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.
for the expectation operator, if the stochastic process
is a weakly stationary (WSS) process, then the following are true:[1]: p. 163 and and 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:[2]: 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 where respectively for a WSS process: The autocovariance of a linearly filtered process
is Autocovariance can be used to calculate turbulent diffusivity.
[4] 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[citation needed].
Reynolds decomposition is used to define the velocity fluctuations
(assume we are now working with 1D problem and
If we choose a correct
, all of the stochastic components of the turbulent velocity will be included in
, 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
can be calculated using the following 3 methods: