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

Abstract

International audienceThe geometric median covariation matrix is a robust multivariate indicator of dispersion which can be extended without any difficulty to functional data. 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. 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 hours 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

Lp and almost sure rates of convergence of averaged stochastic gradient algorithms: locally strongly convex objective

“…The idea is to give non asymptotic results without focusing only on the rate of convergence in quadratic mean. Indeed, recent works (see Cardot and Godichon-Baggioni (2015) and Godichon-Baggioni (2016) for instance), confirm that having L p rates of convergence can be very useful to establish rates of convergence of more complex estimates.…”

“…The aim of this work is to seek inspiration in the demonstration methods introduced by Cardot et al (2017) and improved by Godichon-Baggioni (2016) and Cardot and Godichon-Baggioni (2015) to give convergence results for stochastic gradient algorithms and their averaged versions when the function we would like to minimize is only locally strongly convex. First, we establish almost sure rates of convergence of the estimates in general Hilbert spaces.…”

A Penalized Criterion for Selecting the Number of Clusters for K-Medians

“…In Figures 3 to 5, in order to visualize data points in dimensions higher than 3, we represent data as curves that we call "profiles", gathered it by cluster, and represented the centers of the groups in red. We also represent the 2 first principal components of the data using robust principal component analysis components (RPCA) (Cardot and Godichon-Baggioni, 2017). In Figure 3, we focus on the clustering obtained with K-medians algorithm ("Offline" version) for non contaminated data.…”

Robust functional principal components for sparse longitudinal data

“…Other robust scatter operator estimators have been proposed in the literature. For example, the ρ-scatter operator of Kraus and Panaretos [37], and the geometric median covariation studied in Cardot and Godichon-Baggioni [10]. However, the eigenfunctions of these operators are not guaranteed to remain in the same order as those of the true covariance operator when the latter exists, or of the true scatter operator for elliptical processes.…”