Open Access
2021 Principal regression for high dimensional covariance matrices
Yi Zhao, Brian Caffo, Xi Luo
Author Affiliations +
Electron. J. Statist. 15(2): 4192-4235 (2021). DOI: 10.1214/21-EJS1887


This manuscript presents an approach to perform generalized linear regression with multiple high dimensional covariance matrices as the outcome. In many areas of study, such as resting-state functional magnetic resonance imaging (fMRI) studies, this type of regression can be utilized to characterize variation in the covariance matrices across units. Model parameters are estimated by maximizing a likelihood formulation of a generalized linear model, conditioning on a well-conditioned linear shrinkage estimator for multiple covariance matrices, where the shrinkage coefficients are proposed to be shared across matrices. Theoretical studies demonstrate that the proposed covariance matrix estimator is optimal achieving the uniformly minimum quadratic loss asymptotically among all linear combinations of the identity matrix and the sample covariance matrix. Under certain regularity conditions, the proposed estimator of the model parameters is consistent. The superior performance of the proposed approach over existing methods is illustrated through simulation studies. Implemented to a resting-state fMRI study acquired from the Alzheimer’s Disease Neuroimaging Initiative, the proposed approach identified a brain network within which functional connectivity is significantly associated with Apolipoprotein E ε4, a strong genetic marker for Alzheimer’s disease.

Funding Statement

Zhao was partially supported by NIH grant U54AG065181 and P30AG010133; Caffo by NIH grant R01EB029977 and P41EB031771; and Luo by NIH grant R01EB022911. Data collection and sharing for this project was funded by the Alzheimer’s Disease Neuroimaging Initiative (ADNI, National Institutes of Health Grant U01 AG024904 and Department of Defense award number W81XWH-12-2-0012). ADNI is funded by the National Institute on Aging, the National Institute of Biomedical Imaging and Bioengineering, and through generous contributions from the following: AbbVie, Alzheimer’s Association; Alzheimer’s Drug Discovery Foundation; Araclon Biotech; BioClinica, Inc.; Biogen; Bristol-Myers Squibb Company; CereSpir, Inc.; Cogstate; Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli Lilly and Company; EuroImmun; F. Hoffmann-La Roche Ltd and its affiliated company Genentech, Inc.; Fujirebio; GE Healthcare; IXICO Ltd.; Janssen Alzheimer Immunotherapy Research & Development, LLC.; Johnson & Johnson Pharmaceutical Research & Development LLC.; Lumosity; Lundbeck; Merck & Co., Inc.; Meso Scale Diagnostics, LLC.; NeuroRx Research; Neurotrack Technologies; Novartis Pharmaceuticals Corporation; Pfizer Inc.; Piramal Imaging; Servier; Takeda Pharmaceutical Company; and Transition Therapeutics. The Canadian Institutes of Health Research is providing funds to support ADNI clinical sites in Canada. Private sector contributions are facilitated by the Foundation for the National Institutes of Health ( The grantee organization is the Northern California Institute for Research and Education, and the study is coordinated by the Alzheimer’s Therapeutic Research Institute at the University of Southern California. ADNI data are disseminated by the Laboratory for Neuro Imaging at the University of Southern California.


Download Citation

Yi Zhao. Brian Caffo. Xi Luo. "Principal regression for high dimensional covariance matrices." Electron. J. Statist. 15 (2) 4192 - 4235, 2021.


Received: 1 October 2020; Published: 2021
First available in Project Euclid: 14 September 2021

arXiv: 2007.12740
Digital Object Identifier: 10.1214/21-EJS1887

Primary: 62J99
Secondary: 62H99

Keywords: covariance matrix estimation , generalized linear regression , Heteroscedasticity , shrinkage estimator

Vol.15 • No. 2 • 2021
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