Fast computation of Tukey trimmed regions and median in dimension p > 2

Xiaohui Liu (), Karl Mosler and Pavlo Mozharovskyi ()
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Xiaohui Liu: School of Statistics, Research Center of Applied Statistics of Jiangxi University of Finance and Economics
Pavlo Mozharovskyi: CREST-ENSAI, Université Bretagne Loire

No 2017-71, Working Papers from Center for Research in Economics and Statistics

Abstract: Given data in Rp, a Tukey K-trimmed region, shortly Tukey K-region or just Tukey region, is the set of all points that have at least Tukey depth K 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 application is impeded by the lack of efficient computational procedures in dimension p > 2. We derive a strict bound on the number of facets of a Tukey region and construct a new efficient algorithm to compute the region, which runs much faster than existing ones. The new algorithm is compared with a slower exact algorithm, yielding always the same correct results. Finally, the approach is extended to an algorithm that efficiently calculates the innermost Tukey region and its barycenter, the Tukey median.

Keywords: Tukey depth; Tukey median; halfspace depth; location depth; depth contours; depth regions; computational geometry; efficient implementation (search for similar items in EconPapers)
Pages: 24 pages
Date: 2017-06-12
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