2. Theory Fundamentals of the Multivariate Data
Analysis

2.1. Overview of the Multivariate Quantitative Data Analysis

Multivariate
quantitative data analysis is part of the scientific field of chemometrics.
In a recent review [1] chemometrics
was defined as a process, in which measurements are made, data are collected
and information is obtained. The multivariate quantitative data analysis, which
tries to describe relationships between two groups of variables, also is subject
to this process. A practical implementation of the process could look like this:

1.First, different factors like the analytes of
interest and interfering substances have to be identified, which might influence
the measurements.

2.Then, an experimental design has to be setup, which
defines how many samples have to be measured and how to vary the different
analyte concentrations and other factors for theses samples.

3.Afterwards, these samples are measured, the
responses of the device are recorded, and the raw data are optionally preprocessed.

4.After that, a calibration is performed, which tries
to model a relationship between the factors such as the concentrations of the
analytes, which are generally called independent variables, input variables or
simply x-variables, and the
responses of the device, which are called dependent variables, response
variables or simply y-variables,
ending up in a model. Usually, the quality of the calibration is judged by the
prediction of additional validation data. Thereby the model does not know the
true concentrations of the analytes but predicts these concentrations based on
the input variables (device responses). These predictions are compared with the
true concentrations in a mathematical manner by using a measure of error or in
a graphical manner by using true-predicted plots.

5.Often, an optimization of the calibration or an
interpretation of the established model follows. Finally, the model can be
applied to new measurements in routine analysis (but has to be validated and
updated from time to time).

In the
next sections, several fundamental approaches and steps in multivariate
calibration and their implementations in this work are explained in more
detail.