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Computing halfspace depth contours based on the idea of a circular sequence

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Abstract

This paper presents a new efficient algorithm for exactly computing the halfspace depth contours based on the idea of a circular sequence. Unlike the existing methods, the proposed algorithm segments the unit sphere directly relying on the permutations that correspond to the projections of observations onto some unit directions, without having to use the technique of parametric programming. Some data examples are also provided to illustrate the performance of the proposed algorithm.

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References

  1. Donoho D and Gasko M, Breakdown properties of location estimates based on halfspace depth and projected outlyingness, Ann. Statist., 1992, 20: 1808–3.

    MathSciNet  Google Scholar 

  2. Tyler D, Finite sample breakdown points of projection based multivariate location and scatter statistics, Ann. Statist., 1994, 22: 1024–3.

    Article  MATH  MathSciNet  Google Scholar 

  3. Bai Z and He X, Asymptotic distributions of the maximal depth estimators for regression and multivariate location, Ann. Statist., 1999, 27: 1617–3.

    MathSciNet  Google Scholar 

  4. Zuo Y and Serfling R, General notions of statistical depth function, Ann. Statist., 2000, 28: 461–3.

    Article  MATH  MathSciNet  Google Scholar 

  5. Kong L and Mizera I, Quantile tomography: Using quantiles with multivariate data, Statist. Sinica, 2012, 22: 1589–3.

    MATH  MathSciNet  Google Scholar 

  6. Hallin M, Paindaveine D and Šiman M, Multivariate quantiles and multiple-output regression quantiles: From L 1 optimization to halfspace depth, Ann. Statist., 2010, 38: 635–3.

    Article  MATH  MathSciNet  Google Scholar 

  7. Paindaveine D and Šiman M, On directional multiple-output quantile regression, J. Multivariate Anal., 2011, 102: 193–3.

    Article  MATH  MathSciNet  Google Scholar 

  8. Tukey J, Mathematics and the picturing of data, Proceedings of the International Congress of Mathematicians, 523–531, Canada Mathematical Congress, Montreal, 1975.

    Google Scholar 

  9. Liu R, On a notion of data depth based on random simplices, Ann. Statist., 1990, 18: 191–3.

    Article  MathSciNet  Google Scholar 

  10. Koshevoy H and Mosler K, Zonoid trimming for multivariate distributions, Ann. Statist., 1997, 25: 1998–3.

    Article  MATH  MathSciNet  Google Scholar 

  11. Rousseeuw P and Hubert M, Regression depth (with discussion), J. Amer. Statist. Assoc., 1999, 94: 388–3.

    Article  MATH  MathSciNet  Google Scholar 

  12. Zuo Y, Projection based depth functions and associated medians, Ann. Statist., 2003, 31: 1460–3.

    Article  MATH  MathSciNet  Google Scholar 

  13. Mosler K, Depth statistics, Robustness and Complex Data Structures, Springer Berlin Heidelberg, 2013, 17–2.

    Chapter  Google Scholar 

  14. Ghosh A and Chaudhuri P, On maximum depth and related classifiers, Scand. J. Statist., 2005, 32: 328–3.

    Article  MathSciNet  Google Scholar 

  15. Yeh A and Singh K, Balanced confidence regions based on Tukey’s depth and the bootstrap, J. Roy. Statist. Soc. Ser. B, 1997, 59: 639–3.

    Article  MATH  MathSciNet  Google Scholar 

  16. Chenouri S and Small C, A nonparametric multivariate multisample test based on data depth, Electronic J. Statist., 2012, 6: 760–3.

    Article  MATH  MathSciNet  Google Scholar 

  17. Rousseeuw P and Ruts I, Algorithm AS 307: Bivariate location depth, J. Roy. Statist. Soc. Ser. C, 1996, 45: 516–3.

    MATH  Google Scholar 

  18. Rousseeuw P and Struyf A, Computing location depth and regression depth in higher dimensions, Statist. Comput., 1998, 8: 193–3.

    Article  Google Scholar 

  19. Ruts I and Rousseeuw P, Computing depth contours of bivariate point clouds, Comput. Statist. Data Anal., 1996, 23: 153–3.

    Article  MATH  Google Scholar 

  20. Paindaveine D and Šiman M, Computing multiple-output regression quantile regions, Comput. Statist. Data Anal., 2012a, 56: 840–3.

    Article  MATH  MathSciNet  Google Scholar 

  21. Paindaveine D and Šiman M, Computing multiple-output regression quantile regions from projection quantiles, Comput. Statist., 2012b, 27: 29–3.

    Article  MATH  MathSciNet  Google Scholar 

  22. Edelsbrunner H, Algorithms in Combinatorial Geometry, Springer, Heidelberg}, 1987.

    Book  MATH  Google Scholar 

  23. Dyckerhoff R, Computing zonoid trimmed regions of bivariate data sets, COMPSTAT 2000, Proceedings in Comput. Statist., Ed. by Bethlehem J and van der Heijden P, Physica, Heidelberg, 2000, 295–2.

    Google Scholar 

  24. Cascos I, The expected convex hull trimmed regions of a sample, Comput. Statist., 2007, 22: 557–3.

    Article  MathSciNet  Google Scholar 

  25. Mosler K, Lange T, and Bazovkin P, Computing zonoid trimmed regions of dimension d > 2, Comput. Statist. Data Anal., 2009, 53: 2500–3.

    Article  MATH  MathSciNet  Google Scholar 

  26. Floyd R and Rivest R, ‘Algorithm 489: Select’, Communications of the ACM 1975, 18: 173–3.

    Article  Google Scholar 

  27. Barber C, Dobkin D, and Huhdanpaa H, The quickhull algorithm for convex hulls, ACM Transactions Math. Software, 1996, 22: 469–3.

    Article  MATH  MathSciNet  Google Scholar 

  28. Bremner D, Fukuda K, and Marzetta A, Primal-dual methods for vertex and facet enumeration, Discrete Comput. Geometry, 1998, 20: 333–3.

    Article  MATH  MathSciNet  Google Scholar 

  29. Rousseeuw P and Leroy A, Robust Regression and Outlier Detection, Wiley New York, 1987.

    Book  MATH  Google Scholar 

  30. Liu X, Zuo Y, and Wang Z, Exactly computing bivariate projection depth median and contours, Comput. Statist. Data Anal., 2013, 60: 1–3.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    Xiaohui Liu

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

    Xiaohui Liu

  3. School of Software, Jiangxi University of Science and Technology, Nanchang, 330013, China

    Haiping Ren

  4. School of Mathematics and Statistics, Central South University, Changsha, 417100, China

    Guofu Wang

Authors
  1. Xiaohui Liu
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  2. Haiping Ren
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  3. Guofu Wang
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Corresponding author

Correspondence to Guofu Wang.

Additional information

This research was supported by the National Natural Science Foundation of China under Grant No. 11461029, the Natural Science Foundation of Jiangxi Province under Grant Nos. 20142BAB211014, 20132BAB211015, 20122BAB201023, 20133BCB23014, and the Youth Science Fund Project of Jiangxi provincial education department under Grant Nos. GJJ14350, GJJ14449, KJLD13033.

This paper was recommended for publication by Editor ZOU Guohua.

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Liu, X., Ren, H. & Wang, G. Computing halfspace depth contours based on the idea of a circular sequence. J Syst Sci Complex 28, 1399–1411 (2015). https://doi.org/10.1007/s11424-015-3160-y

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  • DOI: https://doi.org/10.1007/s11424-015-3160-y

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