarily $aD(\cdot ;P)$ must be constant over $\mathbb {S}^{d-1}$.

1.2 B.2. Proof of Theorem 2

By rotational symmetry of P we know that for any $O \in {\mathbb {R}}^{d \times d}$ orthogonal that fixes $\mu$ we have $P = P_{O X}$ in the notation from ($\textrm{D}_1$). Condition ($\textrm{D}_1$) then gives

$$\begin{aligned} aD(x;P) = aD(Ox;P_{O X}) = aD(Ox; P) \quad \text{ for } \text{ all } x \in \mathbb {S}^{d-1}, \end{aligned}$$

and necessarily $aD(\cdot ;P)$ is constant on each set

$$\begin{aligned} \left\{ O x :O \in {\mathbb {R}}^{d \times d} \text{ orthogonal } \text{ such } \text{ that } O\mu = \mu \right\} \subset \mathbb {S}^{d-1}. \end{aligned}$$

This set is precisely the collection of those $y \in \mathbb {S}^{d-1}$ such that $\left\langle y, \mu \right\rangle = \left\langle x, \mu \right\rangle$. Indeed, we have

$$\begin{aligned} \left\langle O x, \mu \right\rangle = \left( O x \right) ^{\textsf{T}}\mu = x^{\textsf{T}}O^{\textsf{T}}\mu = x^{\textsf{T}}O^{-1} \mu = \left\langle x, O^{-1} \mu \right\rangle = \left\langle x, \mu \right\rangle , \end{aligned}$$

where we used that for O orthogonal we have $O^{\textsf{T}}= O^{-1}$ and that $O \mu = \mu$ implies $O^{-1} \mu = \mu$. For the other inclusion, we can assume without loss of generality that $\mu = \left( 1, 0, \dots , 0\right) \in \mathbb {S}^{d-1}$ is the first canonical vector; otherwise, we first rotate the sphere so that $\mu$ maps to that vector. Suppose that $\left\langle y, \mu \right\rangle = \left\langle x, \mu \right\rangle$. Then both x, y are contained in the $(d-2)$-dimensional sphere S obtained as an intersection of $\mathbb {S}^{d-1}$ and the hyperplane $H = \left\{ z \in {\mathbb {R}}^d :\left\langle z, \mu \right\rangle = \left\langle x, \mu \right\rangle \right\}$ orthogonal to $\mu$. Restricting to the $(d-1)$-dimensional affine space H and canonically identifying H with ${\mathbb {R}}^{d-1}$, inside the sphere S we can find an orthogonal matrix $O' \in {\mathbb {R}}^{(d-1) \times (d-1)}$ map** x to y. Extending $O'$ to a $d \times d$ matrix O by appending it in its first row and first column with the vector $\left( 1, 0, \dots , 0\right) \in {\mathbb {R}}^d$ gives the desired orthogonal transform O map** x to y that fixes $\mu$.

To conclude the proof, it remains to define $\xi (t) = aD(x;P)$ for any $x \in \mathbb {S}^{d-1}$ such that $\left\langle x, \mu \right\rangle = t \in [-1,1]$. By our argumentation above, this is a correct definition and $aD(x;P) = \xi (\left\langle x, \mu \right\rangle )$ for all $x \in \mathbb {S}^{d-1}$.

1.3 B.3. Proof of Theorem 3

The central regions (2) corresponding to $aD$ are clearly nested in the sense that if $0 \le \alpha _1 \le \alpha _2$ then for any $P \in {\mathcal {P}}\left( {\mathbb {S}^{d-1}}\right)$ we have $aD_{\alpha _2}(P) \subseteq aD_{\alpha _1}(P)$. At the same time, $aD_{0}(P) = \mathbb {S}^{d-1}$. By assumption ($\textrm{D}_6$) we have that each $aD_{\alpha }(P)$ is spherical convex, which must be either the whole sphere $\mathbb {S}^{d-1}$, or a set contained in some closed hemisphere in $\mathbb {S}^{d-1}$. By the nestedness of the central regions, we can define $\alpha _0 = \inf \left\{ \alpha \ge 0 :aD_{\alpha }(P) \text{ is } \text{ contained } \text{ in } \text{ a } \text{ hemisphere }\right\}$. We consider two cases: either ($\textrm{C}_{1}$) $aD_{\alpha _0}(P)$ is contained in a hemisphere S; or ($\textrm{C}_{2}$) $aD_{\alpha _0}(P)$ is not contained in any hemisphere in $\mathbb {S}^{d-1}$.

In case ($\textrm{C}_{1}$) choose $u \in \mathbb {S}^{d-1}$ to be the direction that gives

$$\begin{aligned} S = \left\{ y \in \mathbb {S}^{d-1}:\left\langle y, u \right\rangle \ge 0 \right\} . \end{aligned}$$

(21)

In case ($\textrm{C}_{2}$) necessarily $aD_{\alpha _0}(P) = \mathbb {S}^{d-1}$. Then we take $\alpha _n = \alpha _0 + 1/n$, and observe that for each $n = 1,2,\dots$ we have that $aD_{\alpha _n}(P)$ is contained in a hemisphere $S_n \subset \mathbb {S}^{d-1}$. The hemispheres $S_n$ do not have to be the same, but if $m \ge n$, then $aD_{\alpha _n}(P) \subseteq aD_{\alpha _m}(P) \subseteq S_m$, meaning that each $S_m$ contains all central regions $aD_{\alpha _n}(P)$ at levels $n \le m$. To each $S_n$ find a direction $u_n \in \mathbb {S}^{d-1}$ such that (21) holds true with S replaced by $S_n$ and u replaced by $u_n$. The sequence $\left\{ u_n \right\} _{n=1}^\infty$ must have a limit point $u \in \mathbb {S}^{d-1}$; define then S directly by (21) using this direction u. Necessarily, we obtain that all $aD_\alpha (P)$ that are contained in a hemisphere must be contained in S in both cases ($\textrm{C}_{1}$) and ($\textrm{C}_{2}$) above.

Take now any two points $x, y \in \mathbb {S}^{d-1}\setminus S$, and suppose that $\alpha ' = aD(x;P) > aD(y;P)$. Consider the central region $aD_{\alpha '}(P)$ that contains x. Since $aD_{\alpha '}(P)$ does not contain y, this region is not the whole sphere $\mathbb {S}^{d-1}$. But, $aD_{\alpha '}(P)$ must be spherical convex, so if it is not $\mathbb {S}^{d-1}$, it must be contained in a hemisphere, in which case $\alpha ' \ge \alpha _0$ by the definition of $\alpha _0$. We get $\alpha ' = \alpha _0$. Using $x \notin S$ again, necessarily case ($\textrm{C}_{2}$) above must hold true (that is, $aD_{\alpha _0}(P)$ is not contained in S). But then, there must exist a hemisphere $S' \subset \mathbb {S}^{d-1}$ such that $aD_{\alpha _0}(P)$ is contained in $S'$, which directly contradicts case ($\textrm{C}_{2}$) as in that situation $aD_{\alpha _0}(P)$ is contained in $S'$. Altogether, we have obtained that for all $x \notin S$, we must have $aD(x;P) = c$ for some constant $c \in [0,\alpha _0]$.

Under the additional condition ($\textrm{D}_4$), clearly $c = \inf _{x \in \mathbb {S}^{d-1}} aD(x;P)$.

1.4 B.4. Proof of Theorem 4

We use the gnomonic projection, described in detail in Appendix A, to express the exact arc distance depth of P as a depth on the tangent line H of $\mathbb {S}^{1}$ that touches the unit circle at its northern pole $(0,1) \in \mathbb {S}^{1}$. For a precise definition of H and the properties of the gnomonic projection please refer to Appendix A. Let $Q_+$ be the pushforward measure of P restricted to $\mathbb {S}^{1}_+$ via the gnomonic projection $\Pi$, and analogously let $Q_-$ be the pushforward of P restricted to $\mathbb {S}^{1}_-$ (notations such as $\mathbb {S}^{1}_+$ and $\mathbb {S}^{1}_-$ come from Appendix A). We suppose that the equator $\mathbb {S}^{1}_0$ has null P-mass. This is made without loss of generality, as by condition ($\textrm{D}_1$) we are allowed to rotate P to a position where $P(\mathbb {S}^{1}_0) = 0$. Denote by $F_+$ and $F_-$ the cumulative distribution functions of $Q_+$ and $Q_-$, respectively, when H is identified with ${\mathbb {R}}$. For $x \in \mathbb {S}^{1}_+$ we can write

$$\begin{aligned} \pi - aD_{arc}(x;P)&= \int _{\mathbb {S}^{1}} \arccos {\left( \left\langle x, y \right\rangle \right) } \,\textrm{d}\,P(y) \\&= \int _{\mathbb {S}^{1}_+} \arccos \left( \left\langle x, y \right\rangle \right) \,\textrm{d}\,P(y) + \int _{\mathbb {S}^{1}_-} \arccos \left( \left\langle x, y \right\rangle \right) \,\textrm{d}\,P(y) \\&= \int _{\mathbb {S}^{1}_+} \arccos \left( \left\langle x, y \right\rangle \right) \,\textrm{d}\,P(y) + \int _{\mathbb {S}^{1}_-} \left( \pi - \arccos \left( - \left\langle x, y \right\rangle \right) \right) \,\textrm{d}\,P(y) \\&= \pi \,P(\mathbb {S}^{1}_-) + \int _{\mathbb {S}^{1}_+} \arccos \left( \left\langle x, y \right\rangle \right) \,\textrm{d}\,P(y) - \int _{\mathbb {S}^{1}_+} \arccos \left( \left\langle x, y \right\rangle \right) \,\textrm{d}\,P(-y) \\&= \pi \,P(\mathbb {S}^{1}_-) + \int _{\mathbb {S}^{1}_+} \arccos \left( \left\langle x, y \right\rangle \right) \,\textrm{d}\,\left( P(y) - P(-y)\right) \\&= \pi \,P(\mathbb {S}^{1}_-) + \int _{H} \arccos \left( \left\langle \Pi ^{-1}\left( \Pi \left( x\right) \right) , \Pi ^{-1}({\widetilde{y}}) \right\rangle \right) \,\textrm{d}\,\left( Q_+({\widetilde{y}}) - Q_-({\widetilde{y}}) \right) \\&= \pi \,P(\mathbb {S}^{1}_-) + \int _{H} \arccos \left( \frac{\left\langle {\widetilde{x}}, {\widetilde{y}} \right\rangle }{\left\| {\widetilde{x}} \right\| \left\| {\widetilde{y}} \right\| } \right) \,\textrm{d}\,\left( Q_+({\widetilde{y}}) - Q_-({\widetilde{y}}) \right) \\&= \pi \,P(\mathbb {S}^{1}_-) + \int _{{\mathbb {R}}} \arccos \left( \frac{1 + x\, y}{\sqrt{\left( 1 + x^2 \right) \left( 1 + y^2 \right) }} \right) \,\textrm{d}\,\left( F_+(y) - F_-(y) \right) . \\ \end{aligned}$$

We used that $\arccos (t) + \arccos (-t) = \pi$, a change of variables $y \mapsto -y$ in the integral over $\mathbb {S}^{1}_-$, and finally the change of variables for the integral with the pushforward measures $Q_+$ and $Q_-$. In the final equalities we denoted ${\widetilde{x}} = \Pi (x) = \left( x,1\right) \in H$ and ${\widetilde{y}} = \left( y,1\right) \in H$. We obtain an explicit form for the arc distance depth in terms of the density of the mass of P projected onto H.

To find local extremes of the arc distance depth, we take a derivative of the final expression above w.r.t. $x \in {\mathbb {R}}$. Given that the function $\arccos$ is bounded and differentiable, we can exchange the derivative and the integral, and after a simplification and rearrangement we obtain

$$\begin{aligned} \frac{\partial aD_{arc}(x;P)}{\partial x} = - \frac{1}{1 + x^2} \int _{{\mathbb {R}}} \textrm{sign}\left( x - y\right) \,\textrm{d}\,\left( F_+(y) - F_-(y) \right) . \end{aligned}$$

In particular, the depth $aD_{arc}\left( \cdot ;P\right)$ has a vanishing derivative at x if and only if

$$\begin{aligned} \int _{{\mathbb {R}}} \textrm{sign}\left( x - y\right) \,\textrm{d}\,F_+(y) = \int _{{\mathbb {R}}} \textrm{sign}\left( x - y\right) \,\textrm{d}\,F_-(y), \end{aligned}$$

or equivalently if

$$\begin{aligned} \int _{(-\infty ,x)} \,\textrm{d}\,F_+(y) - \int _{(x,\infty )} \,\textrm{d}\,F_+(y) = \int _{(-\infty ,x)} \,\textrm{d}\,F_-(y) - \int _{(x,\infty )} \,\textrm{d}\,F_-(y). \end{aligned}$$

Suppose now that $Q_+(H) = Q_-(H) = 1/2$, and that P does not have atoms in $\mathbb {S}^{1}$. Then the preceding formula simplifies further to

$$\begin{aligned} F_+(x) = F_-(x). \end{aligned}$$

In particular, our computation shows that under our assumptions on P, whenever the distribution functions $F_+$ and $F_-$ cross in ${\mathbb {R}}$, the arc distance depth $aD_{arc}(\cdot ;P)$ possesses a critical point.

1.5 B.5. Analysis of Example 1

In the notations from the proof of Theorem 4, we see that $P \in {\mathcal {P}}\left( {\mathbb {S}^{1}}\right)$ is the inverse gnomonic projection of Q. The corresponding distribution $Q_+$ from the proof of Theorem 4 is uniform on the interval $[-K, K]$, and $Q_-$ is uniform on the m points $\left\{ - K - \ell /2 + j \, \ell \right\} _{j=1}^m$. The distribution functions $F_+$ and $F_-$ of $Q_+$ and $Q_-$, respectively, are

$$\begin{aligned} \begin{aligned} F_+(y)&= (y+K)/(2\,K) \quad \text{ for } y \in [-K,K], \\ F_-(y)&= j/m \quad \text{ for } y \in [y_j, y_{j+1}), \end{aligned} \end{aligned}$$

with the notation $y_{m+1} = \infty$. These two distribution functions intersect in m points $\left\{ y_j \right\} _{j=1}^m$ inside the interval $[-K,K]$, see the upper panel of Fig. 1.

The circular distribution $P \in {\mathcal {P}}\left( {\mathbb {S}^{1}}\right)$ is defined on the upper half-circle as the inverse gnomonic projection of $Q_+$ when considered inside the straight line $L_+$; on the lower half-circle it is the inverse gnomonic projection of $Q_-$ inside the antipodally reflected line $L_-$. Both $Q_+$ and $Q_-$ contribute by mass 1/2 to the final measure P. According to our discussion from the proof of Theorem 4, the arc distance depth of such circular distribution P possesses at least m local extremes.

1.6 B.6. Proof of Theorem 5

For $X \sim P$ and $x \in \mathbb {S}^{d-1}$ we have

$$\begin{aligned} \begin{aligned} aD_\delta (-x;P)&= \delta (-1) - {{\,\textrm{E}\,}}\delta \left( - \left\langle x, X \right\rangle \right) = \delta (-1) - c + {{\,\textrm{E}\,}}\delta \left( \left\langle x, X \right\rangle \right) \\&= 2\,\delta (-1) - c - aD_{\delta }(x;P). \end{aligned} \end{aligned}$$

Necessarily, $aD_{\delta }(x;P)$ is maximized if and only if $aD_{\delta }(-x;P)$ takes the minimum value over $\mathbb {S}^{d-1}$, and ($\textrm{D}_4$) is satisfied.

1.7 B.7. Proof of Theorem 6

Take any sequence $\left\{ x_n \right\} _{n=1}^\infty \subset \mathbb {S}^{d-1}$ that converges to $x \in \mathbb {S}^{d-1}$. Since $x_n \rightarrow x$ as $n \rightarrow \infty$, we can define a sequence of orthogonal matrices $\left\{ O_n \right\} _{n=1}^\infty \subset {\mathbb {R}}^{d \times d}$ such that $O_n x_n = x$ for all n, and at the same time $O_n$ converges to the identity matrix $I_d \in {\mathbb {R}}^{d \times d}$ in, say, the Frobenius matrix norm (the choice of the matrix norm is irrelevant in this case, as in finite-dimensional spaces all metrics are equivalent). Defining $P_n \in {\mathcal {P}}\left( {\mathbb {S}^{d-1}}\right)$ as the pushforward measure of P under the map** $\mathbb {S}^{d-1}\rightarrow \mathbb {S}^{d-1}:y \mapsto O_n y$, we obtain that the sequence $\left\{ P_n \right\} _{n=1}^\infty \subset {\mathcal {P}}\left( {\mathbb {S}^{d-1}}\right)$ converges weakly to P. As observed in Liu and Singh (1992), the depth $asD$ satisfies ($\textrm{D}_1$). Due to that, we have that $asD(x_n;P) = asD(x;P_n)$ for all $n = 1, 2, \dots$. The angular simplicial depth of $x \in \mathbb {S}^{d-1}$ w.r.t. any measure $Q \in {\mathcal {P}}\left( {\mathbb {S}^{d-1}}\right)$ can be written as $asD(x;Q) = Q^d\left( S_x \right)$, where $Q^d \in {\mathcal {P}}\left( {\left( \mathbb {S}^{d-1}\right) ^d}\right)$ is the product of d copies of measure Q, and $S_x \subset \left( \mathbb {S}^{d-1}\right) ^d$ is a closed set (e.g., Nagy et al. (2016), proof of Theorem A.6). The portmanteau theorem (Dudley (2002), Theorem 11.1.1) applied to the sequence $\left\{ P_n \right\} _{n=1}^\infty$ and the closed set $S_x$ then guarantees that

$$\begin{aligned} \begin{aligned} {\lim \sup }_{n\rightarrow \infty } ahD(x_n;P)&= {\lim \sup }_{n\rightarrow \infty } ahD(x;P_n) \\&= {\lim \sup }_{n\rightarrow \infty } P_n^d(S_x) \le P^d(S_x) = ahD(x;P), \end{aligned} \end{aligned}$$

as we wanted to show.