 Covariance function

In probability theory and statistics, covariance is a measure of how much two variables change together and the covariance function describes the variance of a random variable process or field. 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(x_{1}), Y(x_{2}))).^{[1]}
Contents
Admissibility
For locations x_{1}, x_{2}, …, 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 nonnegative for all possible choices of N and weights w_{1}, …, w_{N}. A function with this property is called positive definite.
Simplifications with stationarity
In case of a weakly stationary random field, where
for any lag h, the covariance function can be represented by a oneparameter 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 singleargument version of the covariance function can be checked by Bochner's theorem.^{[2]}
Parametric families of covariance functions
A simple stationary parametric covariance function is the "exponential covariance function"
 C(d) = exp( − d / V)
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"
 C(d) = exp( − d^{2} / V)
is a stationary covariance function with smooth sample paths.
The Matérn covariance function and rational quadratic covariance function are two parametric families of stationary covariance functions. The Matérn family includes the exponential and squared exponential covariance functions as special cases.
See also
 Variogram
 Random field
 Stochastic process
 Kriging
 Autocorrelation function
 Correlation function
References
Categories: Geostatistics
 Spatial data analysis
 Covariance and correlation
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