Fast Computation of Tukey Trimmed Regions and Median in Dimension p > 2. Issue 3 (3rd July 2019)

• Record Type: Journal Article
• Authors: Liu, Xiaohui; Mosler, Karl; Mozharovskyi, Pavlo
• Is Part Of: Journal of computational and graphical statistics. Volume 28:Issue 3(2019)
• Page Start: 682
• Page End: 697
• Publication Date: 2019-07-03
• Subjects: Computational geometry -- Depth contours -- Depth regions -- Halfspace depth -- Location depth -- R-package TukeyRegion -- Tukey depth -- Tukey median; Mathematical statistics -- Data processing -- Periodicals; Mathematical statistics -- Graphic methods -- Periodicals; 519.50285
• DOI: 10.1080/10618600.2018.1546595
• Languages: English
• ISSNs: 1061-8600

Abstract

ABSTRACT: Given data in R p, a Tukey κ -trimmed region is the set of all points that have at least Tukey depth κ 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 p > 2. We construct two novel algorithms to compute a Tukey κ -trimmed region, a naïve 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 naïve 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. Supplementary materials for this article are available online.