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, 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))).

Admissibilty
For locations x1, x2, &hellip;, xN &isin; D the variance of every linear combination


 * $$X=\sum_{i=1}^N w_i Z(x_i)$$

can be computed as


 * $$var(X)=\sum_{i=1}^N \sum_{j=1}^N w_i C(x_i,x_j) w_j$$

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, &hellip;, wN. 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(xi, xj) can be computed from Cs(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.