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Identification of the causal relationship between multivariate time series isa ubiquitous problem in data science. Granger causality measure (GCM) andconditional Granger causality measure (cGCM) are widely used statisticalmethods for causal inference and effective connectivity analysis inneuroimaging research. Both GCM and cGCM have frequency-domain formulationsthat are developed based on a heuristic algorithm for matrix decompositions.The goal of this work is to generalize GCM and cGCM measures and theirfrequency-domain formulations by using a theoretic framework for minimumentropy (ME) estimation. The proposed ME-estimation method extends theclassical theory of minimum mean squared error (MMSE) estimation for stochasticprocesses. It provides three formulations of cGCM that include Geweke'soriginal time-domain cGCM as a special case. But all three frequency-domainformulations of cGCM are different from previous methods. Experimental resultsbased on simulations have shown that one of the proposed frequency-domain cGCMhas enhanced sensitivity and specificity in detecting network connectionscompared to other methods. In an example based on in vivo functional magneticresonance imaging, the proposed frequency-domain measure cGCM can significantlyenhance the consistency between the structural and effective connectivity ofhuman brain networks.

Minimum-entropy causal inference and its application in brain network analysis