Granger causality and information flow in multivariate processes

The multivariate versus bivariate measures of Granger causality were considered. Granger causality in the application to multivariate physiological time series has the meaning of the information flow between channels. It was shown by means of simulations and by the example of experimental electroencephalogram signals that bivariate estimates of directionality in case of mutually interdependent channels give erroneous results, therefore multivariate measures such as directed transfer function should be used for determination of the information flow.

Figure 1

Top: Simulated signals. Bottom left: Simulation scheme. Bottom right: In each box DTF as functions of frequency $(0–25\mathrm{Hz})$; the number above the columns indicates output channels, the numbers on the left of the rows indicate destination channels.

Figure 2

Top: Granger causality calculated pairwise; each graph represents the function describing transmission from the channel marked above the row to the channel marked on the left of the row. Granger causality in arbitrary units on vertical axes; graphs on the diagonal contain power spectra; frequency on horizontal axes $(0–25\mathrm{Hz}$ range). Bottom left: Simulation scheme. Bottom right: Resulting flow scheme. The black arrows represent true (simulated) flows; the dotted arrows represent false flows found by the applied method.

Figure 3

Pairwise coherences and resulting flows. Top: Coherence amplitude (black graphs above diagonal) and coherence phase (graphs below diagonal); each graph represents the function for the pair of channels marked on the left of the row and above the column; on the horizontal axes frequency $(0–25\mathrm{Hz})$; on vertical axes coherence amplitudes (0–1 range) or phases ($-\pi –\pi$ range); delay values (in samples) estimated from phases, marked by the numbers shown over the phase graphs. Bottom left: Simulation scheme. Bottom right: Resulting flow scheme. The same convention in drawing arrows as in Fig. 2.

Figure 4

Top: Non-normalized multichannel DTFs for the simulation (Fig. 2). The picture organization is similar to Fig. 3 (on the diagonal power spectra). Bottom: DTFs obtained from surrogate data. Thick line: The average obtained from 100 surrogates. Ninety five percent of surrogate realizations are contained between the thin lines. The plots in both panels are in the same scale in arbitrary units. Frequency on horizontal axes, $0–25\mathrm{Hz}$ range. At left is the resulting flow pattern.

Figure 5

The directions of flows found for experimental EEG signals recorded from 21 channels ($10∕20$ system), for awake state, obtained by the bivariate Granger causality measure (top) and non-normalized DTF function calculated by means of fitting MVAR to all 21 channels simultaneously (bottom). The shade of gray of the arrow represents the strength of the connection ($\text{black}=\text{the}$ strongest). In order to make the pattern of flows clear, the cutoff of relative intensity was applied on the level 0.53 with respect to the strongest flow, which was assigned the value of 1.