The PLSR equations

The PLSR method is, like PCA, a bilinear technique which expresses a data matrices as a matrix product of two matrices:


$\displaystyle {\bf X} = {\bf T}{\bf P}^{T} + {\bf E}
$     (1)
$\displaystyle {\bf Y} = {\bf U}{\bf Q}^{T} + {\bf F}$     (2)

where the scores ($ {\bf T}$ and $ {\bf U}$) are not the same as the scores found in PCA. These equations are often referred to as the outer relations.

In PCR we have performed the latent variable projection in $ {\bf X}$ independently of whether it is relevant for the prediction in $ {\bf Y}$. It is often that the principal components in $ {\bf X}$ do not represent the best directions that are relevant for the prediction of $ {\bf Y}$. In the Partial Least Squares regression (PLSR) scheme we find new latent variables for $ {\bf X}$ that are relevant for the prediction of $ {\bf Y}$.

In addition to the outer relations we also have the inner relation which relates the $ {\bf t}_j$ and $ {\bf u}_j$ scores as follows:

$\displaystyle {\bf u}_j = g_j {\bf t}_j
$

where $ g_j$ is a regression coefficient.

We can write the inner relation as:

$\displaystyle {\bf U} = {\bf T}{\bf G}
$

where $ {\bf G}$ is a diagonal matrix. So we can write:


$\displaystyle {\bf X}$ $\displaystyle =$ $\displaystyle {\bf T}{\bf P}^{T} + {\bf E}$  
$\displaystyle {\bf Y}$ $\displaystyle =$ $\displaystyle {\bf T}{\bf G} {\bf Q}^{T} + {\bf F}$  

The main idea in PLSR is to ensure that the latent vectors in $ {\bf X}$ have maximum relevance for $ {\bf Y}$. We can formulate this as we find a vector $ {\bf t}$ in column space of $ {\bf X}$:

$\displaystyle {\bf t} = {\bf X}{\bf w}
$

and a vector $ {\bf u}$ in the column space of $ {\bf Y}$:

$\displaystyle {\bf u} = {\bf Y}{\bf q}
$

such that the squared covariance between $ {\bf t}$ and $ {\bf u}$ is maximized:

   max$\displaystyle ({\bf u}^{T}{\bf t})^2 =$   max$\displaystyle ({\bf q}^{T}{\bf Y}^{T}{\bf X}{\bf w})^2
$

for $ \vert{\bf q}\vert=\vert{\bf w}\vert=1$

Bjørn Alsberg 2005-02-18