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Granger Causality Maps for Langevin Systems

Lionel Barnett, Benjamin Wahl, Nadine Spychala, Anil K. Seth

TL;DR

The paper resolves key limitations in Granger causality maps for Langevin systems by deriving a direct, continuous-time GC rate for vector Ornstein–Uhlenbeck processes from model parameters, valid even in locally unstable regions. It constructs GC maps over the full phase space and shows invariance of the GC rate to global noise scaling, enabling interpretation for deterministic dynamics via a noise-ghost map. The authors provide a practical CARE-based formulation, with a 1D-source case reducing to a simple quadratic, and demonstrate the method on the Lorenz system to illustrate hole-free, globally defined information transfer. This framework broadens GC analysis to nonlinear stochastic and deterministic dynamics, with potential applications across physics, neuroscience, and AI, and offers a computationally efficient alternative to transfer entropy for certain systems.

Abstract

Wahl et al. (2016, 2017) introduced the idea of Granger causality (GC) maps for Langevin systems: dynamics are localised 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. Their implementation, however, suffered a significant drawback: GCs were approximated from models based on discrete-time stroboscopic sampling of local VOU processes, which is not only computationally inefficient, but more seriously, unfeasible 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 continuous-time 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.

Granger Causality Maps for Langevin Systems

TL;DR

The paper resolves key limitations in Granger causality maps for Langevin systems by deriving a direct, continuous-time GC rate for vector Ornstein–Uhlenbeck processes from model parameters, valid even in locally unstable regions. It constructs GC maps over the full phase space and shows invariance of the GC rate to global noise scaling, enabling interpretation for deterministic dynamics via a noise-ghost map. The authors provide a practical CARE-based formulation, with a 1D-source case reducing to a simple quadratic, and demonstrate the method on the Lorenz system to illustrate hole-free, globally defined information transfer. This framework broadens GC analysis to nonlinear stochastic and deterministic dynamics, with potential applications across physics, neuroscience, and AI, and offers a computationally efficient alternative to transfer entropy for certain systems.

Abstract

Wahl et al. (2016, 2017) introduced the idea of Granger causality (GC) maps for Langevin systems: dynamics are localised 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. Their implementation, however, suffered a significant drawback: GCs were approximated from models based on discrete-time stroboscopic sampling of local VOU processes, which is not only computationally inefficient, but more seriously, unfeasible 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 continuous-time 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.

Paper Structure

This paper contains 8 sections, 2 theorems, 69 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Granger causality rate for vector Ornstein-Uhlenbeck processes
  3. Granger causality maps for Langevin processes
  4. Example: the Lorenz system
  5. Discussion
  6. Stationarity of the finite-horizon prediction error
  7. Proof of Theorem \ref{['the:main']}
  8. Proof of detectability for stable full matrix

Key Result

Theorem 1

For the VOU process eq:vou with $d\boldsymbol{w}(t) \sim \mathcal{N}\!\left({0},{\Sigma\,dt}\right)$ and a partitioning $\boldsymbol{y} = [\boldsymbol{y}_1^\mathrm{\hbox{T}}\; \boldsymbol{y}_2^\mathrm{\hbox{T}}\; \boldsymbol{y}_3^\mathrm{\hbox{T}}]^\mathrm{\hbox{T}}$, if $\Sigma$ is positive-definit has a unique stabilising solution $P_{33}$, and the Granger causality rate from $\boldsymbol{y}_3$

Figures (2)

  • Figure 1: Local stability of Lorenz dynamics. The colour scale corresponds to the largest real part $\lambda(\boldsymbol{y})$ of the eigenvalues of the Jacobian matrix $J(\boldsymbol{y})$; the system is locally unstable where $\lambda(\boldsymbol{y}) \ge 0$ (grey$\,\to\,$red).
  • Figure 2: GC maps of Lorenz dynamics on the attractor. The colour scale measures local Granger-causal graph values $\mathcal{G}_{ij}(\boldsymbol{y})$. The $\mathcal{G}_{ij}$ values above the plots are the global Granger-causal graph values calculated according to \ref{['eq:gcint']}.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Corollary 1
  • proof