Shrinkage Estimation of High-Dimensional Vector Autoregressions for Effective Connectivity in fMRI

We consider the challenge of estimating effective brain connectivity network with a large number of nodes from fMRI data. This involves estimation of a very-high dimensional vector autoregressive (VAR) models commonly used to identify directed brain networks. The conventional least-squares (LS) estimator is not longer consistent when applied on the high-dimensional fMRI data compared to sample size due to large number of fitted parameters, and thus produces unreliable estimates of the brain connectivity. In this paper, we propose an well-conditioned large-dimensional VAR estimator based on shrinkage approach, by incorporating a Ledoit-Wolf (LW) shrinkage-based estimator of the Gramian matrix in the LS-based linear regression fitting of VAR. This allows better-conditioned and invertible Gramian matrix estimate which is an important ingredient in generating a reliable LS estimator, when the data dimension is larger than the sample size. Simulation results show significant superiority of the proposed LW-shrinkage-VAR estimator over the conventional LS estimator under the high-dimensional settings. Application to real resting-state fMRI dataset shows the capability of the proposed method in identifying resting-state brain connectivity networks, with directionality of connections and interesting modular structure, which potentially provide useful insights to neuroscience studies of human brain connectome.