Case IV: experiments on signals of cases II and III with imposed additive Gaussian noise.

<p>We consider noise levels at 5% and 10% of the standard deviation of the original signals in cases II and III. (a) Δ<i>ρ</i> at <i>L</i> = 500 as a function of </p><p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub><mo>/</mo><msub><mi>μ</mi><mo>¯</mo><mi>N</mi></msub></mrow></math></p> for signals of case II. (b) Δ<i>ρ</i> at <i>L</i> = 500 as a function of <p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub><mo>/</mo><msub><mi>μ</mi><mo>¯</mo><mrow><mi>W</mi><mi>b</mi></mrow></msub></mrow></math></p> for signals of case III. Δ<i>ρ</i> = 0 and <p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub><mo>/</mo><msub><mi>μ</mi><mo>¯</mo><mi>N</mi></msub><mo>=</mo><mn>1</mn></mrow></math></p> are shown by dashed, bold lines.<p></p

Schematics of <i>L</i>–point windows and reconstructed phase spaces.

<p>(a) Two time–series <i>x</i>(<i>t</i>) and <i>y</i>(<i>t</i>) and a schematic of the <i>L</i>–point windows with different lengths sweeping the whole span of the time–series. (b) A schematic of the reconstructed phase spaces of the <i>L</i>–point windows corresponding to two time–series. For each <i>E</i>-dimensional Y–central point, <i>Y</i>_<i>c</i>, in the response reconstructed phase space, <i>M</i>_<i>y</i>, a sufficient number of nearest neighbor points are selected (empty circles, <i>Y</i>_<i>j</i>, right) and their distances, <i>d</i>_<i>j</i>, to <i>Y</i>_<i>c</i> are determined. For each neighbor point, its contemporaneous point in the driver reconstructed phase space, <i>M</i>_<i>x</i>, is determined (empty circles, <i>X</i>_<i>j</i>, left). The weighted average of these points, </p><p><math><mrow><mi>X</mi><mo>̂</mo></mrow></math></p>, is compared with <i>X</i>_<i>c</i>, the true contemporaneous point in <i>M</i>_<i>x</i> corresponding to <i>Y</i>_<i>c</i>. The CCM coefficient, <i>ρ</i>(<i>L</i>), is defined as the correlation coefficient between <p><math><mrow><mi>X</mi><mo>̂</mo></mrow></math></p> and <i>X</i>_<i>c</i>, averaged over all possible <i>L</i>–point windows.<p></p

Case III: mixed normal–nonnormal coupling coefficients.

<p>(a) </p><p><math><mrow><mo>Δ</mo><mi>ρ</mi><mo>=</mo><msub><mi>ρ</mi><mrow><mi>X</mi><mo>̂</mo><mo>∣</mo><mi>Y</mi></mrow></msub><mo>−</mo><msub><mi>ρ</mi><mrow><mi>Y</mi><mo>̂</mo><mo>∣</mo><mi>X</mi></mrow></msub></mrow></math></p> at <i>L</i> = 500 as a function of <p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub><mo>/</mo><msub><mi>μ</mi><mo>¯</mo><mrow><mi>W</mi><mi>b</mi></mrow></msub></mrow></math></p>. (b) 1000 independent realizations of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0131226#pone.0131226.e069" target="_blank">Eq (12)</a> and the corresponding Δ<i>ρ</i> at <i>L</i> = 500 as a function of <p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub><mo>/</mo><msub><mi>μ</mi><mo>¯</mo><mrow><mi>W</mi><mi>b</mi></mrow></msub></mrow></math></p>. <p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub></mrow></math></p> and <p><math><mrow><msub><mi>μ</mi><mo>¯</mo><mrow><mi>W</mi><mi>b</mi></mrow></msub></mrow></math></p> are the mean values of <i>η</i>_<i>N</i> and <i>μ</i>_<i>Wb</i> over the span of the time–series. Δ<i>ρ</i> = 0 and <p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub><mo>/</mo><msub><mi>μ</mi><mo>¯</mo><mi>N</mi></msub><mo>=</mo><mn>1</mn></mrow></math></p> are shown by dashed, bold lines.<p></p

Case I: periodic–constant coupling coefficients.

<p>(a) </p><p><math><mrow><msub><mi>ρ</mi><mrow><mi>Y</mi><mo>̂</mo><mo>∣</mo><mi>X</mi></mrow></msub></mrow></math></p> at <i>L</i> = 500 over the range of <i>μ</i> and <i>η</i> given by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0131226#pone.0131226.e031" target="_blank">Eq (10)</a> and for random initial condition. (b) Difference between the CCM coefficients, <p><math><mrow><msub><mi>ρ</mi><mrow><mi>Y</mi><mo>̂</mo><mo>∣</mo><mi>X</mi></mrow></msub><mo>−</mo><msub><mi>ρ</mi><mrow><mi>X</mi><mo>̂</mo><mo>∣</mo><mi>Y</mi></mrow></msub></mrow></math></p>, at <i>L</i> = 500 over the specified range of <i>μ</i> and <i>η</i>.<p></p

Case IV: experiments on signals of cases II and III with imposed temporal uncertainties.

<p>We consider temporal shifts corresponding to 2.5% and 5% of the full length of the time series in cases II and III. (a) Δ<i>ρ</i> at <i>L</i> = 500 as a function of </p><p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub><mo>/</mo><msub><mi>μ</mi><mo>¯</mo><mi>N</mi></msub></mrow></math></p> for shifted signals of case II. (b) Δ<i>ρ</i> at <i>L</i> = 500 as a function of <p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub><mo>/</mo><msub><mi>μ</mi><mo>¯</mo><mrow><mi>W</mi><mi>b</mi></mrow></msub></mrow></math></p> for shifted signals of case III. Δ<i>ρ</i> = 0 and <p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub><mo>/</mo><msub><mi>μ</mi><mo>¯</mo><mi>N</mi></msub><mo>=</mo><mn>1</mn></mrow></math></p> are shown by dashed, bold lines.<p></p

An example of lead–lag switching between two signals of a nonlinear system.

<p>Observation of the leading signals (shown by the colored arrows) in short intervals may result in an incorrect conclusion about the cause–effect relationship. The time–series in this figure are generated by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0131226#pone.0131226.e024" target="_blank">Eq (9)</a>.</p

Case II: normally distributed coupling coefficients.

<p>(a) The difference between the CCM coefficients, </p><p><math><mrow><mo>Δ</mo><mi>ρ</mi><mo>=</mo><msub><mi>ρ</mi><mrow><mi>X</mi><mo>̂</mo><mo>∣</mo><mi>Y</mi></mrow></msub><mo>−</mo><msub><mi>ρ</mi><mrow><mi>Y</mi><mo>̂</mo><mo>∣</mo><mi>X</mi></mrow></msub></mrow></math></p>, at <i>L</i> = 500 as a function of the ratio of the averaged coupling coefficients, <p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub><mo>/</mo><msub><mi>μ</mi><mo>¯</mo><mi>N</mi></msub></mrow></math></p>. (b) Monte Carlo simulation of 1000 independent realizations of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0131226#pone.0131226.e048" target="_blank">Eq (11)</a> to calculate Δ<i>ρ</i> at <i>L</i> = 500 as a function of <p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub><mo>/</mo><msub><mi>μ</mi><mo>¯</mo><mi>N</mi></msub></mrow></math></p>. <p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub></mrow></math></p> and <p><math><mrow><msub><mi>μ</mi><mo>¯</mo><mi>N</mi></msub></mrow></math></p> are the mean values of <i>η</i>_<i>N</i> and <i>μ</i>_<i>N</i> over the span of the time–series. Δ<i>ρ</i> = 0 and <p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub><mo>/</mo><msub><mi>μ</mi><mo>¯</mo><mi>N</mi></msub><mo>=</mo><mn>1</mn></mrow></math></p> are shown by dashed, bold lines.<p></p

Case IV: experiments on signals of cases II and III with imposed additive Gaussian noise.

<p>We consider noise levels at 5% and 10% of the standard deviation of the original signals in cases II and III. (a) Δ<i>ρ</i> at <i>L</i> = 500 as a function of </p><p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub><mo>/</mo><msub><mi>μ</mi><mo>¯</mo><mi>N</mi></msub></mrow></math></p> for signals of case II. (b) Δ<i>ρ</i> at <i>L</i> = 500 as a function of <p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub><mo>/</mo><msub><mi>μ</mi><mo>¯</mo><mrow><mi>W</mi><mi>b</mi></mrow></msub></mrow></math></p> for signals of case III. Δ<i>ρ</i> = 0 and <p><math><mrow><msub><mi>η</mi><mo>¯</mo><mi>N</mi></msub><mo>/</mo><msub><mi>μ</mi><mo>¯</mo><mi>N</mi></msub><mo>=</mo><mn>1</mn></mrow></math></p> are shown by dashed, bold lines.<p></p