The PLSR method is, like PCA, a bilinear technique which expresses a data matrices as a matrix product of two matrices:
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(1) | ||
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where the scores ( and
) 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 independently of whether it is relevant for
the prediction in
. It is often that the principal
components in
do not represent the best directions that are
relevant for the prediction of
. In the PLSR scheme we find
new latent variables for
that are most relevant for the
prediction of
.
In addition to the outer relations we also have the inner relation
which relates the and
scores as follows:
![]() |
(2) |
where is a regression coefficient. We can write the inner
relation as:
![]() |
(3) |
where is a diagonal matrix. So we can write:
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(4) |
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The main idea in PLSR is to ensure that the latent vectors in have maximum relevance for
. We can formulate this as
we find a vector
in column space of
:
and a vector in the column space of
:
![]() |
(5) |
such that the squared covariance between and
is
maximized:
max![]() ![]() |
(6) |
Bjørn Kåre Alsberg 2006-04-06