In probability theory and statistics, given a stochastic process X = ( X t ) , the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. With the usual notation E for the expectation operator, if the process has the mean function μ t = E [ X t ] , then the autocovariance is given by
C X X ( t , s ) = cov ( X t , X s ) = E [ ( X t − μ t ) ( X s − μ s ) ] = E [ X t X s ] − μ t μ s , where t and s are two time periods or moments in time.
Autocovariance is closely related to the more commonly used autocorrelation of the process in question.
In the case of a multivariate random vector X = ( X 1 , X 2 , . . . , X n ) , the autocovariance becomes a square n by n matrix, C X X , with entry i , j given by C X i X j ( t , s ) = cov ( X i , t , X j , s ) and commonly referred to as the autocovariance matrix associated with vectors X t and X s .
If X(t) is a weakly stationary process, then the following are true:
μ t = μ s = μ for all
t,
sand
C X X ( t , s ) = C X X ( s − t ) = C X X ( τ ) where τ = | s − t | is the lag time, or the amount of time by which the signal has been shifted.
When normalizing the autocovariance, C, of a weakly stationary process with its variance, C X X ( 0 ) = σ 2 , one obtains the autocorrelation coefficient ρ :
ρ X X ( τ ) = C X X ( τ ) σ 2 with − 1 ≤ ρ X X ( τ ) ≤ 1 .
The autocovariance of a linearly filtered process Y t
Y t = ∑ k = − ∞ ∞ a k X t + k is
C Y Y ( τ ) = ∑ k , l = − ∞ ∞ a k a l C X X ( τ + k − l ) . 