IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v171y2019icp94-111.html
   My bibliography  Save this article

A generalized spatial sign covariance matrix

Author

Listed:
  • Raymaekers, Jakob
  • Rousseeuw, Peter

Abstract

The well-known spatial sign covariance matrix (SSCM) carries out a radial transform which moves all data points to a sphere, followed by computing the classical covariance matrix of the transformed data. Its popularity stems from its robustness to outliers, fast computation, and applications to correlation and principal component analysis. In this paper we study more general radial functions. It is shown that the eigenvectors of the generalized SSCM are still consistent and the ranks of the eigenvalues are preserved. The influence function of the resulting scatter matrix is derived, and it is shown that its asymptotic breakdown value is as high as that of the original SSCM. A simulation study indicates that the best results are obtained when the inner half of the data points are not transformed and points lying far away are moved to the center.

Suggested Citation

  • Raymaekers, Jakob & Rousseeuw, Peter, 2019. "A generalized spatial sign covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 94-111.
    • Handle: RePEc:eee:jmvana:v:171:y:2019:i:c:p:94-111
    DOI: 10.1016/j.jmva.2018.11.010
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X18302410
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2018.11.010?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
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. C. Croux & C. Dehon & A. Yadine, 2010. "The k-step spatial sign covariance matrix," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 4(2), pages 137-150, September.
    2. Andrew F. Magyar & David E. Tyler, 2014. "The asymptotic inadmissibility of the spatial sign covariance matrix for elliptically symmetric distributions," Biometrika, Biometrika Trust, vol. 101(3), pages 673-688.
    3. Marden, John I., 1999. "Some robust estimates of principal components," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 349-359, July.
    4. N. Locantore & J. Marron & D. Simpson & N. Tripoli & J. Zhang & K. Cohen & Graciela Boente & Ricardo Fraiman & Babette Brumback & Christophe Croux & Jianqing Fan & Alois Kneip & John Marden & Daniel P, 1999. "Robust principal component analysis for functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 8(1), pages 1-73, June.
    5. Dürre, Alexander & Tyler, David E. & Vogel, Daniel, 2016. "On the eigenvalues of the spatial sign covariance matrix in more than two dimensions," Statistics & Probability Letters, Elsevier, vol. 111(C), pages 80-85.
    6. Taskinen, Sara & Koch, Inge & Oja, Hannu, 2012. "Robustifying principal component analysis with spatial sign vectors," Statistics & Probability Letters, Elsevier, vol. 82(4), pages 765-774.
    7. Dürre, Alexander & Vogel, Daniel & Tyler, David E., 2014. "The spatial sign covariance matrix with unknown location," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 107-117.
    8. C. Chatzinakos & L. Pitsoulis & G. Zioutas, 2016. "Optimization techniques for robust multivariate location and scatter estimation," Journal of Combinatorial Optimization, Springer, vol. 31(4), pages 1443-1460, May.
    9. Pavlo Mozharovskyi & Karl Mosler & Tatjana Lange, 2015. "Classifying real-world data with the $${ DD}\alpha $$ D D α -procedure," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 9(3), pages 287-314, September.
    10. Dürre, Alexander & Vogel, Daniel, 2016. "Asymptotics of the two-stage spatial sign correlation," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 54-67.
    11. Seija Sirkiä & Sara Taskinen & Hannu Oja & David Tyler, 2009. "Tests and estimates of shape based on spatial signs and ranks," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(2), pages 155-176.
    12. Dürre, Alexander & Vogel, Daniel & Fried, Roland, 2015. "Spatial sign correlation," Journal of Multivariate Analysis, Elsevier, vol. 135(C), pages 89-105.
    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. Majumdar, Subhabrata & Chatterjee, Snigdhansu, 2022. "On weighted multivariate sign functions," Journal of Multivariate Analysis, Elsevier, vol. 191(C).
    2. Bernard, Gaspard & Verdebout, Thomas, 2024. "On some multivariate sign tests for scatter matrix eigenvalues," Econometrics and Statistics, Elsevier, vol. 29(C), pages 252-260.

    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. Dürre, Alexander & Tyler, David E. & Vogel, Daniel, 2016. "On the eigenvalues of the spatial sign covariance matrix in more than two dimensions," Statistics & Probability Letters, Elsevier, vol. 111(C), pages 80-85.
    2. Majumdar, Subhabrata & Chatterjee, Snigdhansu, 2022. "On weighted multivariate sign functions," Journal of Multivariate Analysis, Elsevier, vol. 191(C).
    3. Xu, Yangchang & Xia, Ningning, 2023. "On the eigenvectors of large-dimensional sample spatial sign covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    4. Dürre, Alexander & Vogel, Daniel & Fried, Roland, 2015. "Spatial sign correlation," Journal of Multivariate Analysis, Elsevier, vol. 135(C), pages 89-105.
    5. Dürre, Alexander & Vogel, Daniel, 2016. "Asymptotics of the two-stage spatial sign correlation," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 54-67.
    6. Guangxing Wang & Sisheng Liu & Fang Han & Chong‐Zhi Di, 2023. "Robust functional principal component analysis via a functional pairwise spatial sign operator," Biometrics, The International Biometric Society, vol. 79(2), pages 1239-1253, June.
    7. Dürre, Alexander & Vogel, Daniel & Tyler, David E., 2014. "The spatial sign covariance matrix with unknown location," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 107-117.
    8. C. Croux & C. Dehon & A. Yadine, 2010. "The k-step spatial sign covariance matrix," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 4(2), pages 137-150, September.
    9. Boente, Graciela & Rodriguez, Daniela & Sued, Mariela, 2019. "The spatial sign covariance operator: Asymptotic results and applications," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 115-128.
    10. Davy Paindaveine & Julien Remy & Thomas Verdebout, 2019. "Sign Tests for Weak Principal Directions," Working Papers ECARES 2019-01, ULB -- Universite Libre de Bruxelles.
    11. Bernard, Gaspard & Verdebout, Thomas, 2024. "On some multivariate sign tests for scatter matrix eigenvalues," Econometrics and Statistics, Elsevier, vol. 29(C), pages 252-260.
    12. J. L. Scealy & Patrice de Caritat & Eric C. Grunsky & Michail T. Tsagris & A. H. Welsh, 2015. "Robust Principal Component Analysis for Power Transformed Compositional Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 136-148, March.
    13. Joni Virta & Niko Lietzén & Henri Nyberg, 2024. "Robust signal dimension estimation via SURE," Statistical Papers, Springer, vol. 65(5), pages 3007-3038, July.
    14. Seija Sirkiä & Sara Taskinen & Hannu Oja & David Tyler, 2009. "Tests and estimates of shape based on spatial signs and ranks," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(2), pages 155-176.
    15. Debruyne, Michiel & Hubert, Mia & Van Horebeek, Johan, 2010. "Detecting influential observations in Kernel PCA," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3007-3019, December.
    16. Pesonen, Maiju & Pesonen, Henri & Nevalainen, Jaakko, 2015. "Covariance matrix estimation for left-censored data," Computational Statistics & Data Analysis, Elsevier, vol. 92(C), pages 13-25.
    17. Michiel Debruyne & Tim Verdonck, 2010. "Robust kernel principal component analysis and classification," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 4(2), pages 151-167, September.
    18. Hervé Cardot & Antoine Godichon-Baggioni, 2017. "Fast estimation of the median covariation matrix with application to online robust principal components analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(3), pages 461-480, September.
    19. Taskinen, Sara & Koch, Inge & Oja, Hannu, 2012. "Robustifying principal component analysis with spatial sign vectors," Statistics & Probability Letters, Elsevier, vol. 82(4), pages 765-774.
    20. Italo R. Lima & Guanqun Cao & Nedret Billor, 2019. "M-based simultaneous inference for the mean function of functional data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(3), pages 577-598, June.

    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:eee:jmvana:v:171:y:2019:i:c:p:94-111. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    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.