Principal regression for high dimensional covariance matrices

Yi Zhao, Brian Caffo, Xi Luo

Research output: Contribution to journalArticlepeer-review


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 combi-nations 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 ex-isting methods is illustrated through simulation studies. Implemented to a resting-state fMRI study acquired from the Alzheimer’s Disease Neuroimag-ing Initiative, the proposed approach identified a brain network within which functional connectivity is significantly associated with Apolipopro-tein E ε4, a strong genetic marker for Alzheimer’s disease.

Original languageEnglish (US)
Pages (from-to)4192-4235
Number of pages44
JournalElectronic Journal of Statistics
Issue number2
StatePublished - 2021


  • Covariance matrix estimation
  • Generalized linear regression
  • Heteroscedasticity
  • Shrinkage estimator

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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