Time-Delay Correlation

Synonyms

Time-lagged correlation

Definition

Time-delay correlation is designed to infer potential causal relationships between perturbations and responses in systems. It extends linear correlation by determining the best correlations among profiles shifted in time. For given time-dependent vectors $X(t)$ and $Y(t)$, the correlation between them with a time lag $\tau$ is defined by

$$ {{S}_{{XY}}}(\tau ) = \left\langle {(X(t) - \bar{X})(Y(t + \tau ) - \bar{Y})} \right\rangle $$

$$ {{r}_{{XY}}}(\tau ) = \frac{{{{S}_{{XY}}}(\tau )}}{{\sqrt {{{{S}_{{XX}}}(\tau ){{S}_{{YY}}}(\tau )}} }} $$

where $\bar{X}$, $\bar{Y}$ is the average value of $X$ and $Y$ across time, respectively, and the angled brackets represent the inner product between the time-shifted profiles. Then the maximum absolute value of ${{r}_{{XY}}}(\tau )$ across time

$$ {{c}_{{ij}}} = \mathop{{\max }}\limits_{\tau } |{{r}_{{XY}}}(\tau )| $$

is used to represent the best correlation between \( X...