Cross-correlation matrix: Difference between revisions

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==Definition==

For possibly distinct random variables ''X''(''s'') and ''XY''(''t'') at different points ''s'' and ''t'' of some space, the correlation function is

:<math>C(s,t) = \operatorname{corr} ( X(s), XY(t) ) ,</math>

where <math>\operatorname{corr}</math> is described in the article on [[correlation]]. In this definition, it has been assumed that the stochastic variablevariables isare scalar-valued. If itthey isare not, then more complicated correlation functions can be defined. For example, if ''X''(''s'') is a [[row and column vectors|vector]] with ''n'' elements and ''Y''(t) is a vector with ''q'' elements, then an ''n''×''nq'' matrix of correlation functions is defined with <math>i,j</math> element

:<math>C_{ij}(s,t) = \operatorname{corr}( X_i(s), X_jY_j(t) ).</math>

SometimesWhen ''n''=''q'', sometimes the [[trace (matrix)|trace]] of this matrix is focused on. If the [[probability distribution]]s hashave 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 &mdash;

*'''translational symmetry''' yields ''C''(''s'',''s''<nowiki>'</nowiki>) = ''C''(''s''&nbsp;&minus;&nbsp;''s''<nowiki>'</nowiki>) where ''s'' and ''s''<nowiki>'</nowiki> are to be interpreted as vectors giving coordinates of the points

*'''rotational symmetry''' in addition to the above gives ''C''(''s'', ''s''<nowiki>'</nowiki>) = ''C''(|''s''&nbsp;&minus;&nbsp;''s''<nowiki>'</nowiki>|) where |''x''| denotes the norm of the vector ''x'' (for actual rotations this is the Euclidean or 2-norm).