2.6.   Calibration of Nonlinear Relationships

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.