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 formula_1 for the expectation operator, if the stochastic process formula_2 has the mean function formula_3, then the autocovariance is given by where formula_4 and formula_5 are two moments in time. Definition for weakly stationary process. If formula_2 is a weakly stationary (WSS) process, then the following are true: and  and where formula_12 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: which is equivalent to Normalization. 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 formula_15 is well-defined , its value must lie in the range formula_16, with 1 indicating perfect correlation and −1 indicating perfect anti-correlation. For a WSS process, the definition is where Properties. Symmetry property. respectively for a WSS process: Linear filtering. The autocovariance of a linearly filtered process formula_21 is 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 formula_24 (assume we are now working with  1D problem and formula_25 is the velocity along formula_26 direction): where formula_25 is the true velocity, and formula_29 is the expected value of velocity. If we choose a correct formula_29, all of the stochastic components of the turbulent velocity will be included in formula_24. To determine formula_29, 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 formula_33 (formula_34, 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 formula_38 is the lag time, and formula_39 is the lag distance. The turbulent diffusivity formula_40 can be calculated using the following 3 methods: