Information flow within stochastic dynamical systems

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 twodimensional 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.PACS numbers: 05.45.-a, 89.70.+c, 89.75.-k, 02.50.-r

The Liang-Kleeman Information Flow: Theory and Applications

“…Here we illustrate this with a simple 2D system, which has been studied in reference [54] for the validation of Equation (52):…”

“…But this time no operator analogous to the F-P operator is sought; instead, we discretize the Fokker-Planck equation and expand x 1\ 2(t+∆t) , namely the first component at t + ∆t with x 2 frozen at t, using the Euler-Bernstein approximation. The complete derivation is beyond the scope of this review; the reader is referred to [54] for details. In the following, the final result is supplied in the form of a proposition.…”

“…Several sets of parameters have been chosen in [54] to study the model behavior. Here we just look at one such choice: B = , and a sample path of x starting from (1, 2).…”

On Thermodynamic Interpretation of Transfer Entropy

“…That correlation is not causation is well-understood. Yet while authors increasingly consider the notions of information transfer and information flow and how they fit with our understanding of correlation and causality [1,18,[29][30][31][32][33][34], several questions nag. Is information transfer, captured by transfer entropy, akin to causal effect?…”

“…Measures for both predictive transfer [1] and causal effect [18] have been inferred to capture information transfer in general, and measures of predictive transfer have been used to infer causality [33,[35][36][37] with the two sometimes (problematically) directly equated (e.g., [29,32,34,[38][39][40]). The study of Lizier and Prokopenko [41] clarified the relationship between these concepts and described the manner in which they should be considered separately.…”