For stochastic processes, including those that arise in statistical mechanics and Euclidean quantum field theory, a correlation function is the correlation between random variables at two different points in space or time. If one considers the correlation function between random variables at the same point but at two different times then one refers to this as the autocorrelation function.
Definitions
Consider a probability density functional P[X(s)] for stochastic variables X(s) at different points s of some space, then the correlation function is
- Failed to parse (syntax error): {\displaystyle C(s,s') = \langle X(s) X(s')\rangle where the statistical averages are taken with respect to the [[measure]] specified by the probability density function. In this definition, it has been assumed that the stochastic variable is a scalar. If it is not, then one can define more complicated correlation functions. For example, if one has a vector <b>X<sub>i</sub>(s)</b>, then one can define the matrix of correlation functions :<math>C_{ij}(s,s') = \langle X_i(s) X_j(s')\rangle or a scalar, which is the trace of this matrix. If the probability density <b>P[X(s)]</b> has any target space symmetries, ie, symmetries in the space of the stochastic variable, then the correlation matrix will have induced symmetries. A caveat, though, in [[quantum field theory]], we sometimes have nonpositive states and in that case, a probabilistic interpretation makes no sense. At any rate, even in ordinary quantum field theory, we need to work with quantum probability instead of classical [[probability]]. ==Overview== ''Ideally, it will make clear that the correlation functions in astronomy, financial market analysis, etc., are all instances of the '''same''' idea; therefore, a disambiguation page is not what this should be.'' ==See also== *[[Correlation]] *[[Spearman's rank correlation coefficient]] *[[Pearson product-moment correlation coefficient]] *[[Correlation function (astronomy)]] *[[Correlation function (statistical mechanics)]] *[[Correlation function (quantum field theory)]] *[[Rate distortion theory#Rate-Distortion_Functions |Mutual information]] {{Statistics-stub}} [[Category:Statistics]]}