Tukey depth

In statistics and computational geometry, the Tukey depth ¹ or half-space depth is a measure of the depth of a point in a fixed set of points. The concept is named after its inventor, John Tukey. Given a set of n points ${\displaystyle {\mathcal {X}}_{n}=\{X_{1},\dots ,X_{n}\}}$ in d-dimensional space, Tukey's depth of a point x is the smallest fraction (or number) of points in any closed halfspace that contains x.

Tukey's depth measures how extreme a point is with respect to a point cloud. It is used to define the bagplot, a bivariate generalization of the boxplot.

For example, for any extreme point of the convex hull there is always a (closed) halfspace that contains only that point, and hence its Tukey depth as a fraction is 1/n.

Sample Tukey's depth of point x, or Tukey's depth of x with respect to the point cloud ${\displaystyle {\mathcal {X}}_{n}}$, is defined as

${\displaystyle D(x;{\mathcal {X}}_{n})=\inf _{v\in \mathbb {R} ^{d},\|v\|=1}{\frac {1}{n}}\sum _{i=1}^{n}\mathbf {1} \{v^{T}(X_{i}-x)\geq 0\},}$

where ${\displaystyle \mathbf {1} \{\cdot \}}$ is the indicator function that equals 1 if its argument holds true or 0 otherwise.

Population Tukey's depth of x wrt to a distribution ${\displaystyle P_{X}}$ is

${\displaystyle D(x;P_{X})=\inf _{v\in \mathbb {R} ^{d},\|v\|=1}P(v^{T}(X-x)\geq 0),}$

where X is a random variable following distribution ${\displaystyle P_{X}}$.

A centerpoint c of a point set of size n is nothing else but a point of Tukey depth of at least n/(d + 1).