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:

$C(x,y):=\operatorname {cov} (Z(x),Z(y)).\,$

The same C(x, y) is called autocovariance in two instances: in time series (to denote exactly the same concept, 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(x₁), Y(x₂))).

Admissibility

For locations x₁, x₂, …, x_N ∈ D the variance of every linear combination

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

can be computed as

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

A function is a valid covariance function if and only if this variance is non-negative for all possible choices of N and weights w₁, …, 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 a covariogram and also a 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 definiteness of this single argument version of the covariance function can be checked by Bochner's theorem.

See also

• Variogram
• Random field
• Stochastic process
• Kriging
• Autocorrelation function