IDEAS home Printed from https://ideas.repec.org/a/jss/jstsof/v047i13.html
   My bibliography  Save this article

An Exact Algorithm for Weighted-Mean Trimmed Regions in Any Dimension

Author

Listed:
  • Bazovkin, Pavel
  • Mosler, Karl

Abstract

Trimmed regions are a powerful tool of multivariate data analysis. They describe a probability distribution in Euclidean d-space regarding location, dispersion, and shape, and they order multivariate data with respect to their centrality. Dyckerhoff and Mosler (2011) have introduced the class of weighted-mean trimmed regions, which possess attractive properties regarding continuity, subadditivity, and monotonicity. We present an exact algorithm to compute the weighted-mean trimmed regions of a given data cloud in arbitrary dimension d. These trimmed regions are convex polytopes in Rd. To calculate them, the algorithm builds on methods from computational geometry. A characterization of a region’s facets is used, and information about the adjacency of the facets is extracted from the data. A key problem consists in ordering the facets. It is solved by the introduction of a tree-based order, by which the whole surface can be traversed efficiently with the minimal number of computations. The algorithm has been programmed in C++ and is available as the R package WMTregions.

Suggested Citation

  • Bazovkin, Pavel & Mosler, Karl, 2012. "An Exact Algorithm for Weighted-Mean Trimmed Regions in Any Dimension," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 47(i13).
    • Handle: RePEc:jss:jstsof:v:047:i13
    DOI: http://hdl.handle.net/10.18637/jss.v047.i13
    as

    Download full text from publisher

    File URL: https://www.jstatsoft.org/index.php/jss/article/view/v047i13/v47i13.pdf
    Download Restriction: no
    File URL: https://www.jstatsoft.org/index.php/jss/article/downloadSuppFile/v047i13/WMTregions_3.2.5.tar.gz
    Download Restriction: no
    File URL: https://www.jstatsoft.org/index.php/jss/article/downloadSuppFile/v047i13/v47i13.R
    Download Restriction: no
    File URL: https://www.jstatsoft.org/index.php/jss/article/downloadSuppFile/v047i13/Indices_0809.dat
    Download Restriction: no
    File URL: https://libkey.io/http://hdl.handle.net/10.18637/jss.v047.i13?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Lawrence, Michael & Temple Lang, Duncan, 2010. "RGtk2: A Graphical User Interface Toolkit for R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 37(i08).
    2. 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.
    3. Karl Mosler, 2003. "Central Regions and Dependency," Methodology and Computing in Applied Probability, Springer, vol. 5(1), pages 5-21, March.
    4. Ignacio Cascos & Ilya Molchanov, 2007. "Multivariate risks and depth-trimmed regions," Finance and Stochastics, Springer, vol. 11(3), pages 373-397, July.
    5. 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.
    6. Dyckerhoff, Rainer & Mosler, Karl, 2011. "Weighted-mean trimming of multivariate data," Journal of Multivariate Analysis, Elsevier, vol. 102(3), pages 405-421, March.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Liu, Xiaohui & Zuo, Yijun, 2015. "CompPD: A MATLAB Package for Computing Projection Depth," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 65(i02).
    2. Karl Mosler, 2023. "Representative endowments and uniform Gini orderings of multi-attribute welfare," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 21(1), pages 233-250, March.
    3. Karl Mosler, 2020. "Commentary on “From unidimensional to multidimensional inequality: a review”," METRON, Springer;Sapienza Università di Roma, vol. 78(1), pages 51-54, April.
    4. Liu, Xiaohui & Zuo, Yijun & Wang, Zhizhong, 2013. "Exactly computing bivariate projection depth contours and median," Computational Statistics & Data Analysis, Elsevier, vol. 60(C), pages 1-11.
    5. Pavel Bazovkin & Karl Mosler, 2015. "A general solution for robust linear programs with distortion risk constraints," Annals of Operations Research, Springer, vol. 229(1), pages 103-120, June.
    6. Wiechers, Christof, 2011. "Construction of uncertainty sets for portfolio selection problems," Discussion Papers in Econometrics and Statistics 4/11, University of Cologne, Institute of Econometrics and Statistics.
    7. Bazovkin, Pavel & Mosler, Karl, 2011. "Stochastic linear programming with a distortion risk constraint," Discussion Papers in Econometrics and Statistics 6/11, University of Cologne, Institute of Econometrics and Statistics.
    8. Bazovkin, Pavel, 2014. "Geometrical framework for robust portfolio optimization," Discussion Papers in Econometrics and Statistics 01/14, University of Cologne, Institute of Econometrics and Statistics.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Dyckerhoff, Rainer & Mozharovskyi, Pavlo, 2016. "Exact computation of the halfspace depth," Computational Statistics & Data Analysis, Elsevier, vol. 98(C), pages 19-30.
    2. Liu, Xiaohui & Rahman, Jafer & Luo, Shihua, 2019. "Generalized and robustified empirical depths for multivariate data," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 70-79.
    3. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    4. Bazovkin, Pavel, 2014. "Geometrical framework for robust portfolio optimization," Discussion Papers in Econometrics and Statistics 01/14, University of Cologne, Institute of Econometrics and Statistics.
    5. repec:spo:wpmain:info:hdl:2441/3qnaslliat80pbqa8t90240unj is not listed on IDEAS
    6. Ra'ul Torres & Rosa E. Lillo & Henry Laniado, 2015. "A Directional Multivariate Value at Risk," Papers 1502.00908, arXiv.org.
    7. Xiaohui Liu, 2017. "Fast implementation of the Tukey depth," Computational Statistics, Springer, vol. 32(4), pages 1395-1410, December.
    8. repec:hal:spmain:info:hdl:2441/3qnaslliat80pbqa8t90240unj is not listed on IDEAS
    9. Torres, Raúl & Lillo, Rosa E. & Laniado, Henry, 2015. "A directional multivariate value at risk," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 111-123.
    10. Chiara GIGLIARANO & Karl MOSLER, 2009. "Measuring middle-class decline in one and many attributes," Working Papers 333, Universita' Politecnica delle Marche (I), Dipartimento di Scienze Economiche e Sociali.
    11. Pavel Bazovkin & Karl Mosler, 2015. "A general solution for robust linear programs with distortion risk constraints," Annals of Operations Research, Springer, vol. 229(1), pages 103-120, June.
    12. Michele, Carlo de & Laniado Rodas, Henry, 2016. "Directional multivariate extremes in environmental phenomena," DES - Working Papers. Statistics and Econometrics. WS 23419, Universidad Carlos III de Madrid. Departamento de Estadística.
    13. Liu, Xiaohui & Zuo, Yijun, 2015. "CompPD: A MATLAB Package for Computing Projection Depth," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 65(i02).
    14. Hamel, Andreas H. & Kostner, Daniel, 2018. "Cone distribution functions and quantiles for multivariate random variables," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 97-113.
    15. Cascos, Ignacio & Ochoa, Maicol, 2021. "Expectile depth: Theory and computation for bivariate datasets," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    16. Xiaohui Liu & Shihua Luo & Yijun Zuo, 2020. "Some results on the computing of Tukey’s halfspace median," Statistical Papers, Springer, vol. 61(1), pages 303-316, February.
    17. Mia Hubert & Peter Rousseeuw & Pieter Segaert, 2015. "Multivariate functional outlier detection," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(2), pages 177-202, July.
    18. Zuo, Yijun & Lai, Shaoyong, 2011. "Exact computation of bivariate projection depth and the Stahel-Donoho estimator," Computational Statistics & Data Analysis, Elsevier, vol. 55(3), pages 1173-1179, March.
    19. Hemant Kulkarni & Jayabrata Biswas & Kiranmoy Das, 2019. "A joint quantile regression model for multiple longitudinal outcomes," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 103(4), pages 453-473, December.
    20. María Edo & Walter Sosa Escudero & Marcela Svarc, 2021. "A multidimensional approach to measuring the middle class," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 19(1), pages 139-162, March.
    21. Gleb A. Koshevoy & Karl Mosler, 2007. "Multivariate Lorenz dominance based on zonoids," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 91(1), pages 57-76, March.
    22. Paola Stolfi & Mauro Bernardi & Lea Petrella, 2016. "Multivariate Method Of Simulated Quantiles," Departmental Working Papers of Economics - University 'Roma Tre' 0212, Department of Economics - University Roma Tre.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:jss:jstsof:v:047:i13. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Christopher F. Baum (email available below). General contact details of provider: http://www.jstatsoft.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.