Generative formalism of causality quantifiers for processes.

The concept of dynamical causal effect (DCE) is generalized and equipped with a formalism which allows one to formulate in a unified manner and interrelate a variety of causality quantifiers used in time series analysis. An elementary DCE from a subsystem Y to a subsystem X is defined within the stochastic dynamical systems framework as a response of a future X state to an appropriate variation of an initial (X,Y)-state distribution or a certain parameter of Y or of the coupling element Y→X; this response is quantified in a probabilistic sense via a certain distinction functional; elementary DCEs are assembled over a set of initial variations via an assemblage functional. To include all those aspects, a "triple brackets formula" for the general DCE is suggested and serves as a first principle to produce specific causality quantifiers as realizations of the general DCE. As an application, transfer entropy and Liang-Kleeman information flow are related surprisingly as opposite limit cases in a family of DCEs; it is shown that their "nats per time unit" may differ drastically. The suggested DCE viewpoint links any formal causality quantifier to "intervention-effect" experiments, i.e., future responses to initial variations, and so provides its dynamical interpretation, opening a way to its further physical interpretations in studies of physical systems.