Linear
modeling of relationships, which is the basis for the most common methods in
chemometrics, is based on many laws of physics showing linear relationships
in a first approach. For example, in the field of analytical chemistry, the
relationship between the concentration of an analyte and the absorption of radiation
by this analyte is linear if certain conditions are fulfilled, known as Beer's
Law. In the same way, the relationship between the concentration of an analyte
in the gaseous phase and the amount of analyte sorbed into an amorphous polymer
can be linear, known as Henry's Law. Yet, both laws of physics are borderline
cases and hardly fulfilled in many real world applications. For example, Beer's
Law is not valid for high concentrations of analytes, interfering analytes and
turbid solutions ending up in a nonlinear relationship. Similarly, Henry's Law
is the borderline case for small concentrations of analyte in the gaseous phase
in contrast to the nonlinear Langmuir sorption for a higher range of the concentration
of analyte. If the relationships between the input variables and the response
variables are nonlinear, widely used linear calibration methods show a systematical
bias. Although it was shown in [39] that under certain
circumstances a linear PLS model can be successfully used if some variables
show a nonlinear relationship, linear models fail in most nonlinear real world
applications. Especially if all independent variables show similar nonlinear
relationships with the dependent variables [41] or if
variables show interactions, which are often observed in mixtures of analytes,
linear models often fail.

Several
approaches can be found in literature dealing with calibrations when nonlinearities
in the data are present. Besides of the application of methods, which are in
principle nonlinear like neural networks, there exist a couple of methods trying
to remove the nonlinearities in the data or extending linear models to cope
with the nonlinearities. The quadratic PLS (QPLS) belongs to the latter whereby
a quadratic term for the inner relation is used instead of a linear term. By
adding squared terms and interaction terms to the input variables, the implicit
nonlinear PLS (INLR) tries to account for the nonlinear relationship. On the
other hand, the Box-Cox transformation for the response variables tries to linearize
the relationship of the data directly. The locally weighted regression (LWR)
and the modeling trees use a piecewise linear approximation whereas the classification
and regression trees (CART) and the GIFI-PLS use discrete variables to approximate
a nonlinear behavior. The methods, which are used in this study, are explained
in detail in chapter 6 when applied to a nonlinear data
set. As neural networks play a major part of this study, a detailed description
of neural networks follows directly in the next section.