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A correlation function is a statistical correlation between random variables 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 autocorrelations. 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 correlations. 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 no observations. Correlation functions used in astronomy, financial analysis, and statistical mechanics differ only in the particular stochastic processes they are applied to. In quantum field theory there are correlation functions over quantum distributions. ==Definition== For random variables ''X''(''s'') and ''X''(''t'') at different points ''s'' and ''t'' of some space, the correlation function is : where 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 : or a scalar, which is the trace of this matrix. If the probability distribution has any target space symmetries, i.e. symmetries in the value space of the stochastic variable (also called internal symmetries), then the correlation matrix will have induced symmetries. Similarly, if there are symmetries of the space (or time) domain in which the random variables exist (also called spacetime symmetries), then the correlation function will have corresponding space or time symmetries. Examples of important spacetime symmetries are — *translational symmetry yields ''C''(''s'',''s'' *rotational symmetry in addition to the above gives ''C''(''s'', ''s'' Higher order correlation functions are often defined. A typical correlation function of order ''n'' is : If the random variable has only one component, then the indices are redundant. If there are symmetries, then the correlation function can be broken up into irreducible representations of the symmetries — both internal and spacetime. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Correlation function」の詳細全文を読む スポンサード リンク
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