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
With the usual notation for the expectation operator, if the stochastic process has the mean function , then the autocovariance is given by: p. 162
where and are two moments in time.
Definition for weakly stationary process
If is a weakly stationary (WSS) process, then the following are true:: p. 163
- for all
- for all
where is the lag time, or the amount of time by which the signal has been shifted.
The autocovariance function of a WSS process is therefore given by:: p. 517
which is equivalent to
It is common practice in some disciplines (e.g. statistics and time series analysis) to normalize the autocovariance function to get a time-dependent Pearson correlation coefficient. However in other disciplines (e.g. engineering) the normalization is usually dropped and the terms "autocorrelation" and "autocovariance" are used interchangeably.
The definition of the normalized auto-correlation of a stochastic process is
If the function is well-defined, its value must lie in the range , with 1 indicating perfect correlation and −1 indicating perfect anti-correlation.
For a WSS process, the definition is
- : p.169
respectively for a WSS process:
- : p.173
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.
Auto-covariance of random vectors