The PLSR equations

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

$${\bf X} = {\bf T}{\bf P}^{T} + {\bf E}$$ (1)

$${\bf Y} = {\bf U}{\bf Q}^{T} + {\bf F}$$ (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:

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

where $g_j$ is a regression coefficient.

We can write the inner relation as:

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

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

$${\bf X} = {\bf T}{\bf P}^{T} + {\bf E}$$

$${\bf Y} = {\bf T}{\bf G} {\bf Q}^{T} + {\bf F}$$

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 ${\bf X}$:

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

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

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

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

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

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