Tukey depth
In statistics and computational geometry, the Tukey depth 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 in d-dimensional space, Tukey's depth of a point x is the smallest fraction 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 halfspace that contains only that point, and hence its Tukey depth as a fraction is 1/n.
Definitions
Sample Tukey's depth of point x, or Tukey's depth of x with respect to the point cloud, is defined aswhere 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 is
where X is a random variable following distribution.