Fast estimation of the median covariation matrix with application to online robust principal components analysis
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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.
KeywordsFunctional data Geometric median \(L_1\)-median Recursive robust estimation Stochastic gradient
Mathematics Subject Classification62G05 62L20
We thank the two anonymous referees for their comments and suggestions that helped us to improve the presentation of the paper. We thank the company Médiamétrie for allowing us to illustrate our methodologies with their data. We also thank Dr. Peggy Cénac for a careful reading of the proofs.
- Bosq D (2000) Linear processes in function spaces, vol 149. Lecture notes in statistics, theory and applications, Springer, New York,Google Scholar
- Cardot, H, Cénac P, Godichon-Baggioni A (2016) Online estimation of the geometric median in Hilbert spaces: non asymptotic confidence balls. Ann Stat arXiv:1501.06930
- Cardot H, Degras D (2015) Online principal components analysis: which algorithm to choose? Tech Rep arXiv:1511.03688
- Kemperman JHB (1987) The median of a finite measure on a Banach space. In: Statistical data analysis based on the \(L_1\)-norm and related methods (Neuchâtel, 1987). North-Holland, Amsterdam, pp 217–230Google Scholar
- Maronna RA, Martin RD, Yohai VJ (2006) Robust statistics. Wiley series in probability and statistics, theory and methods. Wiley, ChichesterGoogle Scholar
- R Development Core Team (2010) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna ISBN 3-900051-07-0Google Scholar