A Study Of Methodology In Granger Causality Analysis

As a time series analysis technique based on the vector autoregres-sion framework, Granger Causality (GC), which derived from econo-metrics, and its information theoretic analog Transfer Entropy (TE) that defined by the form of conditional mutual information, has been applied to the data analysis across the field of neuroscience, clima-tology and physiology over recent years. Due to its statistical intu-itive, easy implementation, and invasive, data driven nature, Granger Causality has been generally used to construct the effective connectivi-ty network among brain regions. Differ from the undirected functional connectivity which based on the statistical dependencies such as Pear-son correlation or mutual information, effective connectivity focus on investigating the causality relationship between signals from brain in-teractions. That kind of causality which based on the temporal prece-dence, aims to measure the change of predictability of time series. The distinction between them is, functional connectivity modeling assesses the instantaneous, undirected, statistical independence, with common techniques including independent components analysis (ICA) and var-ious measures of synchrony or correlation. While effective connectivity reflect an underlying dynamics from considering the neuronal activity of present and past state, in which causes precede effects, with com-mon techniques including Dynamic Causal Modelling (DCM), Granger Causality and Transfer Entropy.This study primarily focuses on Granger Causality and Tranfer Entropy. The main contributions include:We discussed the applicability of signed-path coefficient Granger Causality that implemented by an order-1 vector autoregressive (VAR) model, on the fMRI data analysis. Different from traditional Granger Causality measure which based on the residuals, an new Granger causal-ity method was proposed and applied by an increasing number of re-seachers over these years:the signed-path coefficient method, which derived from the coefficient parameters estimated by a bivariate or multivariate autoregressive model, and the positive or negative coeffi-cients are respectively defined as excitative or inhibitive influence that one brain region cast to another. In this work we proposed a series of simulation results and computations from fMRI data to illustrate that under certain conditions this kind of operation and defination of causality was flawed and untenable since it would inevitablely lead to erroneous conclusions, since the signed path coefficients are not always consistent with the real causal relationship if the actual lag of the data generation process is higher than one but the data is estimated by an underfitted order-1 autoregressive model. Due to the commonly limit-ed length of fMRI data, we should pay attention to the optimum order (not always be one) determined by information criteria as the number of time series changed, and also consider the impact of band-pass filter in preprocessing procedures on the data, both of which will severely affect the final outcome. Thus when adopt the signed-path coefficient method we must be very cautious to avoid uncorrect deduction and interpretation.Via an equivalent relationship between Granger Causality and Transfer Entropy under Gaussian variables we investigated the devi-ation of these two causality measures by comparing the outcome be-tween original and surrogate fMRI/EEG datasets, where the surrogates are generated in a way that the linear correlation between time series preserved. In fMRI data, the deduced causality did not present any sig-nificant change in the surrogates, prove that there did not exist much nonlinearity in causal relationships and it was rational to use Granger Causality which adopted the linear correlation assumption. Neverthe-less, the results of EEG shows a great distortion between original data and surrogates by Tranfer Entropy, means the higher linearity in EEG also induce a great deal of higher order causal relationship, which Granger Causality was unable to capture. Hence under such condition we should apply nonlinear causality measure Tranfer Entropy to avoid severe loss of causal information.To reach an unambiguous scenario of the causality structure un-derlying time series, a shrinkage strategy of parameters is needed to construct a parsimonious model. We propose an advanced search algo-rithm to eliminate the redundant varibles to provide maximum predict-ing power. In econometrics a common method with the name "Subset Regression" was proposed to place zero restrictions on specific param-eters in autoregression, while a parallel strategy also had been intro-duced in physics field under the name "Non-uniform Embedding" in the state space, which aimed to tackle the problem of dimesion curse to give a more exact estimation of tranfer entropy. Both method focus on reconstructing an efficient causality network with highest parsimo-ny, by a traditional greedy search method to select significant vari-ables with the most contribution in the history spanned by time series’ past state. In this work we employed a feature selection tool to im-prove the searching and evaluating performance in building the causal network. Compared to previous approaches, our algorithm showed a higher specificity while preserve the good sensitivity, and promised a shorter computing time.