“Causometry” of processes in arbitrary dynamical systems: Three levels of directional coupling quantifiers

The concept of dynamical causal effect (DCE) which unites numerous causality quantifiers for processes in dynamical systems (DS) as measures of X-responses to some Y-variations is generalised through updating the definition of the factors constituting those variations. Along with initial states and governing parameters, external driving (noise) realisations are included and all three factors are represented with their probability distributions. Three basic levels of DCEs are introduced based on their resolution with respect to each factor: low (three factors resolved), medium (two factors), and high (one factor). It is shown that i) a directional coupling with zero transfer entropy (a medium-level DCE) may be strong in terms of its low-level DCEs; ii) DCEs of the same level may be small or large depending on their Y-variations, e.g., a coupling which induces synronisation is both weak and strong according to different medium-level DCEs; iii) high-level DCEs allow comparison of couplings from different sources which may be inaccessible via medium-level DCEs. The generalised DCE concept applies to arbitrary DS including deterministic (driven) DS and random DS with any kind of noise.