Relation to other linear subspace methods

Instead of the two eigenvalue equations in 4 we can formulate the problem in one single eigenvalue equation:

$$\bB^{-1}\bA \bhw = \rho \bhw$$ (11)

where

$${\bf A} = \begin{bmatrix} {\bf0} & {\bf C}_{xy} \\ {\bf ... ...gin{pmatrix} \mu _x \bhw_x \\ \mu _y \bhw_y \end{pmatrix}.$$ (12)

Solving the eigenproblem in equation 11 with slightly different matrices will give solutions to principal component analysis (PCA), partial least squares (PLS) and multivariate linear regression (MLR). The matrices are listed in table 1.

 

Table: The matrices $\bA$ and $\bB$ for PCA, PLS, CCA and MLR.

	$\bA$	$\bB$
PCA	$\bC_{xx}$	$\bI$
PLS	$\begin{pmatrix} {\bf0} & {\bf C}_{xy}\\ {\bf C}_{yx} & {\bf0} \end{pmatrix}$	$\begin{pmatrix} {\bf I} & {\bf0} \\ {\bf0} & {\bf I} \end{pmatrix}$
CCA	$\begin{pmatrix} {\bf0} & {\bf C}_{xy}\\ {\bf C}_{yx} & {\bf0} \end{pmatrix}$	$\begin{pmatrix} {\bf C}_{xx} & {\bf0} \\ {\bf0} & {\bf C}_{yy} \end{pmatrix}$
MLR	$\begin{pmatrix} {\bf0} & {\bf C}_{xy}\\ {\bf C}_{yx} & {\bf0} \end{pmatrix}$	$\begin{pmatrix} {\bf C}_{xx} & {\bf0} \\ {\bf0} & {\bf I} \end{pmatrix}$