Predicted concentration

That all four methods give a different result for the concentration of analyte underscores the importance of choosing a proper blank but does not tell us which of the methods is correct. In fact, the variation within each method for the reported concentration of analyte indicates that none of these four methods has adequately corrected for the blank. Since the three samples were drawn from the same source, they must have the same true concentration of analyte. Since all four methods predict concentrations of analyte that are dependent on the size of the sample, we can conclude that none of these blank corrections has accounted for an underlying constant source of determinate error. 

Fig. 7. Comparison of various transport schemes for advecting a cone-shaped puff in a rotating windfield after one complete rotation (a), the exact solution (b), obtained by an accurate numerical technique (c), the effect of numerical diffusion where the peak height of the cone has been severely tmncated and (d), where the predicted concentration field is very bumpy, showing the effects of artificial dispersion. In the case of (d), spurious waves are...

The data in the validation set are used to challenge the calibration. We treat the validation samples as if they are unknowns. We use the calibration developed with the training set to predict (or estimate) the concentrations of the components in the validation samples. We then compare these predicted concentrations to the actual concentrations as determined by an independent referee method (these are also called the expected concentrations). In this way, we can assess the expected performance of the calibration on actual unknowns. To the extent that the validation samples are a good representation of all the unknown samples we will encounter, this validation step will provide a reliable estimate of the calibration s performance on the unknowns. But if we encounter unknowns that are significantly different from the validation samples, we are likely to be surprised by the actual performance of the calibration (and such surprises are seldom pleasant). 

The best protection we have against placing an inadequate calibration into service is to challenge the calibration as agressively as we can with as many validation samples as possible. We do this to uncover any weaknesses the calibration might have and to help us understand the calibration s limitations. We pretend that the validation samples are unknowns. We use the calibration that we developed with the training set to predict the concentrations of the validation samples. We then compare these predicted concentrations to the known or expected concentrations for these samples. The error between the predicted concentrations vs. the expected values is indicative of the error we could expect when we use the calibration to analyze actual unknown samples. 

As discussed in the first chapter, it is possible to use almost any kind of data to predict almost any type of property. But to keep things simple, we will continue using the vocabulary of spectroscopy. Accordingly, we will call the data we create absorbance spectra, or simply spectra, and we will call the property we are trying to predict concentration. 

Now that we have calculated K we can use it to predict the concentrations in an unknown sample from its measured spectrum. First, we place the spectrum into a new absorbance matrix, Aunk. We can now use equation [29] to give us a new concentration matrix, Cunk, containing the predicted concentration values for the unknown sample. 

Figure 19 contains plots of the expected (x-axis) vs. predicted (y-axis) concentrations for the fits to training sets A1 and A2. (Notice that the expected concentration values for Al, the factorially designed training set are either 0.0, 0.5, or 1.0, plus or minus the added noise). While there is certainly a recognizable correlation between the expected and predicted concentration values this is not as good a fit as we might have hoped for. 

Figure 19. Expected concentrations (x-axis) vs. predicted concentrations (y-axis) for the fit to training sets Al and A2.

Table 2 also contains the correlation coefficient, r, for each K . If the predicted concentrations for a data set exactly matched the expected concentrations, r would equal 1.0. If there were absolutely no relationship between the predicted and expected concentrations, r would equal 0.0. 

Figure 23 contains plots of the expected vs. predicted concentrations for all of the nonzero intercept CLS results. We can easily see that these results are much better than the results of the first calibrations. It is also apparent that when we predict the concentrations from the spectra in A5, the validation set with the... 

Calculate the sum-squared of errors between the expected and predicted concentrations for the sample that was left out. 

Fortunately, since we also have concentration values for our samples, We have another way of deciding how many factors to keep. We can create calibrations with different numbers of basis vectors and evaluate which of these calibrations provides the best predictions of the concentrations in independent unknown samples. Recall that we do this by examing the Predicted Residual Error Sum-of Squares (PRESS) for the predicted concentrations of validation samples. 

It is often helpful to examine the regression errors for each data point in a calibration or validation set with respect to the leverage of each data point or its distance from the origin or from the centroid of the data set. In this context, errors can be considered as the difference between expected and predicted (concentration, or y-block) values for the regression, or, for PCA, PCR, or PLS, errors can instead be considered in terms of the magnitude of the spectral... 

CLS calibration with non-zero intercept from Al/Cl predicts concentrations for AI... 

CLS calibration from A6/C6 predicts concentrations for A6 CLS calibration from A6/C6 predicts concentrations for A7 CLS calibration from A6/C6 predicts concentrations for A3... 

CLS calibration from Al/Cl predicts concentrations for A3 CLS calibration from Al/Cl predicts concentrations for A4 CLS calibration from Al/Cl predicts concentrations for AS... 

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