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Fast estimation of the median covariation matrix with application to online robust principal components analysis

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  • Hervé Cardot

    (Université de Bourgogne Franche-Comté)

  • Antoine Godichon-Baggioni

    (Université de Bourgogne Franche-Comté)

Abstract

The geometric median covariation matrix is a robust multivariate indicator of dispersion which can be extended to infinite dimensional spaces. We define estimators, based on recursive algorithms, that can be simply updated at each new observation and are able to deal rapidly with large samples of high-dimensional data without being obliged to store all the data in memory. Asymptotic convergence properties of the recursive algorithms are studied under weak conditions in general separable Hilbert spaces. The computation of the principal components can also be performed online and this approach can be useful for online outlier detection. A simulation study clearly shows that this robust indicator is a competitive alternative to minimum covariance determinant when the dimension of the data is small and robust principal components analysis based on projection pursuit and spherical projections for high-dimension data. An illustration on a large sample and high-dimensional dataset consisting of individual TV audiences measured at a minute scale over a period of 24 h confirms the interest of considering the robust principal components analysis based on the median covariation matrix. All studied algorithms are available in the R package Gmedian on CRAN.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:testjl:v:26:y:2017:i:3:d:10.1007_s11749-016-0519-x
    DOI: 10.1007/s11749-016-0519-x
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    References listed on IDEAS

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    1. 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.
    2. Daniel Gervini, 2008. "Robust functional estimation using the median and spherical principal components," Biometrika, Biometrika Trust, vol. 95(3), pages 587-600.
    3. 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.
    4. David Kraus & Victor M. Panaretos, 2012. "Dispersion operators and resistant second-order functional data analysis," Biometrika, Biometrika Trust, vol. 99(4), pages 813-832.
    5. Cardot, Hervé & Cénac, Peggy & Monnez, Jean-Marie, 2012. "A fast and recursive algorithm for clustering large datasets with k-medians," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1434-1449.
    6. Hyndman, Rob J. & Shahid Ullah, Md., 2007. "Robust forecasting of mortality and fertility rates: A functional data approach," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4942-4956, June.
    7. Croux, Christophe & Ruiz-Gazen, Anne, 2005. "High breakdown estimators for principal components: the projection-pursuit approach revisited," Journal of Multivariate Analysis, Elsevier, vol. 95(1), pages 206-226, July.
    8. Godichon-Baggioni, Antoine, 2016. "Estimating the geometric median in Hilbert spaces with stochastic gradient algorithms: Lp and almost sure rates of convergence," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 209-222.
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    Cited by:

    1. Graciela Boente & Matías Salibián-Barrera, 2021. "Robust functional principal components for sparse longitudinal data," METRON, Springer;Sapienza Università di Roma, vol. 79(2), pages 159-188, August.

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