Introduction

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)
$\displaystyle {\bf Y} = {\bf U}{\bf Q}^{T} + {\bf F}$      

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 principal component regression we perform 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 PLSR scheme we find new latent variables for $ {\bf X}$ that are most 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:

$\displaystyle {\bf u}_j = g_j {\bf t}_j$ (2)

where $ g_j$ is a regression coefficient. We can write the inner relation as:

$\displaystyle {\bf U} = {\bf T}{\bf G}$ (3)

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


$\displaystyle {\bf X}$ $\displaystyle =$ $\displaystyle {\bf T}{\bf P}^{T} + {\bf E}$ (4)
$\displaystyle {\bf Y}$ $\displaystyle =$ $\displaystyle {\bf T}{\bf G} {\bf Q}^{T} + {\bf F}$  

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}$:

$\displaystyle {\bf u} = {\bf Y}{\bf q}$ (5)

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

max$\displaystyle ({\bf u}^{T}{\bf t})^2 =$   max$\displaystyle ({\bf q}^{T}{\bf Y}^{T}{\bf X}{\bf w})^2$ (6)

Bjørn Kåre Alsberg 2006-04-06