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	~~Golda~~ is ~~what~~'~~s written~~ on ~~my birth certification tarot card readings~~ ([~~http~~://~~www~~.~~prograd.uff.br~~/~~novo~~/~~facts-about-growing-greater-organic-garden http~~://~~www~~.~~prograd~~.~~uff.br~~/~~novo~~/~~facts~~-~~about-growing-greater-organic-garden~~]~~) although it is not the title on my birth certification~~. ~~One~~ of the ~~issues she enjoys most~~ is ~~canoeing~~ and ~~she~~'~~s been doing it for fairly a whilst~~. ~~Mississippi is exactly where her house is but her spouse desires them psychic love readings~~ (~~[http:~~//~~modenpeople~~.~~co.kr~~/~~modn/qna/292291 simply click~~ the ~~following website page~~]~~) to move~~. ~~My day job~~ is ~~an information officer but I've already utilized for another 1.~~<br><br>~~Take~~ a ~~look at my homepage live psychic reading~~ ([~~http~~://~~www~~.~~Khuplaza~~.~~com~~/~~dent/14869889 click through the following web page~~])		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 covariance 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'':

			:<math>C(x,y):=\operatorname{cov}(Z(x),Z(y)).\,</math>

			The same ''C''(''x'', ''y'') is called the [[autocovariance]] 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''(''x''<sub>1</sub>), ''Y''(''x''<sub>2</sub>))).<ref>{{cite book\|title=Multivariate Geostatistics\|first=Hans\|last=Wackernagel\|year=2003\|publisher=Springer}}</ref>

			== Admissibility ==
			For locations ''x''<sub>1</sub>, ''x''<sub>2</sub>, …, ''x''<sub>''N''</sub> ∈ ''D'' the variance of every linear combination

			:<math>X=\sum_{i=1}^N w_i Z(x_i)</math>

			can be computed as

			:<math>\operatorname{var}(X)=\sum_{i=1}^N \sum_{j=1}^N w_i C(x_i,x_j) w_j.</math>

			A function is a valid covariance function if and only if<ref name="Cressie">{{cite book\|title=Statistics for Spatial Data\|first=Noel A.C.\|last=Cressie\|year=1993\|publisher=Wiley-Interscience}}</ref> this variance is non-negative for all possible choices of ''N'' and weights ''w''<sub>1</sub>, …, ''w''<sub>''N''</sub>. A function with this property is called [[Positive-definite kernel\|positive definite]].

			== Simplifications with stationarity ==
			In case of a weakly [[Stationary process\|stationary]] [[random field]], where

			:<math>C(x_i,x_j)=C(x_i+h,x_j+h)\,</math>

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

			:<math>C_s(h)=C(0,h)=C(x,x+h)\,</math>

			which is called a ''covariogram'' and also a ''covariance function''. Implicitly the ''C''(''x''<sub>''i''</sub>, ''x''<sub>''j''</sub>) can be computed from ''C''<sub>''s''</sub>(''h'') by:

			:<math>C(x,y)=C_s(y-x).\,</math>

			The [[Positive-definite function\|positive definiteness]] of this single-argument version of the covariance function can be checked by [[Bochner's theorem]].<ref name="Cressie"/>

			== Parametric families of covariance functions ==

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

			:<math>
			C(d) = \exp(-d/V)
			</math>

			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"

			:<math>
			C(d) = \exp(-d^2/V)
			</math>

			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 ==
			<references />

			[[Category:Geostatistics]]
			[[Category:Spatial data analysis]]
			[[Category:Covariance and correlation]]

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