  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 are demonstrated via simulation studies. Applying to data
collected in the Human Connectome Project Aging study, the proposed approach
identifies three pairs of brain networks, where functional connectivity within
the resting-state network predicts functional connectivity within the
corresponding task-state network. The three 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.
