Canonical correlation coefficients

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.