Tukey depth

In statistics and computational geometry, the Tukey depth {{cite book |last1=Tukey |first1=John W |title=Mathematics and the Picturing of Data |date=1975 |publisher=Proceedings of the International Congress of Mathematicians |page=523-531}} 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 $\mathcal{X}_n = \{X_1,\dots,X_n\}$ 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.

Definitions

File:Tukey's halfspace depth.pdf

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

$D(x;\mathcal{X}_n) = \inf_{v\in\mathbb{R}^d, \|v \|=1} \frac{1}{n}\sum_{i=1}^n \mathbf{1}\{ v^T (X_i - x) \ge 0\},$

where $\mathbf{1}\{\cdot\}$ 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 $P_X$ is

$D(x; P_X) = \inf_{v\in\mathbb{R}^d, \|v \|=1} P(v^T (X - x) \ge 0),$

where X is a random variable following distribution $P_X$ .

Tukey mean and relation to centerpoint

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).

References

Category:Computational geometry

Tukey depth

Definitions

Tukey mean and relation to centerpoint

See also

References