Cross-correlation matrix: Difference between revisions

Revision as of 20:17, 9 August 2007

Revision as of 12:17, 29 May 2008

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

*'''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).