Centrality Without a Single Multivariate Ranking
In one dimension, observations can be arranged from smallest to largest. In several dimensions, there is no natural analogue of that line: two points may differ in different coordinate directions, and neither need be unambiguously “larger” or “more central” than the other. A depth function supplies a different organizing principle. It assigns each point a score, with larger values representing greater centrality relative to a distribution or data cloud.
The score is useful because it summarizes a point’s position in the whole multivariate configuration. But it is not a complete description of that position. Distinct points can have the same score, and their directions from the center can still matter. Depth therefore provides center-outward layers rather than a unique sequence of all observations.
This definition deliberately does not prescribe one formula. Different depth functions encode centrality differently, so they can respond differently to skewness, tails, or other features of \(P\). A common geometric property is that the depth is quasiconcave in \(x\): for each threshold \(c\), the set of points with depth at least \(c\) is convex. Other useful properties include affine invariance and vanishing depth as \(x\) moves arbitrarily far from the data. These are properties to check for a particular depth, not automatic consequences of assigning a centrality score.
Scores, Ties, and Depth Contours
For a fixed distribution \(P\), the score compares points by their depth values: if \(D(x;P)>D(y;P)\), then \(x\) is more central according to that depth. Equal scores produce ties. Thus a depth score gives a total preorder on points, but generally not a strict total order: it need not distinguish every pair of distinct points. In particular, two observations on different sides of a central region may lie on the same depth contour.
For a threshold \(c\), the corresponding upper level set, or depth region, is \(C_c(P)=\{x:D(x;P)\ge c\}\). Its boundary is often called a depth contour. Higher thresholds select points nearer the deepest region, while lower thresholds include more distant points. When the upper level sets are convex, the contours provide nested central regions with a clear geometric interpretation.
Proof. If \(x\in C_{c_1}(P)\), then \(D(x;P)\ge c_1\ge c_2\), so \(x\in C_{c_2}(P)\). This proves nesting. For convexity, take \(x,y\in C_c(P)\) and \(0\le\lambda\le1\). Quasiconcavity means \(D(\lambda x+(1-\lambda)y;P)\ge\min\{D(x;P),D(y;P)\}\). Since both depths are at least \(c\), the depth at the point between them is also at least \(c\). Therefore \(\lambda x+(1-\lambda)y\in C_c(P)\), which is precisely convexity. \(\square\)
Worked Example: Reading Nested Contours
Consider an illustrative depth surface on \(\mathbb{R}^2\), \(D(x,y)=\exp\{-(x^2+4y^2)\}\). Its largest value is \(1\), attained at \((0,0)\), and it decreases as \(x^2+4y^2\) increases. The contours can be calculated directly.
At threshold \(c=0.8\), the condition \(D(x,y)\ge0.8\) is equivalent to \(\exp\{-(x^2+4y^2)\}\ge0.8\). Taking logarithms gives \(x^2+4y^2\le-\log(0.8)\approx0.2231\). This is an ellipse centered at the origin.
At threshold \(c=0.2\), the same calculation gives \(x^2+4y^2\le-\log(0.2)\approx1.6094\). The second ellipse contains the first because \(0.2231<1.6094\). For example, \((0.3,0)\) has depth \(\exp(-0.09)\approx0.914\), while \((0,\sqrt{0.4})\) has depth \(\exp(-1.6)\approx0.202\). Both are inside the \(0.2\) region, but only the first is inside the \(0.8\) region.
This surface illustrates how a scalar score forms nested layers. It also shows that a contour is not a list: many points, including points in different directions, can have the same depth.
Affine Invariance and Distributional Sensitivity
A depth is affine invariant if changing coordinates by any nonsingular affine transformation preserves each point’s depth, provided the distribution is transformed along with the point. Write \(P_{A,b}\) for the distribution of \(AX+b\) when \(X\sim P\), where \(A\) is a nonsingular \(d\times d\) matrix and \(b\in\mathbb{R}^d\). The defining condition is \(D(Ax+b;P_{A,b})=D(x;P)\). This property makes the score independent of the choice of origin, measurement units, and nonsingular linear coordinate system.
Affine invariance does not mean that a depth ignores the distribution. The distribution is transformed together with the point in the invariance condition; changing \(P\) to a genuinely different shape while holding the point fixed can change its depth. Conversely, a particular depth may ignore some features of \(P\). For example, a depth calculated only from the mean and covariance cannot distinguish distributions that have the same mean and covariance.
Worked Example: Affine Invariance of Mahalanobis Depth
Suppose \(P\) has finite mean \(\mu\) and positive-definite covariance matrix \(\Sigma\). Its Mahalanobis depth is \(D_M(x;P)=1/\{1+(x-\mu)^\mathsf{T}\Sigma^{-1}(x-\mu)\}\). For \(\mu=(0,0)^\mathsf{T}\) and \(\Sigma=\begin{pmatrix}4&0\\0&1\end{pmatrix}\), the point \(x=(2,1)^\mathsf{T}\) has squared Mahalanobis distance \(2^2/4+1^2/1=2\), and hence depth \(1/(1+2)=1/3\). The point \((0,\sqrt{2})^\mathsf{T}\) has squared distance \(0+2=2\) and the same depth.
To verify affine invariance, let \(Y=AX+b\), with \(A\) nonsingular. Then its mean is \(\mu_Y=A\mu+b\), and its covariance is \(\Sigma_Y=A\Sigma A^\mathsf{T}\). Its inverse is \(\Sigma_Y^{-1}=(A^\mathsf{T})^{-1}\Sigma^{-1}A^{-1}\). At \(y=Ax+b\), the centered vector is \(y-\mu_Y=A(x-\mu)\). Substitution gives \((y-\mu_Y)^\mathsf{T}\Sigma_Y^{-1}(y-\mu_Y)=(x-\mu)^\mathsf{T}\Sigma^{-1}(x-\mu)\). The squared distances, and therefore the depths, agree. This proves affine invariance for nonsingular affine transformations.
The contours of this depth are ellipsoids: \(D_M(x;P)\ge c\), for \(0<c\le1\), is equivalent to \((x-\mu)^\mathsf{T}\Sigma^{-1}(x-\mu)\le c^{-1}-1\). This construction reflects location and covariance, but not all aspects of the distribution’s shape.
A different illustration of distributional sensitivity comes from one dimension. For a continuous distribution with cumulative distribution function \(F\), define the central depth \(D_F(x)=\min\{F(x),1-F(x)\}\). It is largest at a median and measures how balanced the probability lying on either side of \(x\) is. Because it uses \(F\) itself, it can distinguish distributions that share their first two moments.
Worked Example: Same Mean and Variance, Different Depth
Compare a standard normal distribution with a logistic distribution whose scale is \(s=\sqrt{3}/\pi\). Both have mean \(0\) and variance \(1\). At \(x=1\), the standard normal central depth is \(1-\Phi(1)\approx1-0.8413=0.1587\).
The logistic cumulative distribution function is \(F(x)=1/\{1+\exp(-x/s)\}\). Here \(1/s=\pi/\sqrt{3}\approx1.8138\), so \(F(1)\approx1/(1+\exp(-1.8138))\approx0.8598\). Its central depth is \(1-F(1)\approx0.1402\). The depths differ despite the matching means and variances: this depth responds to the distributions’ different probability at \(x=1\). The example also underscores that distributional sensitivity is a property of a chosen depth formula, not a promise that every pair of different distributions must receive different scores at every point.
Why Affine Invariance Does Not Mean Distribution-Free
These two ideas answer different questions. Affine invariance asks whether a change of coordinates alters the depth when both the point and its distribution are transformed. Distributional sensitivity asks which features of the distribution affect the score. A depth can be affine invariant and still respond to shape features that are not removed by affine transformations. Another affine-invariant depth may depend only on a small set of distributional summaries and ignore other features.
Mahalanobis depth makes the distinction concrete. Its calculation uses only \(\mu\) and \(\Sigma\). If two distributions have the same values of those quantities, the Mahalanobis depth scores are identical at every \(x\), even if the distributions differ in tail behavior or other aspects of shape. The univariate central depth, by contrast, uses the cumulative distribution function and can reflect such differences. Neither statement establishes that one construction is preferable in every application; it identifies what information each construction uses.
Common Pitfall: Treating Depth as a Unique Rank
- Wrong reasoning: “Each point has a depth score, so the points must have a unique rank from most central to least central.”
- Correction: Distinct points can share a score, and a contour can contain many points in different directions. Depth creates layers, not necessarily a unique ordering of observations.
- Wrong reasoning: “An affine-invariant depth must be unaffected by any change in the distribution.”
- Correction: Affine invariance concerns coordinated changes of point and distribution. It does not assert insensitivity to every change in distributional shape.
Depth contours matter because they provide a way to describe central regions even when observations cannot be naturally listed along one axis. They can support visual summaries and center-outward ordering rules, while preserving the geometry of the multivariate configuration. Their interpretation depends on the particular depth chosen: an affine-invariant score can facilitate comparisons across coordinate systems, while a distribution-sensitive construction may encode features beyond location and covariance. For inference, the score and its properties must be matched to the question; a contour alone does not identify every directional distinction.
Check Your Understanding
Answer each question by relating the score to its geometry and assumptions.
- What does affine invariance require when a point and its distribution undergo the same nonsingular affine transformation?
- Why do upper level sets of a quasiconcave depth form convex regions?
- Can two distinct observations have the same depth? What does that imply for interpreting depth as a ranking?
- Why can Mahalanobis depth fail to distinguish two distributions with different tail behavior?
- How does distributional sensitivity differ from affine invariance?