Cross-correlation matrix: Difference between revisions

Revision as of 19:37, 22 July 2016

Revision as of 19:41, 22 July 2016

Line 13:

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

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 aswith <math>i,j</math> element

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

or a scalar, which isSometimes the [[trace (matrix)|trace]] of this matrix is focused on. 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 &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).