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:

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]

Contents 1 Admissibilty 2 Simplifications with Stationarity 3 See also 4 References

[edit] Admissibilty

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

can be computed as

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, …, w_N. A function with this property is called positive definite.

[edit] Simplifications with Stationarity

In case of a weakly stationary random field, where

for any lag h, the covariance function can represented by a one parameter function

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:

The positive definiteness of this single argument version of the covariance function can be checked by Bochner's theorem.^[2]

[edit] See also

• Variogram
• Random field
• Stochastic process
• Kriging

[edit] References

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