Covariance-on-covariance regression

A covariance-on-covariance regression model is introduced in this manuscript. It is assumed that there exists (at least) a pair of linear projections on outcome covariance matrices and predictor covariance matrices such that a log-linear model links the variances in the projection spaces, as well as additional covariates of interest. An ordinary least square type of estimator is proposed to simultaneously identify the projections and estimate model coefficients. Under regularity conditions, the proposed estimator is asymptotically consistent. The superior performance of the proposed approach over existing methods is demonstrated via simulation studies. Applying to data collected in the Human Connectome Project Aging study, the proposed approach identifies 3 pairs of brain networks, where functional connectivity within the resting-state network predicts functional connectivity within the corresponding task-state network. The 3 networks correspond to a global signal network, a task-related network, and a task-unrelated network. The findings are consistent with existing knowledge about brain function.