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| {{Other uses2|Correlation function}}
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| {{Unreferenced|date=December 2009}}
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| [[File:Comparison_convolution_correlation.svg|thumb|300px|Visual comparison of [[convolution]], [[cross-correlation]] and [[autocorrelation]].]] | |
| A '''correlation function''' is a statistical [[correlation]] between [[random variable]]s at two different points in space or time, usually as a function of the spatial or temporal distance between the points. If one considers the correlation function between random variables representing the same quantity measured at two different points then this is often referred to as an [[autocorrelation function]] being made up of [[autocorrelation]]s. Correlation functions of different random variables are sometimes called '''cross correlation functions''' to emphasise that different variables are being considered and because they are made up of [[cross correlation]]s.
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| Correlation functions are a useful indicator of dependencies as a function of distance in time or space, and they can be used to assess the distance required between sample points for the values to be effectively uncorrelated. In addition, they can form the basis of rules for interpolating values at points for which there are observations.
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| Correlation functions used in [[correlation function (astronomy)|astronomy]], [[financial analysis]], and [[statistical mechanics]] differ only in the particular stochastic processes they are applied to. In [[quantum field theory]] there are [[Correlation function (quantum field theory)|correlation functions over quantum distributions]].
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| ==Definition==
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| For random variables ''X''(''s'') and ''X''(''t'') at different points ''s'' and ''t'' of some space, the correlation function is
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| :<math>C(s,t) = \operatorname{corr} ( X(s), X(t) ),</math> | |
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| where <math>\operatorname{corr}</math> is described in the article on [[correlation]]. In this definition, it has been assumed that the stochastic variable is scalar-valued. If it is not, then more complicated correlation functions can be defined. For example, if ''X''(''s'') is a vector, then a matrix of correlation functions is defined as
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| :<math>C_{ij}(s,t) = \operatorname{corr}( X_i(s), X_j(t) )</math> | |
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| or a scalar, which is the trace of this matrix. If the [[probability distribution]] has any target space symmetries, i.e. symmetries in the space of the stochastic variable (also called '''internal symmetries'''), then the correlation matrix will have induced symmetries. If there are symmetries of the space (or time) in which the random variables exist (also called '''[[spacetime symmetries]]''') then the correlation matrix will have special properties. Examples of important spacetime symmetries are —
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| *'''translational symmetry''' yields ''C''(''s'',''s''<nowiki>'</nowiki>) = ''C''(''s'' − ''s''<nowiki>'</nowiki>) where ''s'' and ''s''<nowiki>'</nowiki> are to be interpreted as vectors giving coordinates of the points
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| *'''rotational symmetry''' in addition to the above gives ''C''(''s'', ''s''<nowiki>'</nowiki>) = ''C''(|''s'' − ''s''<nowiki>'</nowiki>|) where |''x''| denotes the norm of the vector ''x'' (for actual rotations this is the Euclidean or 2-norm).
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| Higher order correlation functions are often defined. A typical correlation function of order ''n'' is
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| :<math>C_{i_1i_2\cdots i_n}(s_1,s_2,\cdots,s_n) = \langle X_{i_1}(s_1) X_{i_2}(s_2) \cdots X_{i_n}(s_n)\rangle.</math> | |
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| If the random variable has only one component, then the indices <math>i_j</math> are redundant. If there are symmetries, then the correlation function can be broken up into [[irreducible representation]]s of the symmetries — both internal and spacetime.
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| The case of correlations of a single random variable can be thought of as a special case of autocorrelation of a stochastic process on a space which contains a single point.
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| ==Properties of probability distributions==
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| With these definitions, the study of correlation functions is similar to the study of [[probability distributions]]. Many stochastic processes can be completely characterized by their correlation functions; the most notable example is the class of [[Gaussian processes]].
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| Probability distributions defined on a finite number of points can always be normalized, but when these are defined over continuous spaces, then extra care is called for. The study of such distributions started with the study of [[random walk]]s and led to the notion of the [[Itō calculus]].
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| The Feynman [[path integral formulation|path integral]] in Euclidean space generalizes this to other problems of interest to [[statistical mechanics]]. Any probability distribution which obeys a condition on correlation functions called [[reflection positivity]] lead to a local [[quantum field theory]] after [[Wick rotation]] to [[Minkowski spacetime]]. The operation of [[renormalization]] is a specified set of mappings from the space of probability distributions to itself. A [[quantum field theory]] is called renormalizable if this mapping has a fixed point which gives a quantum field theory.
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| ==See also==
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| *[[Autocorrelation]]
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| *[[Covariance function]]
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| *[[Pearson product-moment correlation coefficient]]
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| *[[Correlation function (astronomy)]]
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| *[[Correlation function (statistical mechanics)]]
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| *[[Correlation function (quantum field theory)]]
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| *[[Mutual information]]
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| *[[Rate distortion theory#Rate–distortion_functions|Rate distortion theory]]
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| *[[Radial distribution function]]
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| {{DEFAULTSORT:Correlation Function}}
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| [[Category:Covariance and correlation]]
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| [[Category:Time series analysis]]
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| [[Category:Spatial data analysis]]
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| [[Category:Types of function]]
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