Tukey depth

In statistics and computational geometry, the Tukey depth [1] is a measure of the depth of a point in a fixed set of points.

The concept is named after its inventor, John Tukey.

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

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

where X is a random variable following distribution

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).

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Tukey's depth of a point x wrt to a point cloud. The blue region illustrates a halfspace containing x on the boundary. The halfspace is also a most extreme one so that it contains x but as few observations in the point cloud as possible. Thus, the proportion of points contained in this halfspace becomes the value of Tukey's depth for x.