Granger-causality maps of diffusion processes

Granger causality is a statistical concept devised to reconstruct and quantify predictive information flow between stochastic processes. Although the general concept can be formulated model-free it is often considered in the framework of linear stochastic processes. Here we show how local linear model descriptions can be employed to extend Granger causality into the realm of nonlinear systems. This novel treatment results in maps that resolve Granger causality in regions of state space. Through examples we provide a proof of concept and illustrate the utility of these maps. Moreover, by integration we convert the local Granger causality into a global measure that yields a consistent picture for a global Ornstein-Uhlenbeck process. Finally, we recover invariance transformations known from the theory of autoregressive processes.

Figure 1

Granger causality ${\mathcal{I}}_{X_2\to X_1}$ for $f_s=10\text{Hz}$ depending on system parameter $p$; cf. Example 1.

Figure 2

Granger causality ${\mathcal{I}}_{X_2\to X_1}$ of Ornstein-Uhlenbeck process ($p=0$) for various $\mathrm{\Delta }t$; cf. Example 1.

Figure 3

(a) Histogram of stationary probability distribution $P^{*}\left(\mathit{x}\right)$, (b) Granger-causality map ${\mathcal{F}}_{{\tilde{X}}_2\to {\tilde{X}}_1}\left(\mathit{x}\right)$, (c) Granger-causality map ${\mathcal{F}}_{{\tilde{X}}_1\to {\tilde{X}}_2}\left(\mathit{x}\right)$; cf. Example 2.

Figure 4

(a) Histogram of stationary probability distribution $R^{*}\left(\mathit{y}\right)$, (b) Granger-causality map ${\mathcal{F}}_{{\tilde{Y}}_2\to {\tilde{Y}}_1}\left(\mathit{y}\right)$, (c) Granger-causality map ${\mathcal{F}}_{{\tilde{Y}}_1\to {\tilde{Y}}_2}\left(\mathit{y}\right)$; cf. Example 2.

Figure 5

Granger causality ${\mathcal{I}}_{X_2\to X_1}$ and ${\mathcal{I}}_{X_1\to X_2}$ for various $\mu$ and $\beta =1.2$; cf. Example 1.