A solution to this problem could be either to use variable selection methods such as, e.g., stepwise multiple linear regression, or to use full spectrum methods such as principal component regression or partial least squares.
As an example of a multivariable calibration problem of non-full rank, a situation from the feedingstuff industry is described. The quantities of interests are the contents of different chemical components in various raw materials and feedingstuffs.
The traditional chemical methods for analysis are slow and expensive, and therefore unfeasible for process control purposes. Instead one wants to use a near-infrared reflectance instrument that measures the reflectance from light at 701 different wavelengths-believing that is a connection between the reflectance and the chemical composition of the sample.
The calibration problem is solved by using principal component regression. It is well known that the principal components corresponding to the largest eigenvalues accounts for the largest part of the variance. Therefore these components are frequently used in the principal component regression, irrespectively of their predictive value. In a celibration situation like this however, it is more sensible to use those principal components that exhibit the largest predictive value.
The resulting calibration may be expressed as the coefficients to the reflectance measured by the 701 wavelengths. However, the coefficients to neighbouring wavelenghs exhibit fluctuations which can not be anything than "noise". These fluctuations may be diminished by smoothing the coefficients or by smoothing the incoming spectra. The smoothing may be performed by, e.g., filter methods, or by using Fourier-analysis methods. Such a smoothing is necessary because of the many wavelengths, and it would not have been necessary - or possible - on earlier NIR-instruments which typically have less than 20 wavelengths.
It is shown that the best way of smoothing is by using Fourier-analysis methods because these methods remove the high-frequency noise without moving the peeks.
An important requirement for a calibration is that the results must be very robust in order to avoid frequent recalibrations. In the literature frequently recalibration is often recommended, because of the influence of a new harvest, adjustment of the NIR-instrument and so on. Robustification is somehow achieved by using fullspectrum methods, but more robustness may be achieved by the smoothing. By using the neighbouring properties of the incoming spectra we have been able to make calibrations which lasts for more than 3 years. The thesis presents some results of work in this field, with the above example as an illustration of the ideas.
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