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:
- <math>C(x,y):=Cov(Z(x),Z(y)).\,</math>
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))).[1]
Admissibilty
For locations x1, x2, …, xN ∈ D the variance of every linear combination
- <math>X=\sum_{i=1}^N w_i Z(x_i)</math>
can be computed as
- <math>var(X)=\sum_{i=1}^N \sum_{j=1}^N w_i C(x_i,x_j) w_j</math>
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 w1, …, wN. A function with this property is called positive definite.
Simplifications with Stationarity
In case of a weakly stationary random field, where
- <math>C(x_i,x_j)=C(x_i+h,x_j+h)\,</math>
for any lag h, the covariance function can represented by a one parameter function
- <math>C_s(h)=C(0,h)=C(x,x+h)\,</math>
which is called a covariogram and also a covariance function. Implicitly the C(xi, xj) can be computed from Cs(h) by:
- <math>C(x,y)=C_s(y-x)\,</math>
The positive definiteness of this single argument version of the covariance function can be checked by Bochner's theorem.[2]