Figure 1

(Color online) Bias of mutual information with black solid line and conditional mutual information with gray dashed line (cyan on line) for Gaussian time series with (a) $d_X=d_Y=1$ and $d_Z=1,\dots ,8$, $N=4096$, (b) $d_X=d_Y=1$ and $d_Z=1,\dots ,8$, $N=8192$, (c) $d_X=8$, $d_Y=1$ and $d_Z=1,\dots ,8$, $N=4096$. Lines correspond to different correlation coefficient $r=0.2,\dots ,0.9$ with step 0.1. In (d) we have the case of varying correlation coefficients with $d_X=d_Y=1$ and $d_Z=1,\dots ,8$, $N=4096$.Reuse & Permissions

Figure 2

(Color online) (a)-(d) The first four embedding cycles of $I_0$ for the reconstruction aiming to explain Lorenz’s $x$ variable from $x$ and $z$ variables. Each panel shows the mutual information for all candidate components at each iteration, where the solid line represents lags of $x$, the dotted line represents lags of $z$ and the selected lag is highlighted. The selected lag in panel (d) is not included in the embedding vector because the ratio exceeds the threshold value. Panels (f)-(h) show the corresponding to (b)-(d) embedding cycles using $I_1$ and showing the conditional mutual information. The first embedding cycle is omitted because it is identical to (a). Panel (e) has the values of mutual information (criterion $I_0$—black line) and conditional mutual information (criterion $I_1$—gray line, cyan online) for 10 embedding cycles, with the left and right vertical axes corresponding to the values of MI and CMI, respectively.Reuse & Permissions

Figure 3

(Color online) (a) $E_{\mathbf{x}}$ (solid line) and $E_{{\mathbf{x}}^2}$ (dashed line) for the driven Henon $y$ across all $C$, where embedding vectors are suggested by $I_0$ in the upper panel and $I_1$ in the lower. (b) $S_{X\to Y}$ and $R_{X\to Y}$ for the same system across all $C$ and for both criteria (solid line for $I_0$ and dashed for $I_1$, with the left vertical axis corresponding to values of $S$ and the right of $R$). (c) The same as (b) for unidirectionally coupled Henon maps with 20% noise.Reuse & Permissions

Figure 4

(Color online) As Figs. 3b, 3c but for the coupled Rössler–Lorenz system.Reuse & Permissions

Figure 5

(Color online) $R_{X\to Y}$ vs $A$ for the coupled Rössler–Lorenz system using the $I_1$ criterion and for different time series length $N$, as shown in the legend. For each $N$, the dark line is the median and the two gray (cyan on line) lines are the 2.5% and 97.5% percentiles. The coupling strength is $C=0.5$ in (a) and $C=1$ in (b).Reuse & Permissions

Figure 6

(Color online) (a) $S_{X\to Y}$ vs $C$ for the coupled Rössler–Lorenz system using the $I_1$ criterion and for different time horizons $T$, as shown in the legend. (b) The same as (a) but for $R_{X\to Y}$.Reuse & Permissions

Figure 7

Number of components from each time series for mixed embedding vectors with regard to the coupling strength for the Mackey-Glass systems. Solid line denotes the driving system and dashed line the driven system. The panels are ordered for the systems indicated by ${\Delta }_1$ for the driving system and ${\Delta }_2$ for the driven system. The size of the markers indicates the frequency of occurrence of the most frequently selected embedding vectors (same size of the markers for each $C$ value). For example, the size of the marker in panel (e) for $C=0$ regards the largest frequency 100, whereas in the same panel for $C=0.2$ is the smallest frequency in these results equal to 42.Reuse & Permissions

Figure 8

(Color online) Percentage of rejection vs coupling strength $C$ for the surrogate data test of no causality from $x$ to $y$ for the coupled Rössler-Lorenz system $\left(A=0.95,T=3,N=4096\right)$. The measures $S$ and $R$ used as test statistics and the significance levels $\alpha$ are given in the legend.Reuse & Permissions

Figure 9

Number of components from each channel for mixed embedding vectors for patient A. Each row of 2 panels corresponds to a mixed embedding for explaining the signal of the channel indicated by the label on the left of the panels. Each left panel shows the number of components in the mixed embedding that are derived from the channel being explained and the right panel those from its antidiametrical. For example, the upper left panel has the number of components of C3 and the upper right those of C4 used in the embedding explaining C3, both as a function of time. The dashed vertical line shows the seizure onset.Reuse & Permissions

Figure 10

The same as Fig. 9 but for patient B.Reuse & Permissions

Figure 11

The $R$ measure for the mixed embeddings of Fig. 10 for patient B. $R_{C4\to C3}$ is shown in top left panel and $R_{C3\to C4}$ in top right panel. Similarly for the other panels, as indicated by the inset text.Reuse & Permissions