A spectrum consists of absorbance or reflexion energy values for a number m -- up to few thousand -- wavelengths. It can be represented by a point in the m-dimensional space. Each chemical compound has its characteristic spectrum. If mixtures are measured, the spectra of the compounds are superimposed linearly according to the law of Beer and Lambert. The spectra of mixtures of q components therefore lie in a q-dimensional subspace of the m-dimensional space. In chemical reactions, a mixture of initial compounds and products changes smoothly over time. Thus a smooth curve is a q-dimensional subspace can be expected. It will be blurred by measurement errors and other discrepancies, modelled by a random error term.
When several runs of the same reaction are studied, the starting mixture may be different, but the subspace should be the same. In practice, there will be other influences, like small shifts in the instrument, which lead to a systematic difference between the runs. Such shifts can be modelled by additional dimensions, labelled as "nuisance dimensions". Attemps are made to reduce such shifts by applying a "baseline correction" as a preprocessing step to the measured spectra. We have developped a non-linear baseline correction which is applied successfully in practice.
The problem of finding the q-dimensional subspace which shows the development of the reaction is adequately solved by the statistical method of Principal Components (PCA). In order to make the results more useful in practice, it is important to find meaningful directions (axes) in this space, which should correspond to the development of the reaction and the "nuisance" shifts between reactions, respectively. On the basis of such an analysis, it is possible to answer the following practical questions:
Apart from Process Control applications, such methods are used for basic research on the reactions studied, notably the detection of transient compounds and the kinetics of a reaction.
Our research will focus on graphical and numerical methods to find the important directions and any "aberrations" of a process.