Open main menu

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.

Contents

Auto-covariance of stochastic processesEdit

DefinitionEdit

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.2)

where   and   are two moments in time.

Definition for weakly stationary processEdit

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.3)

which is equivalent to

 .

NormalizationEdit

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

 .

PropertiesEdit

Symmetry propertyEdit

 [3]:p.169

respectively for a WSS process:

 [3]:p.173

Linear filteringEdit

The autocovariance of a linearly filtered process  

 

is

 

Calculating turbulent diffusivityEdit

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.

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:
     
  3. If we have velocity information at two fixed (Eulerian) locations:
     
    where   is the distance separated by these two fixed locations.

Auto-covariance of random vectorsEdit

See alsoEdit

ReferencesEdit

  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". Proceedings of the London Mathematical Society. s2-20 (1): 196–212. doi:10.1112/plms/s2-20.1.196. ISSN 1460-244X.