Classical Least Squares, Part VII Spectral Reconstruction of Mixtures

Oct 01, 2011
Volume 26, Issue 10

We continue our discussion of the classical least squares approach to calibration, focusing on the reconstruction of mixtures in this installment.

Before continuing with our discussion, we need to report some errata that a sharp-eyed reader found in our previous column (1). Fortunately, although the worst one is widespread in the column, it is editorial rather than substantive and we suspect that most readers caught it and realized what was intended. This error was the interchange of the axis labels on all those plots where the spectra were plotted as absorbance; the correct axis labels should be that the x-axis represents wavelengths and the y-axis represents absorbance. Unfortunately, we have to confess that it was our fault, apparently the incorrect information was put into the plot generator program and was thus used repeatedly for all those plots. The article was not proofread for that type of error because the plots were computer-generated and we relied on the computer not making a mistake. (Which it didn't!) In addition, because the data was collected using a Fourier-transform near-infrared (FT-NIR) instrument, the correct units for the wavelength scale are wavenumbers, rather than wavelengths.

Another type of error in that column is that for Figures 4c, 5e, and 6e, the wavelength ranges in the figure captions disagree with the wavelength ranges in the spectral plots. In all of those cases, the wavelength range indicated on the plots are the correct ones.

We thank Karl Norris for finding and reporting those errors to us. Karl also pointed out that one of our statements in the prior column (2), while, strictly speaking, correct, might lead someone to misinterpret our intentions. We stated, "Only one spectrum appears, because the spectra are so similar that they completely and exactly overlap each other, thus the spectrum from one reading hides the spectrum of each of the other two." Karl pointed out that someone might interpret that as indicating that the spectra were exactly matched, even down to the noise. Of course that's not the case; a better wording might have been something like ". . . because the spectra are so similar that they overlap each other to within a pixel of the display . . . ," which would have better indicated our intended meaning.

This column is the next continuation of our discussion of the classical least squares (CLS) approach to calibration (1–6).

Where do we stand? We have just finished presenting the data spectra, measured in the NIR spectral region, for a series of clear liquid samples comprising binary and ternary mixtures of three materials that are all miscible in all proportions.

Figure 1: Absorbance spectra of all mixtures: (a) full spectrum, (b) 4500–5000 cm-1, (c) 5000–6500 cm-1, (d) 6500–7500 cm-1, and (e) 7500–9000 cm-1. Note how the original experimental design can be seen in the spectra.
In the previous installments, we presented the spectra of the pure materials, as well as the spectra of the mixtures. However, we did not present these spectra in absorbance format. Because we will be comparing the spectral recreations to the original data spectra in absorbance format, it is appropriate to present the mixture spectra in absorbance format before we do that. These spectra are shown in Figure 1. As previously stated, when plotting absorbance we do not include the spectral region below 4500 cm-1. Therefore, all of our future spectra plots, as well as the calculations to be performed on the spectra, will exclude the spectral region below 4500 cm-1.

Note that the experimental design can be seen in parts b, c, d, and e of Figure 1, similarly to what we noted in the transmission spectra of the mixtures. In Figure 1e, we have indicated the spectra of the pure components, to assist in visualizing the design.

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