Reducing the bias of causality measures

Measures of the direction and strength of the interdependence between two time series are evaluated and modified to reduce the bias in the estimation of the measures, so that they give zero values when there is no causal effect. For this, point shuffling is employed as used in the frame of surrogate data. This correction is not specific to a particular measure and it is implemented here on measures based on state space reconstruction and information measures. The performance of the causality measures and their modifications is evaluated on simulated uncoupled and coupled dynamical systems and for different settings of embedding dimension, time series length, and noise level. The corrected measures, and particularly the suggested corrected transfer entropy, turn out to stabilize at the zero level in the absence of a causal effect and detect correctly the direction of information flow when it is present. The measures are also evaluated on electroencephalograms (EEG) for the detection of the information flow in the brain of an epileptic patient. The performance of the measures on EEG is interpreted in view of the results from the simulation study.

Figure 1

(a) Mean estimated values of MCR ($M$) and CMCR ($M^C$) for both directions from $100$ realizations of the noise-free unidirectionally coupled Henon map, where $m_x=2$, $m_y=2$, and $n=512$. (b) and (c) are as in (a), but for $m_x=2,m_y=4$ and $m_x=4,m_y=2$, respectively.

Figure 2

(a) Mean estimated values of TE ($T$), ETE ($T^E$), and CTE ($T^C$) for both directions from $100$ realizations of the noise-free unidirectionally coupled Henon maps, with $n=512$ and $m_x=m_y=2$. (b) and (c) are as in (a) but for $m_x=2,m_y=4$ and $m_x=4,m_y=2$, respectively.

Figure 3

(a) Mean estimated values of STE ($T^S$), ESTE ($T^{\mathit{SE}}$), and CSTE ($T^{\mathit{SC}}$) for both directions from $100$ realizations of the noise-free unidirectionally coupled Henon maps, with $n=512$ and $m_x=m_y=3$. (b) As in (a) but for $m_x=4,m_y=3$. (c) As in (b) but for $20\%$ noise level.

Figure 4

(a) Mean estimated values of MCR ($M$) and CMCR ($M^C$) for both directions from $100$ realizations of the noise-free unidirectionally coupled Mackey-Glass systems with ${\Delta }_x=30$ and ${\Delta }_y=100$, $n=2048$, and $m_x=m_y=3$. (b) As in (a) but for $m_x=4,m_y=3$.

Figure 5

(a) Mean estimated values of TE ($T$), ETE ($T^E$), and CTE ($T^C$) for both directions from $100$ realizations of the noise-free unidirectionally coupled Mackey-Glass systems (${\Delta }_x=30$, ${\Delta }_y=100$) with $n=2048$ and $m_x=4$ and $m_y=3$. (b) As in (a) but for $20\%$ noise level. (c) and (d) As in (a) but for ${\Delta }_x={\Delta }_y=17$, $m_x=m_y=3$, and $m_x=m_y=5$, respectively.

Figure 6

The panels are as for the first three panels of Fig. 5 but for the symbolic transfer entropy measures.

Figure 7

(a) Mean TE ($T$), ETE ($T^E$), and CTE ($T^C$) (error bars denote the standard deviation) from $100$ realizations of the noise-free uncoupled Mackey-Glass systems with ${\Delta }_1={\Delta }_2=17$, $n=2048$, and for varying $m_x=m_y$. Only the one direction is shown as the two systems are the same. (b) As in (a) but for STE, ESTE, and CTE. (c) As in (a) but for ${\Delta }_1=30,{\Delta }_2=100$; the two directions are shown, as given in the legend, with a measure at each panel. The drawn points and error bars for each measure are slightly displaced along the $x$ axis to facilitate visualization.

Figure 8

Mean TE ($T$) (error bars denote the standard deviation) from $100$ realizations of the coupled stochastic system vs time-series length $\log {}_{2}n$, for the variables $X$ and $Y$, and $m_x=m_y=1$. (b) As in (a) but for CTE ($T^C$). (c) As in (a) but for $m_x=m_y=2$. (d) As in (b) but for CTE ($T^C$) and ETE ($T^E$) and $m_x=m_y=2$. The drawn points and error bars for each measure are slightly displaced along the $x$ axis to facilitate visualization.

Figure 9

(a)Mean CTE ($T^C$) and TE from the time-shifted surrogates, denoted as $T^{\text{ts}}$ (error bars denote the standard deviation) from $100$ realizations of the noise-free Mackey-Glass systems with ${\Delta }_1={\Delta }_2=17$, $n=2048$, and for $m_x=m_y=5$. (b) and (c) are as (a) but for ${\Delta }_1=17,{\Delta }_2=30$ and ${\Delta }_1=30,{\Delta }_2=100$, respectively. The curves and error bars for each measure are slightly displaced along the $x$ axis to facilitate visualization.

Figure 10

(a) MCR profiles ($M$) from both states (early and late preictal) of the first seizure and both directions, for channels as in the caption, with $m_x=m_y=3$. (b) As in (a) but for $m_x=m_y=5$. (c), (d), and (e) are as in (a) but for CMCR ($M^C$), TE ($T$), and CTE ($T^C$), respectively. The preictal periods are indicated by the time in minutes, with reference to time $0$ at seizure onset, and they are separated by a vertical dashed line.

Figure 11

(a) STE profiles ($T^S$) from both states (early and late preictal) of the first seizure and both directions, for channels as in the inset, with $m_x=m_y=5$. (b) As in (a) but for CSTE ($T^{\mathit{SC}}$).

Figure 12

(a) MCR profiles ($M$) from both states (early and late preictal) and both directions of the intracranial data, for channels as in the inset, with $m_x=m_y=5$. (b), (c), and (d) are as in (a) but for CMCR ($M^C$), STE ($T^C$), and CSTE ($T^{\mathit{SC}}$), respectively.