# canonical correlation

Let $X$ be the $(T,n)$ matrix corresponding to the $n$ signals and $Y$ be a $(T,p)$ matrix corresponding to one set of $p$ signals. Time indexes each row of the matrix ($T$ time samples). Let $\Sigma_{11}$ and $\Sigma_{22}$ be the sample covariance matrices of $X$ and $Y$, respectively, and let $\Sigma_{12}=\Sigma_{21}^{\prime}$ be the sample covariance matrix between $X$ and $Y$. For simplicity, we suppose that all signals have zero mean.

Canonical correlation analysis (CCA) finds the linear combinations of the column of $X$ and $Y$ that has the largest correlation; i.e., it finds the weight vectors (loadings) $a$ and $b$ that maximize:

 $\rho=\frac{a^{\prime}\Sigma_{12}b}{\sqrt{a^{\prime}\Sigma_{11}a}\sqrt{b^{% \prime}\Sigma_{22}b}}.$ (1)

We follow the derivations of Johnson and we do a change of basis: $c=\Sigma_{11}^{1/2}a$ and $d=\Sigma_{22}^{1/2}b$.

 $\rho=\frac{c^{\prime}\Sigma_{11}^{-1/2}\Sigma_{12}\Sigma_{22}^{-1/2}d}{\sqrt{c% ^{\prime}c}\sqrt{d^{\prime}d}}$ (2)
 $\rho\leq\frac{\sqrt{c^{\prime}\Sigma_{11}^{-1/2}\Sigma_{12}\Sigma_{22}^{-1/2}% \Sigma_{22}^{-1/2}\Sigma_{21}\Sigma_{11}^{-1/2}c}\sqrt{d^{\prime}d}}{\sqrt{c^{% \prime}c}\sqrt{d^{\prime}d}}=\sqrt{\frac{c^{\prime}\Sigma_{11}^{-1/2}\Sigma_{1% 2}\Sigma_{22}^{-1}\Sigma_{21}\Sigma_{11}^{-1/2}c}{c^{\prime}c}}.$ (3)

The inequality above is an equality when $\Sigma_{22}^{-1/2}\Sigma_{21}\Sigma_{11}^{-1/2}c$ and $d$ are collinear. The right hand side of the expression above is a Rayleigh quotient and it is maximum when $c$ is the eigenvector corresponding to the largest eingenvalue of $\Sigma_{11}^{-1/2}\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}\Sigma_{11}^{-1/2}$ (we obtain the other rows by using the other eigenvalues in decreasing magnitude). This results if the basis of the CCA. We can compute the two canonical variables: $U_{1}=Xa$ and $V_{1}=Yb$.

We can continue this way to find the subsequent vectors

Title canonical correlation CanonicalCorrelation 2013-03-22 19:16:11 2013-03-22 19:16:11 tony_bruguier (26297) tony_bruguier (26297) 4 tony_bruguier (26297) Definition msc 62H20