Modeling Symmetric Positive Definite Matrices with An Application to Functional Brain Connectivity

TL;DR

This paper introduces a novel statistical model for analyzing time-varying symmetric positive definite matrices in neuroscience, enabling better understanding of brain connectivity changes over time.

Contribution

It proposes a matrix-log mean model with heterogeneous noise on a Riemannian manifold, and a local scan statistic method for detecting change points in functional brain connectivity.

Findings

Model accurately detects change points in simulated data.

Method successfully applied to Human Connectome Project data.

Theoretical guarantees for change point recovery.

Abstract

In neuroscience, functional brain connectivity describes the connectivity between brain regions that share functional properties. Neuroscientists often characterize it by a time series of covariance matrices between functional measurements of distributed neuron areas. An effective statistical model for functional connectivity and its changes over time is critical for better understanding the mechanisms of brain and various neurological diseases. To this end, we propose a matrix-log mean model with an additive heterogeneous noise for modeling random symmetric positive definite matrices that lie in a Riemannian manifold. The heterogeneity of error terms is introduced specifically to capture the curved nature of the manifold. We then propose to use the local scan statistics to detect change patterns in the functional connectivity. Theoretically, we show that our procedure can recover all change points consistently. Simulation studies and an application to the Human Connectome Project lend further support to the proposed methodology.