Chemometric methods application in pharmaceutical products and processes analysis and control - Computer-Aided Applications in Pharmaceutical Technology - page 67

(response) data. This problem is resolved by application of the Partial
Least Squares (PLS) technique.

Partial Least Squares Regression (PLS)

PLS (Partial Least Squares or Projection onto Latent Structures) is a
multivariate technique used to develop models for LV variables or factors.
These variables are calculated to maximize the covariance between the
scores of an independent block (X) and the scores of a dependent block
(Y) (Lopes et al., 2004). Both X and Y blocks (data sets) are modeled to
fi nd out the variables in an X matrix that will best describe the Y matrix.
In this way, variability and correlation are addressed at the same time. In
the PLS method, regressions are calculated with the least squares
algorithm. In comparison to the other least squares algorithms (i.e.
classical MLR), PLS is more robust to noise, co-linearity, and high
dimensionality in the data (Ronen et al., 2011). PLS is advantageous, in
comparison to PCR, because of the LV selection according to the
covariance matrix between the data and the investigated parameters
(Roggo et al., 2007). Therefore, the main difference between PLS and
PCA/PCR is that normalized weight vector w a is calculated as the
covariance between the response y and the data matrix X :

[4.17]

Scores and loadings are calculated by successive projections of the data
matrix, as described for PCA. The part of X is explained by a pair of PLS
scores, and loading vectors in each step are removed before the next pair
is calculated. In comparison to PCA, the weight vector is no longer equal
to p a and loading vectors are no longer orthogonal (nor unit vectors).
Score vectors are kept orthogonal, which makes some of the calculation
steps more easily performed. When applying linear PLS to nonlinear
problems, the minor LVs cannot always be discarded, since they may not
only describe noise. Nonlinear structures may be modeled using a
combination of higher-order and lower-order LVs calculated from linear
PLS, but the result of this approach can be an overfi tted model that is too
sensitive to noise in the modeling data (Ronen et al., 2011).
PLS regression can also be used as a supervised classifi cation method
(Rajalahti and Kvalheim, 2011), as described for the PLS-DA method.
There are many methods derived from PCR and PLS, in order
to improve and/or ease interpretation of results. Some of these

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