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Maximum values of correlation coefficients between pairs of linear functions
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c0201401.png" /></td> </tr></table>
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of two sets of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c0201402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c0201403.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c0201404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c0201405.png" /> are canonical random variables (see [[Canonical correlation|Canonical correlation]]). The problem of determining the maximum correlation coefficient between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c0201406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c0201407.png" /> under the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c0201408.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c0201409.png" /> can be solved using Lagrange multipliers. The canonical correlation coefficients are the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014010.png" /> of the equation
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Maximum values of correlation coefficients between pairs of linear functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014011.png" /></td> </tr></table>
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$$
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= \alpha _ {1} X _ {1} + \dots + \alpha _ {s} X _ {s} ,\ \
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= \beta _ {1} X _ {s+1} + \dots + \beta _ {t} X _ {s+t}  $$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014013.png" /> are the covariance matrices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014015.png" />, respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014016.png" /> is the covariance matrix between the variables of the first and second sets. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014017.png" />-th root of the equation is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014019.png" />-th canonical correlation coefficient between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014021.png" />. It is equal to the maximum value of the correlation coefficients between the pair of linear functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014023.png" /> of canonical random variables, each of which has variance one and is uncorrelated with the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014024.png" /> pairs of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014026.png" />. The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014030.png" /> satisfy the equation
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of two sets of random variables  $  X _ {1} \dots X _ {s} $
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and $  X _ {s+1} \dots X _ {s+t} $
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for which  $  U $
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and $  V $
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are canonical random variables (see [[Canonical correlation|Canonical correlation]]). The problem of determining the maximum correlation coefficient between $  U $
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and $  V $
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under the conditions  $  {\mathsf E} U = {\mathsf E} V = 0 $
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and $  {\mathsf E} U  ^ {2} = {\mathsf E} V  ^ {2} = 1 $
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can be solved using Lagrange multipliers. The canonical correlation coefficients are the roots  $  \lambda _ {1} \geq  \dots \geq  \lambda _ {s} > 0 $
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of the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014031.png" /></td> </tr></table>
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$$
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\left |
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\begin{array}{rr}
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- \lambda \Sigma _ {11}  &\Sigma _ {12}  \\
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\Sigma _ {21}  &- \lambda \Sigma _ {22}  \\
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\end{array}
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\right |  = 0 ,
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$$
  
when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020140/c02014032.png" />.
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where  $  \Sigma _ {11} $
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and  $  \Sigma _ {22} $
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are the covariance matrices of  $  X _ {1} \dots X _ {s} $
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and  $  X _ {s+1} \dots X _ {s+t} $,
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respectively, and  $  \Sigma _ {12} = \Sigma _ {21}  ^ {T} $
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is the covariance matrix between the variables of the first and second sets. The  $  r $-
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th root of the equation is called the  $  r $-
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th canonical correlation coefficient between  $  X _ {1} \dots X _ {s} $
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and  $  X _ {s+1} \dots X _ {s+t} $.  
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It is equal to the maximum value of the correlation coefficients between the pair of linear functions  $  U  ^ {(r)} $
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and  $  V  ^ {(r)} $
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of canonical random variables, each of which has variance one and is uncorrelated with the first  $  r - 1 $
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pairs of variables  $  U $
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and  $  V $.  
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The coefficients  $  \alpha  ^ {(r)} = ( \alpha _ {1}  ^ {(r)} \dots \alpha _ {s}  ^ {(r)} )  ^ {T} $,
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$  \beta  ^ {(r)} = ( \beta _ {1}  ^ {(r)} \dots \beta _ {t}  ^ {(r)} )  ^ {T} $
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of  $  U  ^ {(r)} $
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and  $  V  ^ {(r)} $
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satisfy the equation
  
 +
$$
 +
\left (
 +
\begin{array}{rr}
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- \lambda \Sigma _ {11}  &\Sigma _ {12}  \\
 +
\Sigma _ {21}  &- \lambda \Sigma _ {22}  \\
 +
\end{array}
 +
\right )
 +
\left ( \begin{array}{c}
 +
\alpha \\
 +
\beta
 +
\end{array}
 +
\right )  =  0
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$$
  
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when  $  \lambda = \lambda _ {r} $.
  
 
====Comments====
 
====Comments====
 
See also [[Correlation|Correlation]]; [[Correlation coefficient|Correlation coefficient]].
 
See also [[Correlation|Correlation]]; [[Correlation coefficient|Correlation coefficient]].

Revision as of 06:29, 30 May 2020


Maximum values of correlation coefficients between pairs of linear functions

$$ U = \alpha _ {1} X _ {1} + \dots + \alpha _ {s} X _ {s} ,\ \ V = \beta _ {1} X _ {s+1} + \dots + \beta _ {t} X _ {s+t} $$

of two sets of random variables $ X _ {1} \dots X _ {s} $ and $ X _ {s+1} \dots X _ {s+t} $ for which $ U $ and $ V $ are canonical random variables (see Canonical correlation). The problem of determining the maximum correlation coefficient between $ U $ and $ V $ under the conditions $ {\mathsf E} U = {\mathsf E} V = 0 $ and $ {\mathsf E} U ^ {2} = {\mathsf E} V ^ {2} = 1 $ can be solved using Lagrange multipliers. The canonical correlation coefficients are the roots $ \lambda _ {1} \geq \dots \geq \lambda _ {s} > 0 $ of the equation

$$ \left | \begin{array}{rr} - \lambda \Sigma _ {11} &\Sigma _ {12} \\ \Sigma _ {21} &- \lambda \Sigma _ {22} \\ \end{array} \right | = 0 , $$

where $ \Sigma _ {11} $ and $ \Sigma _ {22} $ are the covariance matrices of $ X _ {1} \dots X _ {s} $ and $ X _ {s+1} \dots X _ {s+t} $, respectively, and $ \Sigma _ {12} = \Sigma _ {21} ^ {T} $ is the covariance matrix between the variables of the first and second sets. The $ r $- th root of the equation is called the $ r $- th canonical correlation coefficient between $ X _ {1} \dots X _ {s} $ and $ X _ {s+1} \dots X _ {s+t} $. It is equal to the maximum value of the correlation coefficients between the pair of linear functions $ U ^ {(r)} $ and $ V ^ {(r)} $ of canonical random variables, each of which has variance one and is uncorrelated with the first $ r - 1 $ pairs of variables $ U $ and $ V $. The coefficients $ \alpha ^ {(r)} = ( \alpha _ {1} ^ {(r)} \dots \alpha _ {s} ^ {(r)} ) ^ {T} $, $ \beta ^ {(r)} = ( \beta _ {1} ^ {(r)} \dots \beta _ {t} ^ {(r)} ) ^ {T} $ of $ U ^ {(r)} $ and $ V ^ {(r)} $ satisfy the equation

$$ \left ( \begin{array}{rr} - \lambda \Sigma _ {11} &\Sigma _ {12} \\ \Sigma _ {21} &- \lambda \Sigma _ {22} \\ \end{array} \right ) \left ( \begin{array}{c} \alpha \\ \beta \end{array} \right ) = 0 $$

when $ \lambda = \lambda _ {r} $.

Comments

See also Correlation; Correlation coefficient.

How to Cite This Entry:
Canonical correlation coefficients. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Canonical_correlation_coefficients&oldid=12606
This article was adapted from an original article by I.O. Sarmanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article