PCR Calibration

It is possible to develop the ideas behind PCR and to a lesser extent behind PLS, based on chemical ideas and intuition. Naturally, this is not the only way and both PCR and PLS methods have been developed on pathways that are theoretically oriented. 

While we do not know the components of the corn samples nor their component spectra, nor even how many components there are, we can still assume that Beer-Lambert s law holds and we can write Y=CA (see Chapter 3.1). There is nothing new here. 

For a chemist, it intuitively makes sense to assume that the quality protein content is related to the concentrations of the components in the mixture. The simplest assumption is that the protein content is the weighted average of the concentrations of the relevant individual components. Some components have a high protein content others might even have a negative influence. This is best expressed in a matrix equation  

All we know at present is the vector of qualities q and that q might be approximated by the product Cb. We do not have an idea about the number of components, nc, nor about C or b.  

Now we remember equation (5.49) C=UT, C is the product of U and a transformation matrix T. Introduction of this equation into equation (5.59) results in 

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PRESS for validation data. One of the best ways to determine how many factors to use in a PCR calibration is to generate a calibration for every possible rank (number of factors retained) and use each calibration to predict the concentrations for a set of independently measured, independent validation samples. We calculate the predicted residual error sum-of-squares, or PRESS, for each calibration according to equation [24], and choose the calibration that provides the best results. The number of factors used in that calibration is the optimal rank for that system. 

We compute a PCR calibration in exactly the same way we computed an ILS calibration. The only difference is the data we start with. Instead of directly using absorbance values expressed in the spectral coordinate system, we use the same absorbance values but express them in the coordinate system defined by the basis vectors we have retained. Instead of a data matrix containing absorbance values, we have a data matrix containing the coordinates of each spectrum on each of the axes of our new coordinate system. We have seen that these new coordinates are nothing more than the projections of the spectra onto the basis vectors. These projections are easily computed ... 

Now we are ready to solve for the PCR calibration matrix. We do this exactly the same way we solved for the ILS calibration. First, we post-multiply both sides of equation [59] by ATpreJ. 

Figure 64. Plots of the PCR calibration coefficients calculated for each component with each training set.

Figure 65. Expected concentrations (x-axis) vs. predicted concentrations (y-axis) for PCR calibrations (see text).

Whether this tendency of PLS to reject nonlinearities by pushing them onto the later factors which are usually discarded as noise factors will improve or degrade the prediction accuracy and robustness of a PLS calibration as compared to the same calibration generated by PCR depends very much upon the specifics of the data and the application. If the nonlinearities are poorly correlated to the properties which we are trying to predict, rejecting them can improve the accuracy. On the other hand, if the rejected nonlinearities contain information that has predictive value, then the PLS calibration may not perform as well as the corresponding PCR calibration that retains more of the nonlinearities and therefore is able to exploit the information they contain. In short, the only sure way to determine if PLS or PCR is better for a given calibration is to try both of them and compare the results. 

PRESS of PCR calibration from Al/Cl for A3 PRESS of PCR calibration from A2/C2 for A3... 

PCR calibration coefficients from Al/Cl PCR calibration coefficients from A2/C2... 

PCR calibration from Al/Cl predicts concentrations for Al PCR calibration from Al/Cl predicts concentrations for A3 PCR calibration from Al/Cl predicts concentrations for A4 PCR calibration from Al/Cl predicts concentrations for AS PCR calibration from A2/C2 predicts concentrations for A2 PCR calibration from A2/C2 predicts concentrations for A3 PCR calibration from A2/C2 predicts concentrations for A4 PCR calibration from A2/C2 predicts concentrations for AS... 

We chose the number of PCs in the PCR calibration model rather casually. It is, however, one of the most consequential decisions to be made during modelling. One should take great care not to overfit, i.e. using too many PCs. When all PCs are used one can fit exactly all measured X-contents in the calibration set. Perfect as it may look, it is disastrous for future prediction. All random errors in the calibration set and all interfering phenomena have been described exactly for the calibration set and have become part of the predictive model. However, all one needs is a description of the systematic variation in the calibration data, not the... 

This should remind the reader of e.g. equation (5.28). The vector qpcR is nothing but the projection of the quality vector q into the space U. The PCR calibration is good if the vector q is close to the space U, it is bad otherwise. 

The Matlab function PCR ca librat ion. m performs the PCR calibration according to equation (5.62). Note that we use ne=12 eigenvectors in the above calculations. This is the optimal number for prediction, as we show in Cross Validation (p.303j. The reader is invited to play with this number and observe the effect. 

Figure 5-60. PCR calibration of the corn data using 12 factors qpcR versus the actual qualities q.

For the sake of completeness, we introduce an alternative at this stage and present a more theoretical but quicker path for the development of the PCR calibration and prediction equations. 

The function PLS ca librat ion, m is the PLS equivalent to PCR calibration, m. The iterative loop implements equations (5.73)-(5.77). The prognostic vector vprog is introduced in the next section. 

In the polyethylene example. Figure 12.22 shows the regression coefficient spectrum for the PCR calibration to the percentage HDPE content in the polyethylene blends. Two particularly recognizable features in... 

Determining tablet hardness usually involves the destructive diametral crushing of each individnal tablet. However, Kirsch and Drennen developed a nondestructive method based on the slope of NIR spectra for intact tablets. Tablet hardness ranged from 1 to 7kp and was predicted with errors of 0.2-0.7kp by then-calibration models. Their approach, the robustness of which was confirmed by the resnlts, was claimed to surpass PCR calibration models in many respects as a result of its being unaffected by the absorption of individual bands. 

So far, the number of basis vectors that should be used in the calibration model has not been discussed. It is standard practice during PCR calibration modeling to use one principal component (PC), two PCs, three PCs, and so on. The error from this prediction is used to calculate the RMSEP figure of merit. Plots of RMSEC and RMSEP against the number of PCs used in the calibration model are used to determine the optimum number of factors. Usually, a continuous decrease in RMSEC is observed as more PCs are added into the calibration model however, the predictive performance of the calibration model often reaches a minimum RMSEP at the optimum number of factors and begins to increase thereafter. 

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