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 X n = { X 1 , … , X n } {\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.
Definitions
Sample Tukey's depth of point x, or Tukey's depth of x with respect to the point cloud X n {\displaystyle {\mathcal {X}}_{n}} , is defined as
D ( x ; X n ) = inf v ∈ R d , ‖ v ‖ = 1 1 n ∑ i = 1 n 1 { v T ( X i − x ) ≥ 0 } , {\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 1 { ⋅ } {\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 P X {\displaystyle P_{X}} is
D ( x ; P X ) = inf v ∈ R d , ‖ v ‖ = 1 P ( v T ( X − x ) ≥ 0 ) , {\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 P X {\displaystyle 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)
References