For
multivariate calibration of the concentrations of one or more analytes
(variables y) and the device
responses (independent variables x)
the simplest and the most common approach is the assumption of a linear
relationship. A model for this linear relationship can be setup by a multiple
linear regression (MLR), which is also known as inverse least-squares regression
(ILS). This calibration can be seen as an extension of the univariate linear
regression. For each response variable, a linear equation is formulated:

_{}

(5)

with
b_{ij} as regression coefficients
and e_{j} as residuals.
Using matrix notation equation (5) can
be expressed as:

_{}

(6)

The
regression coefficients can be estimated by

_{}

(7)

For
the inversion of the variance-covariance matrix at least as many samples as device responses have to be measured. This inversion of the variance-covariance matrix of the independent variables, which is needed not only by
MLR but also by most linear regression techniques to identify the model
parameters, causes several problems. If the variables show a collinear behavior
or if the variables are highly correlated, the resulting variance-covariance
matrix will be ill-conditioned leading to unreliable model parameters und thus
producing unstable calibration models.

Several
methods have been developed to overcome these problems like principal component
regression (PCR), partial least squares regression (PLS or PLSR), ridge regression
(RR) and many more. Among these methods, "PLSR is the de facto standard
for constructing a multivariate model" [29].