Fast robust location and scatter estimation: a depth-based method

TL;DR

This paper introduces a depth-based algorithm called FDB for robust location and scatter estimation that is computationally efficient and maintains the robustness of traditional methods like MCD, especially suitable for high-dimensional data.

Contribution

The paper proposes a novel depth-based method, FDB, replacing the subset selection in MCD with a depth-induced trimmed region, improving efficiency while preserving robustness.

Findings

FDB is computationally more efficient than traditional MCD.

FDB achieves comparable robustness to MCD in high-dimensional settings.

Extensive simulations and real-data applications validate the effectiveness of FDB.

Abstract

The minimum covariance determinant (MCD) estimator is ubiquitous in multivariate analysis, the critical step of which is to select a subset of a given size with the lowest sample covariance determinant. The concentration step (C-step) is a common tool for subset-seeking; however, it becomes computationally demanding for high-dimensional data. To alleviate the challenge, we propose a depth-based algorithm, termed as \texttt{FDB}, which replaces the optimal subset with the trimmed region induced by statistical depth. We show that the depth-based region is consistent with the MCD-based subset under a specific class of depth notions, for instance, the projection depth. With the two suggested depths, the \texttt{FDB} estimator is not only computationally more efficient but also reaches the same level of robustness as the MCD estimator. Extensive simulation studies are conducted to assess the empirical performance of our estimators. We also validate the computational efficiency and robustness of our estimators under several typical tasks such as principal component analysis, linear discriminant analysis, image denoise and outlier detection on real-life datasets. A R package \textit{FDB} and potential extensions are available in the Supplementary Materials.