Time-varying linear and nonlinear parametric model for Granger causality analysis

Statistical measures such as coherence, mutual information, or correlation are usually applied to evaluate the interactions between two or more signals. However, these methods cannot distinguish directions of flow between two signals. The capability to detect causalities is highly desirable for understanding the cooperative nature of complex systems. The main objective of this work is to present a linear and nonlinear time-varying parametric modeling and identification approach that can be used to detect Granger causality, which may change with time and may not be detected by traditional methods. A numerical example, in which the exact causal influences relationships, is presented to illustrate the performance of the method for time-varying Granger causality detection. The approach is applied to EEG signals to track and detect hidden potential causalities. One advantage of the proposed model, compared with traditional Granger causality, is that the results are easier to interpret and yield additional insights into the transient directed dynamical Granger causality interactions.

Figure 1

Time-varying Granger causalities $C_{x\to y}^{\mathrm{LGC}}\left(t\right)$ [thick blue (upper) curve] and $C_{y\to x}^{\mathrm{LGC}}\left(t\right)$ [black (lower) curve] from model (25) with time-varying parameters given in Eq. (26) are shown above. The thin blue and black step functions show that the corresponding estimated time-invariant Granger causalities are $C_{x\to y}^{\mathrm{LGC}}$ and $C_{y\to x}^{\mathrm{LGC}}$.

Figure 2

The EEG signals, for a sort of seizure activity of an epileptic patient, recorded during 20 s, with a sampling rate of 250 Hz, for both electrode channel $C_3^{*}$ (above) and $C_4^{*}$ (below).

Figure 3

The time-varying nonlinear Granger causality $C_{C_3^{*}\to C_4^{*}}^{\mathrm{NGC}}\left(t\right)$ [blue (upper) curve] and $C_{C_4^{*}\to C_3^{*}}^{\mathrm{NGC}}\left(t\right)$ [black (lower) curve] are plotted vs the time courses for EEG signals shown in Fig. 2.

Figure 4

The cross correlations $c_1\left(\tau \right)$ and $c_2\left(\tau \right)$ are shown vs $\tau$ between EEG signals $C_3^{*}$ and $C_4^{*}$ given in Fig. 2.