Tukey depth explained

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. Given a set of n points

l{X} _{n} = \{X _{1,...,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

Sample Tukey's depth of point x, or Tukey's depth of x with respect to the point cloud

l{X} _{n} , is defined as D(x; l{X} _{n)} = inf _{v\in R ^{d,} \|v \| =1} (1)/(n) \sum ^{n} _{i =1} 1 \{ v ^{T} (X _{i} - x) \ge 0 \},

where

1 \{ ⋅ \} 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} isD(x; P _{X)} = inf _{v\in 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).

See also

• Centerpoint (geometry)

Notes and References

1. Book: Tukey . John W . Mathematics and the Picturing of Data . 1975 . Proceedings of the International Congress of Mathematicians . 523-531.