Continuity Scaling: A Rigorous Framework for Detecting and Quantifying Causality Accurately
Illustration of causal relation between two sets of dynamical variables. (a) Existence of causation from in to in , where each correspondence from to is one-to-one, represented by the line or the arrow, respectively, in the upper and the middle panels. In this case, a change in results in a direct change in (the lower panel) with and denoting the neighborhood size of the resulting variable and of the causal variable , respectively. (b) Absence of causation from to , where every point on each trajectory, , in could be the correspondent point from in (the upper panel) and thus every point in belongs to the largest -neighborhood of (the middle panel). In this case, does not depend on (the lower panel). Also refer to the supplemental animation for illustration.