Fast asymptotic algorithm for real-time causal connectivity analysis of multivariate systems and signals

The purpose of this work is to optimize the current state-of-the-art asymptotic algorithm of connectivity Granger-causality measures in the frequency-domain, such as the Directed Transfer Function (DTF), Partial Directed Coherence (PDC), and their variants. These measures stem from the modeling of multidimensional time series by multivariate autoregressive model. Surrogate and asymptotic analysis are the most frequently used methods to quantify the statistical significance of such derived interactions, a critical step for validation of the results. The current asymptotic algorithms run fairly fast on low-dimensional datasets but become impractical for high-dimensional datasets due to the involved computational time and memory demand. This is a huge limitation in the application of these connectivity measures to the fields dealing with numerous concurrently acquired signals from probing of complex systems such as the human brain. Here, we optimized the current algorithms for the fast asymptotic analysis of these connectivity measures and achieved a reduction of their computation time by at least three orders of magnitude. The optimizations were accomplished by decreasing the dimension of the involved matrices, eliminating the complicated functions (e.g., eigenvalue estimation and Cholesky factorization), and variable separation. The superior performance of the proposed optimized algorithm is shown with simulation examples.