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. With the usual notation E for the expectation operator, if the process has the mean function , then the autocovariance is given by
where t and s are two time periods or moments in time.
Autocovariance is closely related to the autocorrelation of the process in question.
In the case of a multivariate random vector , the autocovariance becomes a square n × n matrix with entries given by and commonly referred to as the autocovariance matrix associated with vectors and .
If X(t) is a weakly stationary process, then the following are true:
- for all t, s
where 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 , one obtains the autocorrelation coefficient
The autocovariance of a linearly filtered process
Calculating turbulent diffusivity
Autocovariance can be used to calculate turbulent diffusivity. Turbulence in a flow can cause the fluctuation of velocity in space and time. Thus, we are able to identify turbulence through the statistics of those fluctuations.
Reynolds decomposition is used to define the velocity fluctuations (assume we are now working with 1D problem and is the velocity along direction):
where is the true velocity, and is the expected value of velocity. If we choose a correct , all of the stochastic components of the turbulent velocity will be included in . To determine , a set of velocity measurements that are assembled from points in space, moments in time or repeated experiments is required.
If we assume the turbulent flux (, and c is the concentration term) can be caused by a random walk, we can use Fick's laws of diffusion to express the turbulent flux term:
The velocity autocovariance is defined as
where is the lag time, and is the lag distance.
The turbulent diffusivity can be calculated using the following 3 methods:
- If we have velocity data along a Lagrangian trajectory:
- If we have velocity data at one fixed (Eulerian) location:
- If we have velocity information at two fixed (Eulerian) locations:
where is the distance separated by these two fixed locations.