Covariance-on-Covariance Regression

TL;DR

This paper introduces a novel covariance-on-covariance regression model that identifies relationships between outcome and predictor covariance matrices, with applications in brain network analysis demonstrating its effectiveness.

Contribution

It proposes a new regression framework linking covariance matrices through linear projections and a log-linear model, with an estimator proven to be asymptotically consistent.

Findings

Successfully identified brain network pairs predicting functional connectivity.

Demonstrated superior performance over existing methods in simulations.

Findings align with current understanding of brain function.

Abstract

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.