View source for Canonical correlation

The correlation between linear functions of two sets of random variables determined by the maximization of this correlation subject to certain constraints. In the theory of canonical correlations the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020130/c0201301.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020130/c0201302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020130/c0201303.png" />, are linearly transformed into the so-called canonical random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020130/c0201304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020130/c0201305.png" /> so that a) all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020130/c0201306.png" /> have mathematical expectation zero and variance one; b) within each of the two sets the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020130/c0201307.png" /> are uncorrelated; c) each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020130/c0201308.png" /> in the first set is correlated with just one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020130/c0201309.png" /> of the second set; d) the non-zero correlation coefficients between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020130/c02013010.png" />'s of different sets have (consecutively) maximum values, subject to the requirement of zero correlation with previous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020130/c02013011.png" />'s.

In the particular case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020130/c02013012.png" />, the canonical correlation is the multiple correlation between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020130/c02013013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020130/c02013014.png" />. The transformation to the canonical random variables corresponds to the algebraic problem of reducing a quadric to canonical form. In multivariate statistical analysis the method of canonical correlations is used, in the study of interdependence of two sets of components of a vector of observations, to realize a transition to a new coordinate system in which the correlation between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020130/c02013015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020130/c02013016.png" /> becomes transparently clear. As a result of the analysis of canonical correlations it may turn out that the interdependence between the two sets is completely described by the correlation between several canonical random variables.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Hotelling,   "Relations between two sets of variates"  ''Biometrika'' , '''28'''  (1936)  pp. 321–377</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T.W. Anderson,   "Introduction to multivariate statistical analysis" , Wiley  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.K. Ord,   "The advanced theory of statistics" , '''3''' , Griffin  (1983)</TD></TR></table>