Approximate computation of projection depths

Abstract

Data depth is a concept in multivariate statistics that measures the centrality of a point in a given data cloud in R d . If the depth of a point can be represented as the minimum of the depths with respect to all one-dimensional projections of the data, then the depth satisfies the so-called projection property. Such depths form an important class that includes many of the depths that have been proposed in literature. For depths that satisfy the projection property an approximate algorithm can easily be constructed since taking the minimum of the depths with respect to only a finite number of one-dimensional projections yields an upper bound for the depth with respect to the multivariate data. Such an algorithm is particularly useful if no exact algorithm exists or if the exact algorithm has a high computational complexity, as is the case with the halfspace depth or the projection depth. To compute these depths in high dimensions, the use of an approximate algorithm with better complexity is surely preferable. Instead of focusing on a single method we provide a comprehensive and fair comparison of several methods, both already described in the literature and original.

Uniform convergence rates for the approximated halfspace and projection depth

“…Therefore, the general guidelines for the choice of n are relevant also in the situation when the sample depth is approximated. For instance, if the theoretical bound in Table 1 is already too high for the practical application in mind, the simple approximation of the halfspace depth is certainly not a good idea, and more sophisticated methodologies must be employed; for an empirical comparison of several of such approximation procedures we refer to [16]. In any case, for the sample depth it must be kept in mind that according to the first example of Section 4.…”