Farnaz Rezaei

Date of Award

Summer 2020

Document Type


Degree Name

Master of Science (MS)




Connectivity Granger-causality measures in the frequency domain, such as the Directed Transfer Function (DTF) and Partial Directed Coherence (PDC) and their variants, constitute a family πœ™ of measures that stem from the modeling of multidimensional time series by multivariate autoregressive (MVAR) models. πœ™ measures have become popular for evaluation of causal interactions in neuronal networks. Surrogate and asymptotic statistical analysis are the two most frequently used methods to quantify the statistical significance of the derived interactions, a critical step for validation of the results. Each method has its own pros and cons, with the recently published asymptotic methodology being faster. The state-of-the-art asymptotic methods, introduced by Baccala et al., run fairly fast on low-dimensional datasets but become impractical for high-dimensional datasets due to the involved computational time and memory demand; the amount of calculations increases exponentially with the number of time series to be analyzed. This is a huge limitation in the application of πœ™ measures to fields that deal with a large number of concurrently acquired time series from probing of complex systems such as the human brain. In this study, we optimized the original algorithms for fast asymptotic analysis of πœ™ measures and achieved a reduction of their computation speed by at least three orders of magnitude, thus allowing computation of connectivity measures and their significance in real-time from a plurality of concurrently recorded biological signals. The optimizations were accomplished by a decrease of the dimension of the involved matrices, reduction of the calculation time of complex functions (e.g. eigenvalue estimation and Cholesky factorization), and variable separation. The superior performance of the proposed optimized algorithms in the estimation of the statistical significance and confidence interval of πœ™ measures of causal interactions is shown with simulation examples.