Information flow within stochastic dynamical systems

TL;DR

This paper develops a rigorous formalism for measuring information transfer in stochastic dynamical systems, revealing properties like transfer asymmetry and clarifying causality beyond correlation.

Contribution

It introduces a formal measure of information transfer applicable to stochastic systems, extending causality analysis beyond traditional correlation methods.

Findings

The measure exhibits transfer asymmetry.

In certain cases, no information transfer occurs despite high correlation.

Application to a two-dimensional system confirms expected transfer behavior.

Abstract

Information flow or information transfer is an important concept in dynamical systems which has applications in a wide variety of scientific disciplines. In this study, we show that a rigorous formalism can be established in the context of a generic stochastic dynamical system. The resulting measure of of information transfer possesses a property of transfer asymmetry and, when the stochastic perturbation to the receiving component does not rely on the giving component, has a form same as that for the corresponding deterministic system. An application with a two-dimensional system is presented, and the resulting transfers are just as expected. A remarkable observation is that, for two highly correlated time series, there could be no information transfer from one certain series, say $x_2$, to the other ($x_1$). That is to say, the evolution of $x_1$ may have nothing to do with $x_2$, even though $x_1$ and $x_2$ are highly correlated. Information transfer analysis thus extends the traditional notion of correlation analysis by providing a quantitative measure of causality between time series.