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Fast implementation of the Tukey depth

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  • Xiaohui Liu

    (Jiangxi University of Finance and Economics
    Jiangxi University of Finance and Economics)

Abstract

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 $$p \ge 3$$ p ≥ 3 . 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 $$O\left( n^{p-1}\log (n)\right) $$ O n p - 1 log ( n ) , 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.

Suggested Citation

  • Xiaohui Liu, 2017. "Fast implementation of the Tukey depth," Computational Statistics, Springer, vol. 32(4), pages 1395-1410, December.
    • Handle: RePEc:spr:compst:v:32:y:2017:i:4:d:10.1007_s00180-016-0697-8
    DOI: 10.1007/s00180-016-0697-8
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    References listed on IDEAS

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    1. Marc Hallin & Davy Paindaveine & Miroslav Siman, 2008. "Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth," Working Papers ECARES 2008_042, ULB -- Universite Libre de Bruxelles.
    2. Jun Li & Juan A. Cuesta-Albertos & Regina Y. Liu, 2012. "DD -Classifier: Nonparametric Classification Procedure Based on DD -Plot," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(498), pages 737-753, June.
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    4. Mosler, Karl & Lange, Tatjana & Bazovkin, Pavel, 2009. "Computing zonoid trimmed regions of dimension d>2," Computational Statistics & Data Analysis, Elsevier, vol. 53(7), pages 2500-2510, May.
    5. Kong, Linglong & Zuo, Yijun, 2010. "Smooth depth contours characterize the underlying distribution," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2222-2226, October.
    6. Cuesta-Albertos, J.A. & Nieto-Reyes, A., 2008. "The random Tukey depth," Computational Statistics & Data Analysis, Elsevier, vol. 52(11), pages 4979-4988, July.
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    Cited by:

    1. Xiaohui Liu & Karl Mosler & Pavlo Mozharovskyi, 2017. "Fast computation of Tukey trimmed regions and median in dimension p > 2," Working Papers 2017-71, Center for Research in Economics and Statistics.
    2. Wei Shao & Yijun Zuo, 2020. "Computing the halfspace depth with multiple try algorithm and simulated annealing algorithm," Computational Statistics, Springer, vol. 35(1), pages 203-226, March.
    3. Hamel, Andreas H. & Kostner, Daniel, 2022. "Computation of quantile sets for bivariate ordered data," Computational Statistics & Data Analysis, Elsevier, vol. 169(C).
    4. Cerdeira, J. Orestes & Silva, Pedro C., 2021. "A centrality notion for graphs based on Tukey depth," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    5. Ramsay, Kelly & Durocher, Stéphane & Leblanc, Alexandre, 2019. "Integrated rank-weighted depth," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 51-69.

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