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

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Definition

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With the usual notation   for the expectation operator, if the stochastic process   has the mean function  , then the autocovariance is given by[1]: p. 162 

  (Eq.1)

where   and   are two instances in time.

Definition for weakly stationary process

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If   is a weakly stationary (WSS) process, then the following are true:[1]: p. 163 

  for all  

and

  for all  

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 

  (Eq.2)

which is equivalent to

 .

Normalization

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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

 .

Properties

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Symmetry property

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 [3]: p.169 

respectively for a WSS process:

 [3]: p.173 

Linear filtering

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The autocovariance of a linearly filtered process  

 

is

 

Calculating turbulent diffusivity

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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   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

  or  

where   is the lag time, and   is the lag distance.

The turbulent diffusivity   can be calculated using the following 3 methods:

  1. If we have velocity data along a Lagrangian trajectory:
     
  2. If we have velocity data at one fixed (Eulerian) location[citation needed]:
     
  3. If we have velocity information at two fixed (Eulerian) locations[citation needed]:
     
    where   is the distance separated by these two fixed locations.

Auto-covariance of random vectors

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See also

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References

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  1. ^ a b Hsu, Hwei (1997). Probability, random variables, and random processes. McGraw-Hill. ISBN 978-0-07-030644-8.
  2. ^ Lapidoth, Amos (2009). A Foundation in Digital Communication. Cambridge University Press. ISBN 978-0-521-19395-5.
  3. ^ a b Kun Il Park, Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018, 978-3-319-68074-3
  4. ^ Taylor, G. I. (1922-01-01). "Diffusion by Continuous Movements" (PDF). Proceedings of the London Mathematical Society. s2-20 (1): 196–212. doi:10.1112/plms/s2-20.1.196. ISSN 1460-244X.

Further reading

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