Granger causality maps for Langevin systems

Wahl et al. [Phys. Rev. E 93, 022213 (2016)10.1103/PhysRevE.93.022213] introduced Granger causality (GC) maps for Langevin systems: Dynamics are localized linearly at each point in phase space as vector Ornstein-Uhlenbeck (VOU) processes, for which GCs may in principle be calculated, thus constructing a GC map on phase space. These maps may, furthermore, be averaged over phase space to yield a systemwide global GC value. The implementation, however, suffered some significant drawbacks: GCs were approximated from models based on discrete-time stroboscopic sampling of local VOU processes, which is not only computationally inefficient but, more seriously, infeasible on regions of phase space where local dynamics are unstable, leaving "holes" in the GC maps. We solve these problems by deriving an analytical expression for GC rates associated with a VOU process which, under quite general conditions, yields a meaningful solution even in the unstable case. Applied to GC maps, this not only "fills in the holes" but also furnishes a computationally efficient method of calculation devolving to solution of algebraic Riccati equations which, in the case of a univariate source, become simple quadratic equations. We show, furthermore, that the GC rate for VOU processes is invariant under rescaling of the overall fluctuations intensity, so that GC maps may effectively be calculated for deterministic nonlinear dynamical systems, with a residual "ghost of noise" represented by a variance-covariance map.