Fig. 1

Definition of causality. a. the attractors M _X and M _Y are reconstructed from variables X and Y by lagged-coordinates and they are topological equivalent. b. The predicted time-series (points) \( {Q}_k^{\prime }\ \left(k=1,2,\cdots \right) \) are located in the nearest neighborhood of Q _k  (k = 1, 2, ⋯) which implies causality (see the above part). The predicted time-series (points) \( {Q}_k^{\prime }\ \left(k=1,2,\cdots \right) \) are outside the nearest neighborhood of Q _k  (k = 1, 2, ⋯) which implies no causality (see the below part). Here, the nearest neighborhood is measured by a ball with a small radius r (see the gray district)