Fast Estimation of the Median Covariation Matrix with Application to Online Robust Principal Components Analysis

TL;DR

This paper introduces recursive algorithms for fast, online estimation of the median covariation matrix, enabling robust principal components analysis in high-dimensional, large-scale data with applications to outlier detection.

Contribution

It proposes new recursive estimators for the median covariation matrix with proven asymptotic properties, suitable for online robust PCA in high-dimensional settings.

Findings

Recursive algorithms are computationally efficient for large data.

Robust PCA performs well compared to traditional methods.

Algorithms are implemented in the R package Gmedian.

Abstract

The 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.