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 $ 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|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