Directionality of coupling from bivariate time series: How to avoid false causalities and missed connections

We discuss some problems encountered in inference of directionality of coupling, or, in the case of two interacting systems, in inference of causality from bivariate time series. We identify factors and influences that can lead to either decreased test sensitivity or false detections and propose ways to cope with them in order to perform tests with high sensitivity and a low rate of false positive results.

Figure 1

(a) Cross predictability $P(X\to Y)$ (solid line) and $P(Y\to X)$ (dashed line), (b) relative average distance of the mutual nearest neighbors $S^{\left(k\right)}(\mathbf{Y}\mid \mathbf{X})$ (solid line) and $S^{\left(k\right)}(\mathbf{X}\mid \mathbf{Y})$ (dashed line), and (c) coarse-grained transinformation rate $i(X\to Y)$ (solid line) and $i(Y\to X)$ (dashed line) for the Rössler system (1) driving the Lorenz system (2), as functions of the coupling strength $ϵ$.

Figure 2

(a) Cross predictability $P(X\to Y)$ (solid line) and $P(Y\to X)$ (dashed line), (b) relative average distance of the mutual nearest neighbors $S^{\left(k\right)}(\mathbf{Y}\mid \mathbf{X})$ (solid line) and $S^{\left(k\right)}(\mathbf{X}\mid \mathbf{Y})$ (dashed line), and (c) coarse-grained transinformation rate $i(X\to Y)$ (solid line) and $i(Y\to X)$ (dashed line) for the unidirectionally coupled Hénon system (3, 4), as functions of the coupling strength $ϵ$.

Figure 3

(a) Cross predictability $P(X\to Y)$ (solid line) and $P(Y\to X)$ (dashed line), (b) relative average distance of the mutual nearest neighbors $S^{\left(k\right)}(\mathbf{Y}\mid \mathbf{X})$ (solid line) and $S^{\left(k\right)}(\mathbf{X}\mid \mathbf{Y})$ (dashed line), and (c) coarse-grained transinformation rate $i(X\to Y)$ (solid line) and $i(Y\to X)$ (dashed line) for the unidirectionally coupled Rössler systems (5, 6), as functions of the coupling strength $ϵ$.

Figure 4

(a) Two largest Lyapunov exponents of the drive $\left\{X\right\}$ (constant lines) and the response $\left\{Y\right\}$ (decreasing lines), (b) averaged conditional mutual information $I(x;y_{\tau }\mid y)$ (solid line) and $I(y;x_{\tau }\mid x)$ (dashed line), and (c) averaged CMI $I(x;y_{\tau }\mid \vec{Y})$ (solid line) and $I(y;x_{\tau }\mid \vec{X})$ (dashed line), where the vectors $\vec{X}$ and $\vec{Y}$ are the original components of the integrated systems, for the unidirectionally coupled Rössler systems (5, 6), as functions of the coupling strength $ϵ$. The Lyapunov exponents are measured in nats per a time unit; the conditional mutual information in all figures is measured in units of nats.

Figure 5

(a) Averaged conditional mutual information $I(x;y_{\tau }\mid \vec{Y})$ (solid line) and $I(y;x_{\tau }\mid \vec{X})$ (dashed line), using the time-delay embedding vectors $\vec{X}=[x_1\left(t\right),x_1(t-\eta ),x_1(t-2\eta )]$ and analogously for $\vec{Y}$, (b) averaged CMI $I\mathbf{(}{\varphi }_1\left(t\right);{\varphi }_2(t+\tau )\mid {\varphi }_2\left(t\right)\mathbf{)}$ (solid line) and $I\mathbf{(}{\varphi }_2\left(t\right);{\varphi }_1(t+\tau )\mid {\varphi }_1\left(t\right)\mathbf{)}$ (dashed line), and (c) averaged CMI $I({\varphi }_1;{\Delta }_{\tau }{\varphi }_2\mid {\varphi }_2)$ (solid line) and $I({\varphi }_2;{\Delta }_{\tau }{\varphi }_1\mid {\varphi }_1)$ (dashed line), for the unidirectionally coupled Rössler systems (5, 6), as functions of the coupling strength $ϵ$.

Figure 6

Histograms of estimates of $I({\varphi }_1;{\Delta }_{\tau }{\varphi }_2\mid {\varphi }_2)$ (solid lines) and $I({\varphi }_2;{\Delta }_{\tau }{\varphi }_1\mid {\varphi }_1)$ (dashed lines) from 1000 realizations of (a) the unidirectionally coupled Rössler systems (5, 6), the coupling strength $ϵ=0.05$, and the number of samples $N=1024$; (b) fast Fourier transform (FFT) surrogate data for the data used in (a), $N=1024$; (c) the same as in (a), but the number of samples $N=512$; and (d) FFT surrogate data for the data used in (c), $N=512$.

Figure 7

Histograms of estimates of $I({\varphi }_1;{\Delta }_{\tau }{\varphi }_2\mid {\varphi }_2)$ (solid lines) and $I({\varphi }_2;{\Delta }_{\tau }{\varphi }_1\mid {\varphi }_1)$ (dashed lines) from 1000 realizations of (a) the unidirectionally coupled Rössler systems (5, 6) with the frequency ratio 1:5, the coupling strength $ϵ=0.1$, and the number of samples $N=1024$; (b) FFT surrogate data for the data used in (a), $N=1024$; (c) the uncoupled Rössler systems (5, 6) with the frequency ratio 1:5, $ϵ=0$, $N=1024$; (d) the uncoupled Rössler systems (5, 6) with the frequency ratio 1.015:0.985, $N=1024$. In all the cases $a_1=a_2=0.15$.

Figure 8

(a) Convergence with the time series length of mean CMI estimates for $I({\varphi }_1;{\Delta }_{\tau }{\varphi }_2\mid {\varphi }_2)$ (thin solid line) and $I({\varphi }_2;{\Delta }_{\tau }{\varphi }_1\mid {\varphi }_1)$ (thin dashed line) for the uncoupled Rössler systems (5, 6) with the frequency ratio 1:5, $a_2=0.15$; $I({\varphi }_2;{\Delta }_{\tau }{\varphi }_1\mid {\varphi }_1)$ (thick dotted line) for the same systems with the frequency ratio 1:5 and the coupling strength $ϵ=0.1$ and $I({\varphi }_2;{\Delta }_{\tau }{\varphi }_1\mid {\varphi }_1)$ (thick solid line) for the systems with the frequency ratio 5:1, $a_1=0.15$ and $ϵ=0.1$. (b) Convergence of the CMI means for the FFT surrogate data of the Rössler systems (5, 6) with the frequency ratio 1:5, $a_2=0.15$ and the coupled ($ϵ=0.1$, thick lines) and the uncoupled ($ϵ=0$, thin lines) cases, $I({\varphi }_1;{\Delta }_{\tau }{\varphi }_2\mid {\varphi }_2)$ (solid lines), and $I({\varphi }_2;{\Delta }_{\tau }{\varphi }_1\mid {\varphi }_1)$ (dashed or dotted lines). (c) Convergence with the time series length of mean CMI estimates for $I({\varphi }_1;{\Delta }_{\tau }{\varphi }_2\mid {\varphi }_2)$ (thin solid line) and $I({\varphi }_2;{\Delta }_{\tau }{\varphi }_1\mid {\varphi }_1)$ (thin dashed line) for the uncoupled Rössler systems (5, 6) with the frequency ratio 1:5, $a_2=0.72$; $I({\varphi }_2;{\Delta }_{\tau }{\varphi }_1\mid {\varphi }_1)$ (thick dotted line) for the same systems with the frequency ratio 1:5 and the coupling strength $ϵ=0.11$ and $I({\varphi }_2;{\Delta }_{\tau }{\varphi }_1\mid {\varphi }_1)$ (thick solid line) for the systems with the frequency ratio 5:1, $a_1=0.72$ and $ϵ=0.11$. (d) Convergence of the CMI means for the FFT surrogate data of the Rössler systems (5, 6) with the frequency ratio 1:5, $a_2=0.72$, and the coupled ($ϵ=0.11$, thick lines) and uncoupled ($ϵ=0$, thin lines) cases, $I({\varphi }_1;{\Delta }_{\tau }{\varphi }_2\mid {\varphi }_2)$ (solid lines), and $I({\varphi }_2;{\Delta }_{\tau }{\varphi }_1\mid {\varphi }_1)$ (dashed or dotted lines).

Figure 9

(a) Sensitivity as function of time series length $N$ for the tests using the coupled Rössler systems (5, 6) with the frequency ratio 1:1 (dash-dotted line), 5:1 (solid line), and 1:5 (dashed line) for $a_1=a_2=0.15$. The Hilbert phases were used. (b) Sensitivity of the tests for the coupled Rössler systems (5, 6) with the frequency ratio 5:1 (solid line) and 1:5 (dashed line) for $a_1=a_2=0.15$, as a function of the series length $N$. The marked event phases were used. (c) Sensitivity as function of time series length $N$ for the tests using the coupled Rössler systems (5, 6) with the frequency ratio 5:1, $a_1=0.72$, $a_2=0.15$ (solid line), and 1:5, $a_1=0.15$, $a_2=0.72$ (dashed line). The Hilbert phases were used. (d) The same as in (c), but the marked event phases were used.

Figure 10

Sensitivity (a),(c) and the rate of false positives (b),(d) as functions of time series length $N$ for the tests using the coupled Rössler systems (5, 6) with the frequency ratio 1:1 for different amounts of noise in the data. The portions of noise are in (a),(b) 0% (dotted line), 10% (dashed line), 20% (dash-dotted line), and 30% (solid line); in (c),(d) 50% (dashed line), 70% (dash-dotted line), and 100% (solid line).

Figure 11

Tests of causal influences of cardiac oscillations on respiratory oscillations measured by $I({\varphi }_C;{\Delta }_{\tau }{\varphi }_R\mid {\varphi }_R)$ (a),(c) and of the influence of the respiratory rhythm on the cardiac oscillations given by $I({\varphi }_R;{\Delta }_{\tau }{\varphi }_C\mid {\varphi }_C)$ (b),(d). Values from the tested data are marked by the vertical lines, the surrogate ranges are illustrated by the histograms obtained from 2500 surrogate realizations. (a),(b) The test of bias using data from two animals; (c),(d) a real test for one of the animals.