Fast Robust Location and Scatter Estimation: A Depth-based Method

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 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 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. An R package FDB and additional results are available in the supplementary materials.

KEYWORDS:

• Computational efficiency
• High-dimensional data
• Outliers
• Robustness
• Statistical depth

Supplementary Materials

• We provide an R-package named FDB and R codes of the FDB algorithm proposed in this article.

• Our narrative supplement provides proofs of theoretical results, additional simulation results as well as preliminary explorations of potential extensions.

Acknowledgments

We are very grateful to three anonymous referees, an associate editor, and the Editor for their valuable comments that have greatly improved the manuscript. The first two authors contribute equally to the paper.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

The research was supported by the National Natural Science Foundation of China (grant no. NSFC 11901573, 12171033) and the Beijing Natural Science Foundation (Z200001).