Effect of measurement noise on Granger causality

Most of the signals recorded in experiments are inevitably contaminated by measurement noise. Hence, it is important to understand the effect of such noise on estimating causal relations between such signals. A primary tool for estimating causality is Granger causality. Granger causality can be computed by modeling the signal using a bivariate autoregressive (AR) process. In this paper, we greatly extend the previous analysis of the effect of noise by considering a bivariate AR process of general order $p$. From this analysis, we analytically obtain the dependence of Granger causality on various noise-dependent system parameters. In particular, we show that measurement noise can lead to spurious Granger causality and can suppress true Granger causality. These results are verified numerically. Finally, we show how true causality can be recovered numerically using the Kalman expectation maximization algorithm.

Figure 1

Granger causality spectra for a bivariate AR(3) process: (a) causality from $X\to Y$ and (b) causality from $Y\to X$. The true causality spectra are represented by solid lines, while the spectra for noisy data are represented by dashed lines.

Figure 2

Granger causality spectra for a bivariate AR(4) process: (a) causality from $X\to Y$ and (b) causality from $Y\to X$. The true causality spectra are represented by solid lines, while the spectra for noisy data are represented by dashed lines.

Figure 3

Granger causality spectra for the bivariate AR(3) process in Fig. 1: (a) causality from $X\to Y$ and (b) causality from $Y\to X$. The true causality spectra are represented by solid lines, while the spectra for data denoised by the KEM algorithm are represented by dashed lines.

Figure 4

Granger causality spectra for the bivariate AR(4) process in Fig. 2: (a) causality from $X\to Y$ and (b) causality from $Y\to X$. The true causality spectra are represented by solid lines, while the spectra for data denoised by the KEM algorithm are represented by dashed lines.