Figure 3. Sketch of the cross map smoothness learned by a neural network (NN).

(a) and (b) Illustrations for the neural network's approximation ability for smooth map and unsmooth map. Here the map surface in (a) is assumed to be x = y₁ + y₂ and the surface in (b) is simply generated by random points. (c) and (d) The prediction error (or the smoothness of Φ) for cases in (a) and (b) respectively, where the leave-one-out scheme is used to calculate errors. (e) Assume that x causally influences y, the information of x has been encoded in M_y and consequently Φ: M_y → M_x maps a neighborhood of y to a neighborhood of x, implying Φ_yx is smooth. Thus a neural network can be trained to approximate the map based on the measured data on M_x and M_y. (f) Assume that y has no impact on x, then M_x has no information from y. Training a neural network to approximate the unsmooth map Φ: M_x → M_y will fail.