Fig. 14.

(a) Power spectra of X₁, X₂, X₃. It can be seen that X₁, X₂, X₃ have almost same powers across all frequencies and have an obvious peak at f = 29.5 Hz. (b) $N\left(f\right)=N_{X_3\stackrel{\mathrm{D}}{\to }{X}_2}\left(f\right)\times N_{X_2\stackrel{\mathrm{D}}{\to }{X}_1}\left(f\right)$ based on (44) and N_T(f) = N_X3→X1 (f) based on the estimated AR model of the pair (X₁, X₃). One can see that N_T(f) (i.e., the total causality from X₃ to X₁) is very close to N(f) (i.e., summation of causalities along all routes from X₃ to X₁) before 40 Hz. But after 40 Hz, N(f) > N_T(f). For a general model, the exact relationship between N(f) and N_T(f) is not known.