Robust self-tuning semiparametric PCA for contaminated elliptical distribution

TL;DR

This paper introduces a robust, self-tuning semiparametric PCA method that effectively handles contaminated elliptical distributions and outliers, outperforming traditional PCA and Tyler's M-estimator.

Contribution

It proposes a novel semiparametric PCA approach that is robust to both heavy-tailed and non-elliptical outliers, with a data-driven tuning procedure for adaptability.

Findings

The method is robust to heavy-tailed elliptical distributions.

It adapts to different outlier levels via a data-driven tuning.

Simulation and data analysis show its superior performance.

Abstract

Principal component analysis (PCA) is one of the most popular dimension reduction methods. The usual PCA is known to be sensitive to the presence of outliers, and thus many robust PCA methods have been developed. Among them, the Tyler's M-estimator is shown to be the most robust scatter estimator under the elliptical distribution. However, when the underlying distribution is contaminated and deviates from ellipticity, Tyler's M-estimator might not work well. In this article, we apply the semiparametric theory to propose a robust semiparametric PCA. The merits of our proposal are twofold. First, it is robust to heavy-tailed elliptical distributions as well as robust to non-elliptical outliers. Second, it pairs well with a data-driven tuning procedure, which is based on active ratio and can adapt to different degrees of data outlyingness. Theoretical properties are derived, including the influence functions for various statistical functionals and asymptotic normality. Simulation studies and a data analysis demonstrate the superiority of our method.