For a random field or Stochastic process on a domain , a covariance function gives the covariance of the values of the random field at the two locations and :

The same is called autocovariance in two instances: in time series (to denote exactly the same concept, but where is time), and in multivariate random fields (to refer to the covariance of a variable with itself, as opposed to the cross covariance between two different variables at different locations, ). [1]

Admissibilty

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For locations   the variance of every linear combinations

 

can be computed by

 

A function is a valid covariance function if and only if [2] this variance is non-negative for all possible choices of N and weights  . A function with this property is called positive definite.


Simplifications with Stationarity

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In case of a second order stationary random field, where

 

for any lag  , the covariance function can represented by a one parameter function

 

which is called covariogram or also covariance function. Implicitly the   can be computed from   by:

 

The positive definitness of the single argument version of the covariance function can be checked by Bochner's theorem. [3]


See also

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Variogram Random Field Stochastic Process Kriging


References

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  1. ^ Wackernagel, Hans (2003). Multivariate Geostatistics. Springer.
  2. ^ Cressie, Noel A.C. (1993). Statistics for Spatial Data. Wiley-Interscience.
  3. ^ Cressie, Noel A.C. (1993). Statistics for Spatial Data. Wiley-Interscience.