  Identification of the causal relationship between multivariate time series is
a ubiquitous problem in data science. Granger causality measure (GCM) and
conditional Granger causality measure (cGCM) are widely used statistical
methods for causal inference and effective connectivity analysis in
neuroimaging research. Both GCM and cGCM have frequency-domain formulations
that are developed based on a heuristic algorithm for matrix decompositions.
The goal of this work is to generalize GCM and cGCM measures and their
frequency-domain formulations by using a theoretic framework for minimum
entropy (ME) estimation. The proposed ME-estimation method extends the
classical theory of minimum mean squared error (MMSE) estimation for stochastic
processes. It provides three formulations of cGCM that include Geweke's
original time-domain cGCM as a special case. But all three frequency-domain
formulations of cGCM are different from previous methods. Experimental results
based on simulations have shown that one of the proposed frequency-domain cGCM
has enhanced sensitivity and specificity in detecting network connections
compared to other methods. In an example based on in vivo functional magnetic
resonance imaging, the proposed frequency-domain measure cGCM can significantly
enhance the consistency between the structural and effective connectivity of
human brain networks.
