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Robustifying principal component analysis with spatial sign vectors

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  • Taskinen, Sara
  • Koch, Inge
  • Oja, Hannu
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    Abstract

    In this paper, we apply orthogonally equivariant spatial sign covariance matrices as well as their affine equivariant counterparts in principal component analysis. The influence functions and asymptotic covariance matrices of eigenvectors based on robust covariance estimators are derived in order to compare the robustness and efficiency properties. We show in particular that the estimators that use pairwise differences of the observed data have very good efficiency properties, providing practical robust alternatives to classical sample covariance matrix based methods.

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    File URL: http://www.sciencedirect.com/science/article/pii/S0167715212000028
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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 82 (2012)
    Issue (Month): 4 ()
    Pages: 765-774

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    Handle: RePEc:eee:stapro:v:82:y:2012:i:4:p:765-774

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    Related research

    Keywords: Affine equivariance; Efficiency; Influence function; Robustness; Spatial sign vector;

    References

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    1. Frahm, Gabriel, 2009. "Asymptotic distributions of robust shape matrices and scales," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1329-1337, August.
    2. Lutz Dümbgen & David E. Tyler, 2005. "On the Breakdown Properties of Some Multivariate M-Functionals," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 32(2), pages 247-264.
    3. Klaus Nordhausen & Hannu Oja, . "Multivariate L1 Statistical Methods: The Package MNM," Journal of Statistical Software, American Statistical Association, vol. 43(i05).
    4. Taskinen, Sara & Croux, Christophe & Kankainen, Annaliisa & Ollila, Esa & Oja, Hannu, 2006. "Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 359-384, February.
    5. Marden, John I., 1999. "Some robust estimates of principal components," Statistics & Probability Letters, Elsevier, vol. 43(4), pages 349-359, July.
    6. Tyler, David E., 2010. "A note on multivariate location and scatter statistics for sparse data sets," Statistics & Probability Letters, Elsevier, vol. 80(17-18), pages 1409-1413, September.
    7. Daniel Gervini, 2008. "Robust functional estimation using the median and spherical principal components," Biometrika, Biometrika Trust, vol. 95(3), pages 587-600.
    8. Sirkiä, Seija & Taskinen, Sara & Oja, Hannu, 2007. "Symmetrised M-estimators of multivariate scatter," Journal of Multivariate Analysis, Elsevier, vol. 98(8), pages 1611-1629, September.
    9. Paindaveine, Davy, 2008. "A canonical definition of shape," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2240-2247, October.
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