Classical Least Squares, Part VIII: Comparison of CLS Values with Known Values

Feb 01, 2012
Volume 27, Issue 2

A comparison of the experimental results to the theoretical expectations showed appreciable discrepancies. We consider possible problems in the execution of the experiment as the cause of these discrepancies.

This column is an installment in a series of our discussion of the classical least squares (CLS) approach to calibration (1–7). Where do we stand as of now? At this point, we have developed the theory of the CLS approach to calibration (1,2), described the measurement procedures and the data (3,4), and shown that when we apply the mathematical theory to the data we can reconstruct the spectra of the mixtures (5).

However, the theory of CLS only considers the relationship of the spectrum of a mixture to the spectra of the components of that mixture; the theory tells us nothing about the relationship of the spectrum to other properties of the mixture. In particular, it is silent on the question of the concentrations of the various mixture components. The numerical results of the CLS calculations are fractions, each representing the fraction that each mixture component contributes to the total mixture spectrum.

Table I: Actual values for the various mixtures in the experimental design
Therefore, the remaining task is to verify whether the coefficients calculated for the spectra in fact represent the concentrations of the various components in the different mixtures. We begin this task by presenting the actual concentrations in the experimental mixtures made. The values in the experimental design, as we presented them in our May 2011 installment as Figure 5 (5), are the target values for the various mixtures comprising the design. The actual mixtures were made up gravimetrically to be close to the target values, but the true weight percentages were calculated from the actual weights of the various components added to the mixtures. Table I presents those actual values.

In our previous columns (5,6), we found that by computing the regression coefficients for the spectra of the pure components using the CLS algorithm and then applying those coefficients to those same pure component spectra, we were able to reproduce the spectra of mixtures of those materials very well. Our interest now is to ascertain whether the coefficients we calculated do, in fact, represent the concentrations of the corresponding materials.

Theoretically, any of the spectral regions having the absorbance bands that we previously noted should give equivalent results, the same results, and the correct results. It therefore makes sense to use both the entire spectral region we measured and each of the individual spectral regions that we previously separated out and plotted.

Table II: Results from sample with nominal composition: 25% toluene, 25% dichloromethane, 50% n-heptane. All tabled values are wt %.
Table II presents the coefficients calculated from the various spectral regions for one of the ternary mixtures (25% toluene, 25% dichloromethane, and 50% n-heptane). We see that there is some variation in the percentage of each component calculated at the different wavelength ranges. Thus, we also calculated the mean percentage of each mixture component and include that as an entry in the table.

Table III: Gravimetric and spectrally calculated compositions for all samples
By "splitting the difference" this way, we expect that the mean composition we calculate will be a better estimate of the "true" calculated value in any of the individual calculated values. We are now ready to present the results for all of the different mixtures that our experimental design specified. These are given in Table III, which also contains the actual gravimetric values for the composition. We use the term "actual" because even though the experimental design illustrated in Figure 5 of the May 2011 installment (5) specified the design points to use, the individual samples were made up according to the "dispense approximately and measure exactly" paradigm for preparing samples.

What do we observe in the results of Table III?

Although sporadic readings (such as toluene in sample 9) showed good agreement between the gravimetric and spectrally calculated values, for the most part the two readings disagreed markedly, often by several percent or more. We therefore have to conclude that this isn't working very well.

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