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.