Autocovariance

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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[edit]

Definition[edit]

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 moments in time.

Definition for weakly stationary process[edit]

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[edit]

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[edit]

Symmetry property[edit]

[3]: p.169

respectively for a WSS process:

[3]: p.173

Linear filtering[edit]

The autocovariance of a linearly filtered process

is

Calculating turbulent diffusivity[edit]

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[edit]

See also[edit]

References[edit]

  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.