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Fast implementation of the Tukey depth

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

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References

  • Cuesta-Albertos J, Nieto-Reyes A (2008) The random Tukey depth. Comput Stat Data Anal 52:4979–4988

    Article  MATH  MathSciNet  Google Scholar 

  • Donoho DL, Gasko M (1992) Breakdown properties of location estimates based on halfspace depth and projected outlyingness. Ann Stat 20:1808–1827

    MATH  MathSciNet  Google Scholar 

  • Edelsbrunner H (1987) Algorithms in combinatorial geometry. Springer, Heidelberg

    Book  MATH  Google Scholar 

  • Hallin M, Paindaveine D, Šiman M (2010) Multivariate quantiles and multiple-output regression quantiles: from \(L_{1}\) optimization to halfspace depth. Ann Stat 38:635–669

    Article  MATH  Google Scholar 

  • Härdle W, Simar L (2007) Applied multivariate statistical analysis. Springer, Heidelberg

    MATH  Google Scholar 

  • Johnson T, Kwok I, Ng R (1998) Fast computation of 2-dimensional depth contours. In: Agrawal R, Stolorz P (eds) Proceedings of the 4th international conference on knowledge discovery and data mining. AAAI Press, New York, pp 224–228

  • Kong L, Mizera I (2012) Quantile tomography: using quantiles with multivariate data. Stat Sin 22:1589–1610

    MATH  MathSciNet  Google Scholar 

  • Kong L, Zuo Y (2010) Smooth depth contours characterize the underlying distribution. J Multivar Anal 101:2222–2226

    Article  MATH  MathSciNet  Google Scholar 

  • Koshevoy H, Mosler K (1997) Zonoid trimming for multivariate distributions. Ann Stat 25:1998–2017

    Article  MATH  MathSciNet  Google Scholar 

  • Li J, Cuesta-Albertos JA, Liu RY (2012) DD-classifier: nonparametric classification procedure based on DD-plot. J Am Stat Assoc 107(498):737–753

    Article  MATH  MathSciNet  Google Scholar 

  • Liu X, Zuo Y (2014a) Computing halfspace depth and regression depth. Commun Stat-Simul Comput 43:969–985

    Article  MATH  MathSciNet  Google Scholar 

  • Liu X, Zuo Y (2014b) Computing projection depth and its associated estimators. Stat Comput 24(1):51–63

    Article  MATH  MathSciNet  Google Scholar 

  • Mosler K (2013) Depth statistics. In Robustness and Complex Data Structures. Springer, Berlin, pp 17–34

  • Mosler K, Lange T, Bazovkin P (2009) Computing zonoid trimmed regions of dimension \(d > 2\). Comput Stat Data Anal 53:2500–2510

    Article  MATH  MathSciNet  Google Scholar 

  • Mozharovskyi P (2014) Contributions to depth-based classification and computation of the Tukey depth. PhD thesis. University of Cologne

  • Rousseeuw PJ, Ruts I (1996) Algorithm AS 307: bivariate location depth. J Appl Stat 45:516–526

    Article  MATH  Google Scholar 

  • Rousseeuw PJ, Struyf A (1998) Computing location depth and regression depth in higher dimensions. Stat Comput 8:193–203

    Article  Google Scholar 

  • Tsay RS (2010) Analysis of financial time series, 3rd edn. Wiley, Hoboken

    Book  MATH  Google Scholar 

  • Tukey JW (1975) Mathematics and the picturing of data. In Proceedings of the international congress of mathematicians on canadian mathematical congress, Montreal, pp 523–531

  • Yeh A, Singh K (1997) Balanced confidence regions based on Tukey’s depth and the bootstrap. J R Stat Soc Ser B 59:639–652

    Article  MATH  MathSciNet  Google Scholar 

  • Zuo YJ (2003) Projection based depth functions and associated medians. Ann Stat 31:1460–1490

    Article  MATH  MathSciNet  Google Scholar 

  • Zuo YJ, Serfling R (2000) General notions of statistical depth function. Ann Stat 28:461–482

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author thanks Prof. Mosler, K. and Dr. Mozharovskyi, P. for their valuable discussions during the preparation of this manuscript. The author also greatly appreciates two anonymous reviewers for their careful reading and insightful comments, which led to many improvements in this paper. This research is supported by NSFC of China (No. 11601197, 11461029, 71463020), NSF of Jiangxi Province (No. 20161BAB201024, 20151BAB211016), and the Key Science Fund Project of Jiangxi provincial education department (No. GJJ150439).

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Authors and Affiliations

  1. School of Statistics, Jiangxi University of Finance and Economics, Nanchang, 330013, Jiangxi, China

    Xiaohui Liu

  2. Research Center of Applied Statistics, Jiangxi University of Finance and Economics, Nanchang, 330013, Jiangxi, China

    Xiaohui Liu

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  1. Xiaohui Liu
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Correspondence to Xiaohui Liu.

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I am the sole author of this manuscript. This research involves no human participants and/or animals, and has no conflict of interest.

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Liu, X. Fast implementation of the Tukey depth. Comput Stat 32, 1395–1410 (2017). https://doi.org/10.1007/s00180-016-0697-8

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  • DOI: https://doi.org/10.1007/s00180-016-0697-8

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