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))).

Admissibilty

For locations x₁, x₂, …, x_N ∈ 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 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

<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(x_i, x_j) can be computed from C_s(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.

See also

• Variogram
• Random field
• Stochastic process
• Kriging