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

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.