# Tukey depth

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Computational geometry concept

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In [statistics](/source/Statistics) and [computational geometry](/source/Computational_geometry), the **Tukey depth** [1] 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](/source/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](/source/Half-space_(geometry)) 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](/source/Bagplot), a bivariate generalization of the [boxplot](/source/Boxplot).

For example, for any extreme point of the [convex hull](/source/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

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.

*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](/source/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)](/source/Centerpoint_(geometry))

## References

1. **[^](#cite_ref-1)** Tukey, John W (1975). *Mathematics and the Picturing of Data*. Proceedings of the International Congress of Mathematicians. p. 523-531.

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