  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.
