Difference between revisions of "Canonical correlation coefficients"
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| − | of | + | 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 | |
| + | and X _ {s+1} \dots X _ {s+t} | ||
| + | for which U | ||
| + | and V | ||
| + | are canonical random variables (see [[Canonical correlation|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 | ||
| − | + | $$ | |
| + | \left | | ||
| + | \begin{array}{cc} | ||
| + | - \lambda \Sigma _ {11} &\Sigma _ {12} \\ | ||
| + | \Sigma _ {21} &- \lambda \Sigma _ {22} \\ | ||
| + | \end{array} | ||
| + | \right | = 0 , | ||
| + | $$ | ||
| − | + | 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 | ||
| + | $$ | ||
| + | \left ( | ||
| + | \begin{array}{cc} | ||
| + | - \lambda \Sigma _ {11} &\Sigma _ {12} \\ | ||
| + | \Sigma _ {21} &- \lambda \Sigma _ {22} \\ | ||
| + | \end{array} | ||
| + | \right ) | ||
| + | \left ( \begin{array}{c} | ||
| + | \alpha \\ | ||
| + | \beta | ||
| + | \end{array} | ||
| + | \right ) = 0 | ||
| + | $$ | ||
| + | when \lambda = \lambda _ {r} . | ||
====Comments==== | ====Comments==== | ||
See also [[Correlation|Correlation]]; [[Correlation coefficient|Correlation coefficient]]. | See also [[Correlation|Correlation]]; [[Correlation coefficient|Correlation coefficient]]. | ||
Latest revision as of 08:00, 25 April 2022
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}{cc} - \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}{cc} - \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.
Canonical correlation coefficients. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Canonical_correlation_coefficients&oldid=12606