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While the order of the + (,) and (,), does not matter in the classical case, as they are merely numbers and hence commute, the ordering is vital in the quantum analogue of these correlation functions. [4] The first order correlation function, measured at the same time and position gives us the intensity i.e. () (,) =.
A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. [1] 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 ...
For example, in order to measure the higher-order analogues of pair distribution functions, coherent x-ray sources are needed. Both the theory of such analysis [ 12 ] [ 13 ] and the experimental measurement of the needed X-ray cross-correlation functions [ 14 ] are areas of active research.
For example, scaled correlation is designed to use the sensitivity to the range in order to pick out correlations between fast components of time series. [16] By reducing the range of values in a controlled manner, the correlations on long time scale are filtered out and only the correlations on short time scales are revealed.
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
In statistical theory, one long-established approach to higher-order statistics, for univariate and multivariate distributions is through the use of cumulants and joint cumulants. [1] In time series analysis, the extension of these is to higher order spectra, for example the bispectrum and trispectrum.
The full correlation is composed of singlets, doublets, triplets, and higher-order correlations, all uniquely defined by the cluster-expansion approach. Each blue sphere corresponds to one particle operator and yellow circles/ellipses to correlations. The number of spheres within a correlation identifies the cluster number.
[23] [24] In such cases, higher order correlation functions are needed to further describe the structure. Higher-order distribution functions g ( k ) {\displaystyle \textstyle g^{(k)}} with k > 2 {\displaystyle \textstyle k>2} were less studied, since they are generally less important for the thermodynamics of the system; at the same time, they ...