Finddelay

Cross-covariance

\[\gamma _{XY}(\tau )=\operatorname {E} \left[\left(X_{t}-\mu _{X}\right)\left(Y_{t+\tau }-\mu _{Y}\right)\right],
\]

Cross-correlation

In signal processing,

\[\delta_{x,y}[n] = \sum _{m=-\infty }^{\infty }x^{*}[m]\ y[m+n].
\]

In time series analysis, as applied in statistics, the cross-correlation between two time series is the normalized cross-covariance function.
Cross correlation coefficient

\[\rho _{XY}(\tau )={\frac {1}{\sigma _{X}\sigma _{Y}}}\operatorname {E} \left[\left(X_{t}-\mu _{X}\right)\left(Y_{t+\tau }-\mu _{Y}\right)\right]={\frac {1}{\sigma _{X}\sigma _{Y}}}\gamma _{XY}(\tau ),
\]

where \(\mu _{X}\) and \(\sigma _{X}\) are the mean and standard deviation of the process \((X_{t})\), which are constant over time due to stationarity; and similarly for \((Y_{t})\), respectively. That the cross-covariance and cross-correlation are independent of \(t\) is precisely the additional information (beyond being individually wide-sense stationary) conveyed by the requirement that {\displaystyle (X_{t},Y_{t})} {\displaystyle (X_{t},Y_{t})} are jointly wide-sense stationary.

Cross correlation （pearson）
Covariance
coefficient of determination

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