  This paper considers joint learning of multiple sparse Granger graphical
models to discover underlying common and differential Granger causality (GC)
structures across multiple time series. This can be applied to drawing
group-level brain connectivity inferences from a homogeneous group of subjects
or discovering network differences among groups of signals collected under
heterogeneous conditions. By recognizing that the GC of a single multivariate
time series can be characterized by common zeros of vector autoregressive (VAR)
lag coefficients, a group sparse prior is included in joint regularized
least-squares estimations of multiple VAR models. Group-norm regularizations
based on group- and fused-lasso penalties encourage a decomposition of multiple
networks into a common GC structure, with other remaining parts defined in
individual-specific networks. Prior information about sparseness and sparsity
patterns of desired GC networks are incorporated as relative weights, while a
non-convex group norm in the penalty is proposed to enhance the accuracy of
network estimation in low-sample settings. Extensive numerical results on
simulations illustrated our method's improvements over existing sparse
estimation approaches on GC network sparsity recovery. Our methods were also
applied to available resting-state fMRI time series from the ADHD-200 data sets
to learn the differences of causality mechanisms, called effective brain
connectivity, between adolescents with ADHD and typically developing children.
Our analysis revealed that parts of the causality differences between the two
groups often resided in the orbitofrontal region and areas associated with the
limbic system, which agreed with clinical findings and data-driven results in
previous studies.
