Robust estimation of principal components from depth-based multivariate rank covariance matrix

TL;DR

This paper introduces a robust and efficient PCA method using depth-based spatial ranks to estimate the eigenvectors of a new Depth Covariance Matrix, improving robustness and efficiency over existing methods.

Contribution

It proposes a novel depth-based covariance matrix for PCA, deriving its asymptotic properties and demonstrating superior robustness and efficiency compared to traditional and existing robust PCA methods.

Findings

The DCM-based PCA is robust to deviations from normality.

It outperforms SCM and Tyler's shape matrix in efficiency.

Effective in high-dimensional data analysis and outlier detection.

Abstract

Analyzing principal components for multivariate data from its spatial sign covariance matrix (SCM) has been proposed as a computationally simple and robust alternative to normal PCA, but it suffers from poor efficiency properties and is actually inadmissible with respect to the maximum likelihood estimator. Here we use data depth-based spatial ranks in place of spatial signs to obtain the orthogonally equivariant Depth Covariance Matrix (DCM) and use its eigenvector estimates for PCA. We derive asymptotic properties of the sample DCM and influence functions of its eigenvectors. The shapes of these influence functions indicate robustness of estimated principal components, and good efficiency properties compared to the SCM. Finite sample simulation studies show that principal components of the sample DCM are robust with respect to deviations from normality, as well as are more efficient than the SCM and its affine equivariant version, Tyler's shape matrix. Through two real data examples, we also show the effectiveness of DCM-based PCA in analyzing high-dimensional data and outlier detection, and compare it with other methods of robust PCA.