Covariance function

In probability theory and statistics, covariance is a measure of how much two variables change together, and the covariance function, or kernel, describes the spatial or temporal covariance of a random variable process or field. For a random field or 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 the function in two instances: in time series (to denote exactly the same concept except that x and y refer to locations in time rather than in space), 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))).

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 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.

In case of a weakly stationary random field, where

for any lag h, the covariance function can be 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.

A simple stationary parametric covariance function is the "exponential covariance function"

where V is a scaling parameter, and d=d(x,y) is the distance between two points. Sample paths of a Gaussian process with the exponential covariance function are not smooth. The "squared exponential covariance function"

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