Covariance function

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the covariance function describes how much two <a href="/facts/Random_variable/TwTBXnLT">random variables</a> change together (their <a href="/facts/Covariance/hILocGNH">covariance</a>) with varying spatial or temporal separation. For a <a href="/facts/Random_field/8N7Id48z">random field</a> or <a href="/facts/Stochastic_process/Ng1n1dnB">stochastic process</a> 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:

 
 
 
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 {\displaystyle C(x,y):=\operatorname {cov} (Z(x),Z(y))=\mathbb {E} {\Big [}{\big (}Z(x)-\mathbb {E} [Z(x)]{\big )}{\big (}Z(y)-\mathbb {E} [Z(y)]{\big )}{\Big ]}.\,}

The same C(x, y) is called the <a href="/facts/Autocovariance/3uM1sVVV">autocovariance</a> function in two instances: in <a href="/facts/Time_series/fSXPR817">time series</a> (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 <a href="/facts/Cross_covariance/nBl7HhGY">cross covariance</a> between two different variables at different locations, Cov(Z(x1), Y(x2))).

Covariance function open-in-new

Covariance function