Covariance function

For a random field or Stochastic process $Z(x)$ on a domain $D$, a covariance function $C(x,y)$ gives the covariance of the values of the random field at the two locations $x$ and $y$:

$C(x,y):=Cov(Z(x),Z(y)).$

The same $C(x,y)$ is called autocovariance in two instances: in time series (to denote exactly the same concept, but where $x$ 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, $Cov[Z(x_{1}),Y(x_{2})]$).

Admissibilty

For locations $x_{1},x_{2},\ldots ,x_{N}\in D$ the variance of every linear combinations

$X=\sum _{i=1}^{N}w_{i}Z(x_{i})$

can be computed by

$var(X)=\sum _{i=1}^{N}\sum _{i=1}^{N}w_{i}C(x_{i},x_{j})w_{j}$

A function is a valid covariance function if and only if this variance is non-negative for all possible choices of N and weights $w_{1},\ldots ,w_{N}$. A function with this property is called positive definite.

Simplifications with Stationarity

In case of a weakly stationary random field, where

$C(x_{i},x_{j})=C(x_{i}+h,x_{j}+h)$

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

$C_{s}(h)=C(0,h)=C(x,x+h)$

which is called covariogram or also covariance function. Implicitly the $C(x_{i},x_{j})$ can be computed from $C_{s}(h)$ by:

$C(x,y)=C_{s}(y-x)$

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

See also

• Variogram
• Random field
• Stochastic process
• Kriging