Dynamical Systems, Information Flow, and Causality

A dynamical system is one that evolves with time either stochastically or deterministically. The evolution is described by a set of differential equations. The instantaneous state of a dynamical system is represented as a point in some phase space. As the system evolves the point moves describing a trajectory in the phase space. In 1981, Floris Takens established that (under certain conditions) the shape and structure of the trajectory could be deduced from the time series of any one of its co-ordinates.

This insight is helpful in determining causality where other methods such as Granger causality fail. Recall that Variable X is said to Granger cause Y if the predictability of Y (in some idealized model) declines when X is removed from the universe of all possible causative variables.

In a deterministic dynamical system,  if X is a cause for Y, information about X will be redundantly present in Y itself and cannot formally be removed. We can exploit Takens theorem via a cross mapping technique originated by Sugihara to detect causal influences.

Another way to detect causal influence is through information flow. Liang formulated a measure of the rate of information flow between two time series, formulated as a discussion in terms of marginal entropy.

For a more detailed discussion  please see http://situ.com/sidd/EDM/

References:

F. Takens, in Dynamical Systems and Turbulence, D. A. Rand, L. S. Young, Eds. (Springer-Verlag, New York, 1981), pp. 366-381.

“Detecting Causality in Complex Ecosystems,” George Sugihara et al. Science 338 , 496 (2012), DOI: 10.1126/science.1227079

“Causal feedbacks in climate change,” van Nes et al.,Nature Climate Change, 2015, DOI: 10.1038/NCLIMATE2568

“Local predictability and information flow in complex dynamical systems,” X. San Liang, Physica D 2013, DOI: 10.1016/j.physd.2012.12.011

“Unraveling the cause-effect relation between time series,” X. San Liang, Phys. Rev. E 90, 052150 (2014), DOI: 10.1103/PhysRevE.90.052150