Data depth for the uniform distribution

Abstract

Given a set $X$ of $k$ points and a point $z$ in the $n$-dimensional euclidean space, the Tukey depth of $z$ with respect to $X$, is defined as $m/k$, where $m$ is the minimum integer such that $z$ is not in the convex hull of some set of $k-m$ points of $X$. If $z$ belongs to the closed region $B$ delimited by an ellipsoid, define the continuous depth of $z$ with respect to $B$ as the quotient $V(z)/\text{ Vol }(B)$, where $V(z)$ is the minimum volume of the intersection of $B$ with the halfspaces defined by any hyperplane passing through $z$, and $\text{ Vol }(B)$ is the volume of $B$. We consider $z$ a random variable and prove that, if $z$ is uniformly distributed in $B$, the continuous depth of $z$ with respect to $B$ has expected value $1/2^{n+1}$. This result implies that if $z$ and $X$ are uniformly distributed in $B$, the expected value of Tukey depth of $z$ with respect to $X$ converges to $1/2^{n+1}$ as the number of points $k$ goes to infinity. These findings have applications in ecology, namely within the niche theory, where it is useful to explore and characterize the distribution of points inside species niche.

Appendix

1.1 Proof of Theorem 1

Denote by $V_n$ the volume of the unit closed $n$-ball $B_n$. It is well known that $V_n=(2\pi /n)\, V_{n-2}$, with $V_1=2$ and $V_2=\pi$.

Given $R\in \mathbb{R }^+$ and a continuous function $f:[0,R] \rightarrow \mathbb{R }$ set

$$\begin{aligned} I_n[f,R]:=\int \!\!\!\int \limits _{B_n(R)} \!\! \cdots \!\!\! \int f(\Vert x\Vert ) \, dx_1dx_2\cdots dx_n, \end{aligned}$$

(1)

where $B_n(R)$ denotes the $n$-ball centred at the origin of radius $R$. Using hyperspherical coordinates we obtain by straightforward computations,

$$\begin{aligned} I_n[f,R]= n \,V_n \int \limits _0^R f(\rho ) \,\rho ^{n-1} \,d\rho . \end{aligned}$$

(2)

The volume of the hyperspherical cap orthogonal to $z=(0,\ldots ,0,z_n)$ is

$$\begin{aligned} V(z)= \int \!\!\!\!\!\!\!\!\!\!\!\int \limits _{B_{n-1} \left( \sqrt{1-z_n^2}\right) }\!\!\!\!\!\!\!\!\!\!\!\cdots \!\!\int \left( \sqrt{1-\Vert y\Vert ^2}-\Vert z\Vert \right) \,dy_1\, dy_2\,\cdots \,dy_{n-1}. \end{aligned}$$

Set $\Theta _n=\int \!\!\!\int \limits _{B_n(1)}\!\!\cdots \!\!\int V(z)\,dz_1\, dz_2\,\cdots \,dz_n$. If $z$ is uniformly distributed in $B_n \equiv B_n(1)$, the expected value of the continuous depth of $z$ with respect to $B_n$ is

$$\begin{aligned} \mathrm{E}\!\left[ d_z^c(B_n)\right] =\frac{1}{V_n} \int \!\!\!\!\int \limits _{B_n(1)}\!\!\!\!\cdots \!\!\int \frac{V(z)}{V_n}\,dz_1\, dz_2\,\cdots \,dz_n = \frac{\Theta _n}{V_n^2}, \end{aligned}$$

and we have to show that

$$\begin{aligned} \Theta _n=\frac{V_n^2}{2^{n+1}},\quad n\ge 1. \end{aligned}$$

(3)

For $n=1, V(z)=1-|z|$ and we obtain

$$\begin{aligned} \Theta _1=2 \int \limits _0^1 (1-z) \,dz=\frac{V_1^2}{2^2}. \end{aligned}$$

For $n\ge 2$, we have by (2),

$$\begin{aligned} \Theta _n&= I_n\left[ I_{n-1}\left[ \sqrt{1-\rho ^2} - r, \sqrt{1-r^2}\right] ,1\right] \\&= (n-1)n \, V_{n-1} V_n \int \limits _0^1\int \limits _0^{\sqrt{1-r^2}} \left( \sqrt{1-\rho ^2}- r\right) \,\rho ^{n-2} \, d\rho \,\, r^{n-1}\, dr, \end{aligned}$$

where $\rho =\Vert y\Vert$ and $r=\Vert z\Vert$.

Given nonnegative integers $k,\ell$, set $\Omega _{k,\ell }=\int \limits _0^1 (1- \rho ^2)^{\frac{k}{2}} \,\rho ^\ell \,d\rho$. Changing the order of integration and simplifying, we obtain

$$\begin{aligned} \Theta _n=\frac{n-1}{n+1}\,V_{n-1}\, V_n \, \Omega _{n+1,n-2}. \end{aligned}$$

In particular, we get

$$\begin{aligned} \Theta _2&= \frac{1}{3} V_1 \,V_{2}\, \Omega _{3,0}=\frac{2\pi }{3}\, \int \limits _0^1 (1- \rho ^2)^{3/2}\, d\rho =\frac{\pi ^2}{8}=\frac{V_2^2}{2^3},\\ \Theta _3&= \frac{2}{4}\, V_2 \,V_{3}\, \Omega _{4,1}= \frac{2\pi ^2}{3}\, \int \limits _0^1 (1- \rho ^2)^{2}\,\rho \,d\rho =\frac{\pi ^2}{9}=\frac{V_3^2}{2^4}. \end{aligned}$$

Let $\Gamma [z]$ denote the usual gamma function. Using relation $\Gamma [z+1]=z\Gamma [z]$ along with relation

$$\begin{aligned} \Omega _{k,\ell }=\frac{1}{2}\frac{\Gamma \left[ \frac{k+2}{2}\right] \,\Gamma \left[ \frac{\ell +1}{2}\right] }{\Gamma \left[ \frac{k+\ell +3}{2}\right] },\qquad k,\ell \ge 0, \end{aligned}$$

(see, for instance, Abramowitz et al. 1972, Section 6.2) we get for every $n\ge 4$,

$$\begin{aligned} \Omega _{n+1,n-2}= \Omega _{n-1,n-4}\frac{(n+1)(n-3)}{4\, (n-1)n}. \end{aligned}$$

(4)

By (4) we derive, for $n\ge 4$,

$$\begin{aligned} \frac{\Theta _n}{\Theta _{n-2}}&= \frac{(n-1)/(n+1)}{(n-3)/(n-1)}\,\, \frac{V_n}{V_{n-2}} \,\, \frac{V_{n-1}}{V_{n-3}}\,\, \frac{\Omega _{n+1,n-2}}{\Omega _{n-1,n-4}}\\&= \frac{(n-1)^2}{(n+1)(n-3)} \,\, \frac{2\pi }{n} \,\, \frac{2\pi }{n-1} \,\, \frac{(n+1)(n-3)}{4\, n(n-1)}\\&= \frac{\pi ^2}{n^2}. \end{aligned}$$

The proof of (3) now follows by induction, reminding that $V_{n-2}=n V_n/(2\pi )$.