cross-covariance
Sign in to saveIn probability and statistics, given two stochastic processes \left\{X_t\right\} and \left\{Y_t\right\}, the cross-covariance is a function that gives the covariance of one process with the other at pairs of time points. With the usual notation \operatorname E for the expectation operator, if the processes have the mean functions \mu_X(t) = \operatorname \operatorname E[X_t] and \mu_Y(t) = \operatorname E[Y_t], then the cross-covariance is given by
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Encyclopedic overview
10 sectionsContents
- Cross-covariance of random vectors
- Cross-covariance of stochastic processes
- Definition
- Definition for jointly WSS processes
- Uncorrelatedness
- Cross-covariance of deterministic signals
- Properties
- See also
- References
- External links
In probability and statistics, given two stochastic processes \left\{X_t\right\} and \left\{Y_t\right\}, the cross-covariance is a function that gives the covariance of one process with the other at pairs of time points. With the usual notation \operatorname E for the expectation operator, if the processes have the mean functions \mu_X(t) = \operatorname \operatorname E[X_t] and \mu_Y(t) = \operatorname E[Y_t], then the cross-covariance is given by \operatorname{K}_{XY}(t_1,t_2) = \operatorname{cov} (X_{t_1}, Y_{t_2}) = \operatorname{E}[(X_{t_1} - \mu_X(t_1))(Y_{t_2} - \mu_Y(t_2))] = \operatorname{E}[X_{t_1} Y_{t_2}] - \mu_X(t_1) \mu_Y(t_2).\,
Cross-covariance is related to the more commonly used cross-correlation of the processes in question.
Excerpted from Wikipedia’s “cross-covariance” article, available under the CC BY-SA 4.0 licence.