An information-theoretic framework for conditional causality analysis of brain networks

Identifying directed network models for multivariate time series is a ubiquitous problem in data science. Granger causality measure (GCM) and conditional GCM (cGCM) are widely used methods for identifying directed connections between time series. Both GCM and cGCM have frequency-domain formulations to characterize the dependence of time series in the spectral domain. However, the original methods were developed using a heuristic approach without rigorous theoretical explanations. To overcome the limitation, the minimum-entropy (ME) estimation approach was introduced in our previous work (Ning & Rathi, 2018) to generalize GCM and cGCM with more rigorous frequency-domain formulations. In this work, this information-theoretic framework is further generalized with three formulations for conditional causality analysis using techniques in control theory, such as state-space representations and spectral factorizations. The three conditional causal measures are developed based on different ME estimation procedures that are motivated by equivalent formulations of the classical minimum mean squared error estimation method. The relationship between the three formulations of conditional causality measures is analyzed theoretically. Their performance is evaluated using simulations and real neuroimaging data to analyze brain networks. The results show that the proposed methods provide more accurate network structures than the original approach. This paper introduces a theoretical framework for causal inference in brain networks using time series measurements based on the principle of minimum-entropy regression. Three types of conditional causality measures are derived based on varying formulations of minimum-entropy regressions. The standard time-domain conditional Granger causality measure is formulated as a special case but with a different expression of the frequency-domain measure. The methods were evaluated using simulations and real resting-state functional MRI data of human brains and compared with standard Granger causality measures and directed transfer functions. Two new formulations of minimum-entropy-based causality measures showed better performance than other methods. The algorithms developed from this work may provide new insights to understand information flow in brain networks.