Data depths satisfying the projection property

Abstract

Data depth is a concept that measures the centrality of a point in a given data cloud x 1, x 2,...,x n ∈ ℝ or in a multivariate distribution P X on ℝ d d . Every depth defines a family of so–called trimmed regions. The α–trimmed region is given by the set of points that have a depth of at least α. Data depth has been used to define multivariate measures of location and dispersion as well as multivariate dispersion orders. If the depth of a point can be represented as the minimum of the depths with respect to all unidimensional projections, we say that the depth satisfies the (weak) projection property. Many depths which have been proposed in the literature can be shown to satisfy the weak projection property. A depth is said to satisfy the strong projection property if for every α the unidimensional projection of the α–trimmed region equals the α–trimmed region of the projected distribution. After a short introduction into the general concept of data depth we formally define the weak and the strong projection property and give necessary and sufficient criteria for the projection property to hold. We further show that the projection property facilitates the construction of depths from univariate trimmed regions. We discuss some of the depths proposed in the literature which possess the projection property and define a general class of projection depths, which are constructed from univariate trimmed regions by using the above method. Finally, algorithmic aspects of projection depths are discussed. We describe an algorithm which enables the approximate computation of depths that satisfy the projection property.