PCR Prediction

However, we need to determine the row vector us corresponding to the new sample. 

The new spectrum, ys, is shown as the grey row underneath Y and the corresponding us as the grey row underneath U. 

The spectrum of the new sample is the product of us and the matrix SV. us contains the coordinates of spectrum ys in the eigenvector space spanned by SV. Rearranging equation (5.64), us is computed as  

Inserting this equation into equation (5.60) and substitution of b by equation (5.61) leads to  

The matrices U, S, V and the vector b are determined by the calibration set Y and q thus the product VtS Utq is also completely determined by the calibration. It is a column vector of dimension nix 1 (nl is the number of wavelengths at which the spectra are taken), we call it the prognostic vector, Vprog. The quality qs for any new sample can be predicted by the product of its spectrum ys and the prognostic vector vprog  

---

The routine PCR ca librat ion. m also returns a prognostic vector vprog. It is used for prediction and its function is explained in the next Chapter PCR Prediction. 

In this last study, a two-fold rise in BCR-ABLl expression by RQ-PCR predicted a mutation in a kinase domain of BCR-ABLl in 61% of patients either at the time of the rise in BCR-ABLl expression or within 3 months of the rise (78). However, another study did not predict the presence of mutations in patients who had a single two-fold or greater rise in BCR-ABLl transcripts. Rather, the development of rising BCR-ABLl transcript levels was necessary to reliably pick up a mutation. The authors concluded that a serial rise was more reliable than a single rise (79). Eleven mutations were detected in 10 out of 82 patients in this study. 

Table 8.6 Results obtained from building a PCR predictive model forthe c/ s-butadiene content in styrene-butadiene copolymers...

RNA from barley probed with the DNA fragment obtained by PCR predicts a 1800 b mRNA for the aminotransferase. 

The goal of PCR is to extract intrinsic effects in the data matrix X and to use these effects to predict the values of Y. 

For many applications, quantitative band shape analysis is difficult to apply. Bands may be numerous or may overlap, the optical transmission properties of the film or host matrix may distort features, and features may be indistinct. If one can prepare samples of known properties and collect the FTIR spectra, then it is possible to produce a calibration matrix that can be used to assist in predicting these properties in unknown samples. Statistical, chemometric techniques, such as PLS (partial least-squares) and PCR (principle components of regression), may be applied to this matrix. Chemometric methods permit much larger segments of the spectra to be comprehended in developing an analysis model than is usually the case for simple band shape analyses. 

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. 

Now, we are ready to apply PCR to our simulated data set. For each training set absorbance matrix, A1 and A2, we will find all of the possible eigenvectors. Then, we will decide how many to keep as our basis set. Next, we will construct calibrations by using ILS in the new coordinate system defined by the basis set. Finally, we will use the calibrations to predict the concentrations for our validation sets. 

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

The prediction step for PLS is also slightly different than for PCR. It is also done on a rank-by-rank basis using pairs of special and concentration factors. For each component, the projection of the unknown spectrum onto the first spectral factor is scaled by a response coefficient to become a corresponding projection on the first concentration factor. This yields the contribution to the total concentration for that component that is captured by the first pair of spectral and concentration factors. We then repeat the process for the second pair of factors, adding its concentration contribution to the contribution from the first pair of factors. We continue summing the contributions from each successive factor pair until all of the factors in the basis space have been used. 

Just as we did for PCR, we must determine the optimum number of PLS factors (rank) to use for this calibration. Since we have validation samples which were held in reserve, we can examine the Predicted Residual Error Sum of Squares (PRESS) for an independent validation set as a function of the number of PLS factors used for the prediction. Figure 54 contains plots of the PRESS values we get when we use the calibrations generated with training sets A1 and A2 to predict the concentrations in the validation set A3. We plot PRESS as a function of the rank (number of factors) used for the calibration. Using our system of nomenclature, the PRESS values obtained by using the calibrations from A1 to predict A3 are named PLSPRESS13. The PRESS values obtained by using the calibrations from A2 to predict the concentrations in A3... 

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. 

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... 

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... 

Because of peak overlappings in the first- and second-derivative spectra, conventional spectrophotometry cannot be applied satisfactorily for quantitative analysis, and the interpretation cannot be resolved by the zero-crossing technique. A chemometric approach improves precision and predictability, e.g., by the application of classical least sqnares (CLS), principal component regression (PCR), partial least squares (PLS), and iterative target transformation factor analysis (ITTFA), appropriate interpretations were found from the direct and first- and second-derivative absorption spectra. When five colorant combinations of sixteen mixtures of colorants from commercial food products were evaluated, the results were compared by the application of different chemometric approaches. The ITTFA analysis offered better precision than CLS, PCR, and PLS, and calibrations based on first-derivative data provided some advantages for all four methods. ... 

The purpose of Partial Least Squares (PLS) regression is to find a small number A of relevant factors that (i) are predictive for Y and (u) utilize X efficiently. The method effectively achieves a canonical decomposition of X in a set of orthogonal factors which are used for fitting Y. In this respect PLS is comparable with CCA, RRR and PCR, the difference being that the factors are chosen according to yet another criterion. 

Principal covariates regression (PCovR) is a technique that recently has been put forward as a more flexible alternative to PLS regression [17]. Like CCA, RRR, PCR and PLS it extracts factors t from X that are used to estimate Y. These factors are chosen by a weighted least-squares criterion, viz. to fit both Y and X. By requiring the factors to be predictive not only for Y but also to represent X adequately, one introduces a preference towards the directions of the stable principal components of X. 

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... 

Fig. 36.10. Prediction error (RMSPE) as a function of model complexity (number of factors) obtained from leave-one-out cross-validation using PCR (o) and PLS ( ) regression.

Principal component regression (PCR) is an extension of PCA with the purpose of creating a predictive model of the Y-data using the X or measurement data. For example, if X is composed of temperatures and pressures, Y may be the set of compositions that results from thermodynamic considerations. Piovoso and Kosanovich (1994) used PCR and a priori process knowledge to correlate routine pressure and temperature measurements with laboratory composition measurements to develop a predictive model of the volatile bottoms composition on a vacuum tower. 

---