文章基本信息

标题：Principal regression for high dimensional covariance matrices
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作者：Yi Zhao ; Brian Caffo ; Xi Luo 等
期刊名称：Electronic Journal of Statistics
印刷版ISSN：1935-7524
出版年度：2021
卷号：15
期号：2
页码：4192-4235
DOI：10.1214/21-EJS1887
语种：English
出版社：Institute of Mathematical Statistics
摘要：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.
关键词：62H99; 62J99; covariance matrix estimation; generalized linear regression; Heteroscedasticity; shrinkage estimator