autocovariance

In probability theory and statistics, given a stochastic process, the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. Autocovariance is closely related to the autocorrelation of the process in question.

== Auto-covariance of stochastic processes == === Definition === With the usual notation \operatorname{E} for the expectation operator, if the stochastic process \left\{X_t\right\} has the mean function \mu_t = \operatorname{E}[X_t], then the autocovariance is given by {{Equation box 1 |indent = : |title= |equation = {{NumBlk||\operatorname{K}_{XX}(t_1,t_2) = \operatorname{cov}\left[X_{t_1}, X_{t_2}\right] = \operatorname{E}[(X_{t_1} - \mu_{t_1})(X_{t_2} - \mu_{t_2})] = \operatorname{E}[X_{t_1} X_{t_2}] - \mu_{t_1} \mu_{t_2}|}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}} where t_1 and t_2 are two instances in time.