5 research outputs found
Fast implementation of the Tukey depth
Tukey depth function is one of the most famous multivariate tools serving
robust purposes. It is also very well known for its computability problems in
dimensions . In this paper, we address this computing issue by
presenting two combinatorial algorithms. The first is naive and calculates the
Tukey depth of a single point with complexity ,
while the second further utilizes the quasiconcave of the Tukey depth function
and hence is more efficient than the first. Both require very minimal memory
and run much faster than the existing ones. All experiments indicate that they
compute the exact Tukey depth.Comment: 16 pages, 13 figure
Fast computation of Tukey trimmed regions and median in dimension
Given data in , a Tukey -trimmed region is the set of
all points that have at least Tukey depth w.r.t. the data. As they are
visual, affine equivariant and robust, Tukey regions are useful tools in
nonparametric multivariate analysis. While these regions are easily defined and
interpreted, their practical use in applications has been impeded so far by the
lack of efficient computational procedures in dimension . We construct
two novel algorithms to compute a Tukey -trimmed region, a na\"{i}ve
one and a more sophisticated one that is much faster than known algorithms.
Further, a strict bound on the number of facets of a Tukey region is derived.
In a large simulation study the novel fast algorithm is compared with the
na\"{i}ve one, which is slower and by construction exact, yielding in every
case the same correct results. Finally, the approach is extended to an
algorithm that calculates the innermost Tukey region and its barycenter, the
Tukey median