Cross-correlation matrix: Difference between revisions

Revision as of 16:28, 4 December 2013

Revision as of 16:33, 4 December 2013

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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

:<math>C_{ij}(s,s't) = \operatorname{corr}( X_i(s), X_j(s't) )</math>

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 &mdash;