Figure 2. Simulation results.

(A) Traces of the time series we considered in Example 1 when a_i∼U[0,1], b_i = 2, c_i = 5, i = 1,…5. x ₂,x ₃,x ₄ and x ₅ are shifted upward for visualization purpose. (B) The causality structure is plotted. (C) Comparison of the partial Granger causality F ⁽¹⁾ and the conditional Granger causality F ⁽²⁾ when a_i∼U[0,1]. It is obvious that F ⁽²⁾ fails to pick up the correct relationship while the inferred relationship from F ⁽¹⁾ is consistent with the true structure (B). (D) Comparison of the partial Granger causality in the time domain and frequency domain when a_i∼U[0,1], i = 1,…5 in Example 1. The blue line represents the case of time domain, and the red line is the integral of the frequency domain in the interval [−π, π].