Validity of Time Reversal for Testing Granger Causality

Abstract

Abstract-Inferring causal interactions from observed data is a challenging problem, especially in the presence of measurement noise. To alleviate the problem of spurious causality, Haufe et al. (2013) proposed to contrast measures of information flow obtained on the original data against the same measures obtained on time-reversed data. They show that this procedure, time-reversed Granger causality (TRGC), robustly rejects causal interpretations on mixtures of independent signals. While promising results have been achieved in simulations, it was so far unknown whether time reversal leads to valid measures of information flow in the presence of true interaction. Here we prove that, for linear finiteorder autoregressive processes with unidirectional information flow between two variables, the application of time reversal for testing Granger causality indeed leads to correct estimates of information flow and its directionality. Using simulations, we further show that TRGC is able to infer correct directionality with similar statistical power as the net Granger causality between two variables, while being much more robust to the presence of measurement noise.

Decoding of top-down cognitive processing for SSVEP-controlled BMI

“…In addition, there was strong crosstalk between the left and right anterior prefrontal areas (BA 10L and BA 10R), particularly in the top-down SSVEP condition. All of these directed functional connectivity measures were reliably confirmed by additional observations using time-reversed data4546 (see the Supplementary information). However, neither the intermediate nor the bottom-up SSVEP conditions showed true directed functional connectivity because their time-reversal results in Granger causality were not significantly different from those of the original data, which is indicative of spurious causal relations, such as a volume conduction effect4546.…”

“…Statistical assessment of the connectivity was performed using the surrogate approach (1000 surrogates, p  < 0.05). In order to avoid spurious causal relations, we conducted the same Granger causality analysis using time-reversed data4546. That is, the reversed temporal order of all data points in the same EEG dataset was used to double-check the robustness of the inferred Granger-causal connectivity measures.…”

Frequency-specific directed interactions in the human brain network for language

“…Next, we computed frequencyresolved Granger causality (GC) to quantify directed rhythmic neuronal interactions between brain areas for language that are known to be anatomically connected (9)(10)(11). Because the interpretation of connectivity estimated from neuromagnetic recordings is highly confounded by spatial leakage of source activity (12), we statistically compared, across the sample of 102 participants, the estimated GC with an estimate of GC after time reversal of the signals (13). This allowed us to conservatively discard a substantial subset of the predefined connections for which the direction and/or the strength of the estimated Granger causal interaction is likely Significance The brain's remarkable capacity for language requires bidirectional interactions between functionally specialized brain regions.…”

Noninvasive Electromagnetic Source Imaging and Granger Causality Analysis: An Electrophysiological Connectome (eConnectome) Approach

“…Haufe et al [67] have shown in a simulation study (sensor space) that conventional Granger causality analysis and surrogate testing by phase shuffling, might not be as effective as measures such as phase-slope index (PSI) [68], [69]. Haufe et al have also proposed an time-reversed scheme for Granger causality analyses and have shown that this improves the results and makes it as good as PSI performance [67], [70]. A later study by Vinck et al demonstrated that correlated and uncorrelated noise might have different effects on the causality analysis results [57].…”