Monotonicity properties of spatial depth

Abstract

We investigate the monotonicity properties of the spatial depth for multivariate data. We show that the spatial depth does not decrease monotonically with respect to the deepest point. Moreover, an answer to the conjecture of Gao (2003) is provided.

Introduction

Statistical data depth is a mapping that assigns to a given point $x$ in ${\mathbb{R}}^d$, $d\ge 1$, and a probability distribution $P$ on ${\mathbb{R}}^d$, a non-negative number that characterizes how much “centrally located” $x$ is with respect to $P$. The point maximizing the depth generalizes the median to ${\mathbb{R}}^d$-valued data, and the loci of points of high depth form the central regions of the distribution $P$. A low depth value indicates that $x$ is atypical, or far from this centre. A vast body of literature on data depth and its applications is available — for a general account see Zuo and Serfling (2000), for an overview of some applications see Liu et al. (1999).

The spatial depth is a depth function that builds upon the notion of spatial (also called geometric) quantiles for multivariate data, considered by Chaudhuri (1996) and Koltchinskii (1997). These quantiles, and the associated spatial signs and ranks, are indispensable in modern nonparametric statistics of multivariate data (see, e.g., Möttönen and Oja (1995), Möttönen et al. (1997), or the book by Oja 2010).

A depth associated with the spatial quantiles was first considered by Vardi and Zhang (2000). In its current form, the spatial depth was defined by Serfling (2002), and soon after, independently, also by Gao (2003). For $x\in {\mathbb{R}}^d$ and a probability distribution $P$ on ${\mathbb{R}}^d$ the spatial depth is given by

$$D(x;P)=1-∥E\frac{x-X}{∥x-X∥}∥,$$

where $X\sim P$, and $∥\cdot ∥$ stands for the Euclidean norm. In the definition and throughout this note, we use the convention $0∕0=0$.

The spatial depth function has a number of attractive properties. Its maximal value is attained at the spatial median, a robust location parameter well known in statistics. It is invariant with respect to translations and orthogonal rotations of the data. Unlike for many other depth functions, the computation of $D$ is extremely simple and fast, also in high-dimensional spaces. Finally, it is well applicable also to infinite-dimensional (functional) data Chakraborty and Chaudhuri (2014a), Chakraborty and Chaudhuri (2014b). The spatial depth and its variants have been successfully used in many practical tasks, see, e.g., Chen et al. (2009), Li et al. (2013), Sguera et al. (2014), Dutta et al. (2016), and Serfling and Wijesuriya (2017).

In the paragraph after Proposition 2 in Gao (2003), a conjecture regarding the behaviour of $D(\cdot ;P)$ on rays emanating from the centre of symmetry of $P$ is formulated.

Conjecture. For any $P$ angularly symmetric¹ around $0$,

$$D(x;P)>D(ax;P)\text{for any }x\in {\mathbb{R}}^d\text{ and }a>1\text{.}$$

For any $P$ angularly symmetric around $\theta \in {\mathbb{R}}^d$, the spatial depth $D(\cdot ;P)$ is maximized at $\theta$ (Gao, 2003 Proposition 2). Thus, condition (C) is in fact a weaker version of the following property, considered by Liu (1990, Theorem 3) and Zuo and Serfling (2000, property P3 and Definition 2.1) :

• For any probability distribution $P$ on ${\mathbb{R}}^d$ such that $D(\theta ;P)={sup}_{x\in {\mathbb{R}}^d}D(x;P)$,

  $$D(x;P)\le D(\theta +\alpha (x-\theta );P)\text{holds for any }x\in {\mathbb{R}}^d\text{ and }\alpha \in [0,1]\text{.}$$

This condition, frequently called Monotonicity relative to the deepest point, is standardly recognized as desirable for depth functions. Geometrically, it means that the upper level sets $\left\{x:D(x;P)\ge c\right\}$ of the spatial depth $D$ for $c\ge 0$ form a collection of nested sets, star-shaped relative to the spatial median² . In particular, they are always connected, and the depth induces a reasonable centre-outwards ordering of the data, as required in applications. See also Liu (1990, Remark A).

In the present note we provide two examples of probability distributions that demonstrate that the conjecture ofGao (2003) does not hold, in general. As a consequence, the spatial depth does not satisfy property P3. In Section 2.1 we present an atomic angularly symmetric distribution $P$ on ${\mathbb{R}}^2$ such that (C) is not satisfied for $P$. In Section 2.2 we extend this result to show that also for $P$ absolutely continuous, (C) can be violated.

It is important to mention that these examples are not in conflict with the monotonicity property of the spatial distribution, as asserted by Koltchinskii (1997, Proposition 2.4), see also Chakraborty and Chaudhuri (2014b, Theorem 3.1). There, a different version of monotonicity is considered, in no direct relation with (C) or P3. Recall that the spatial distribution of $x\in {\mathbb{R}}^d$ with respect to a probability distribution $P$ on ${\mathbb{R}}^d$ is given by

$$\mathbf{S}(x;P)=E\frac{x-X}{∥x-X∥}.$$

This map is a special case of the $M$-distribution, studied in detail by Koltchinskii (1997). The spatial depth can be written as $D(x;P)=1-∥\mathbf{S}(x;P)∥$. For the spatial distribution, the following monotonicity property is established in Koltchinskii (1997, Proposition 2.4).

Lemma 1

For any probability distribution $P$ on ${\mathbb{R}}^d$ and any $x_1,x_2\in {\mathbb{R}}^d$ it holds true that $〈x_2-x_1,\mathbf{S}(x_2;P)-\mathbf{S}(x_1;P)〉\ge 0$.

Herein, $〈x,y〉$ stands for the inner product of the vectors $x,y\in {\mathbb{R}}^d$.

Using Lemma 1, it is possible to devise a bound on the spatial depth, see Section 2.3. Exploiting this bound, we provide in Section 2.3 a discussion that illustrates the difference between the monotonicity properties of the spatial depth, and the monotonicity of the spatial distribution.

For the sake of clarity, some computational details and a gif animation related to the examples are provided in the online Supplementary Material accompanying this paper.