General Guidelines for Calibration Data Sets

In the preceding sections of this chapter, several multivariate regression methods have been presented. All of these methods require standard mixmre spectra. The construction of the mixtures will now be addressed in general, but more detailed discussion can be found elsewhere [23]. 

It is not a trivial task to make a good calibration set. The set should be designed so that each constituent spans the anticipated concentrations. As the number of constituents increases, this becomes progressively more difficult. The temptation is to omit constituents or use pure constituents, so that one mixture is 100% of a single constituent. This should be avoided unless it is possible that the constituent could indeed be pure in an unknown. It is more likely that a constituent could be missing from an unknown, in which case a mixture with a missing constiment is acceptable. The reason that pure samples should not be used is that none of the constituent interactions will be present in the spectra. Therefore, more than one parameter of the mixmre will be missing, which can distort the model considerably. 

A more subtle temptation is to generate the mixmres by successive dilution. Two or more mixtures that vary in concentration by only a factor will in reality produce the same spectral vectors. That is, the spectra are identical except that one has a different scale to the other. Mathematically, the two spectra represent the same ratios of components and they are not different. The two spectra force the absorbance matrix A to be collinear, but more important, the concentration ratios in the C matrix are also collinear. The A matrix will have noise associated with each spectrum, and as the noise is not the same in each spectrum, the A matrix will be only approximately collinear. C will be precisely collinear, and if C is inverted (even through pseudoinversion), the result will be indeterminate that is, it is equivalent to division by zero. As soon as this happens, the regression fails and the model is invalid. 

Another subtle problem is under- or oveirepresentation of constiments. For example, if there is a three-constiment solution that contains constiments A, B, and C, A will be overrepresented if it is present in 12 out of 12 mixtures, and C will be underrepresented if it is only in 4 out of 12 mixtures. Ideally, A, B, and C should be in all 12 mixtures, but it could be anticipated that C will be absent in some unknowns. The model as constructed will predict A well but C rather poorly, as it has insufficient data to model C. If C is the only component anticipated to be missing from an unknown solution, A and B should be in all 12 calibration mixtures and C should be in nine or 10. 

For ELS-based models, the solvent concentration can be omitted from the constituent matrix C. The solvent is a constiment, but it is generally not of interest. The constiment concentrations can be randomized and span the anticipated unknown concentrations, but care must be taken to ensure that the sum of all the constiments is not the same. That is, in any given mixture, the sum of all the concentrations should not be a constant. If the concentrations sum to a constant, the model only spans a plane in three or more dimensions (one dimension for each constiment). If the constituents are a binary mixture, the model will describe a line (a two-dimensional model). The model will have no depth , and any unknown mixture 

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