Covariance function

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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))).[1]

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[edit] Admissibilty

For locations x1, x2, …, xN ∈ 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[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.


[edit] 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.[2]


[edit] See also

[edit] References

  1. ^ Wackernagel, Hans (2003). Multivariate Geostatistics. Springer. 
  2. ^ a b Cressie, Noel A.C. (1993). Statistics for Spatial Data. Wiley-Interscience.