Principal Components Regression PCR

Principal component regression is simply an extension of the PCA data compression method described earlier (Section 8.2.6.2), where a regression step is added after the data 

Unlike MLR and CLS, where the matrix inversions (XLX) 1 and (ClC) 1 can be very unstable in some situations, PCR involves a very stable matrix inversion of (T T) 1 because the PCA scores are orthogonal to one another (recall Equation 8.21). 

The PCR model can be easily condensed into a set of regression coefficients (bpcR) using the following equation  

The model can then be applied to provide the Y-value of an unknown sample from its array of X-values (xp)  

The calibration fit metrics previously discussed, RMSEE and r, can be used to express the fit of a PCR model. In this case, the RMSEE can be calculated directly from the estimated Y-residuals (f), which are calculated according to the following equation  

In the example we deal with a collection of ns= 80 com samples for which we have the NIR spectra, measured at nl=700 wavelengths, and 80 corresponding qualities (protein contents). 

The Matlab script Main PCR.m first reads in the complete corn data. Then it execute stepwise all the tasks that are described in the following. 

In order to test PCR and later PLS, we remove a random selection of 10 test samples from the total data set the 10 test spectra are collected row-wise in the matrix Ys and the corresponding known qualities in a column vector qs,known. The remaining spectra are organised in the same way in the matrix Y of dimensions 70x700. For each one of the samples we also know the protein content we collect these qualities in the vector q with 70 entries. In the following, Y and q serve as the calibration set that is used later to predict the unknown qualities qs for the test set Ys. The predicted qs can then be compared with the known qualities qs,known. 

There is a fair amount of structure in the NIR spectra but the differences between the spectra are rather subtle. Obviously, the protein content cannot easily be read from any of the peaks. 

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Some methods that paitly cope with the above mentioned problem have been proposed in the literature. The subject has been treated in areas like Cheraometrics, Econometrics etc, giving rise for example to the methods Partial Least Squares, PLS, Ridge Regression, RR, and Principal Component Regression, PCR [2]. In this work we have chosen to illustrate the multivariable approach using PCR as our regression tool, mainly because it has a relatively easy interpretation. The basic idea of PCR is described below. 

Sections 9A.2-9A.6 introduce different multivariate data analysis methods, including Multiple Linear Regression (MLR), Principal Component Analysis (PCA), Principal Component Regression (PCR) and Partial Least Squares regression (PLS). 

Other chemometrics methods to improve caUbration have been advanced. The method of partial least squares has been usehil in multicomponent cahbration (48—51). In this approach the concentrations are related to latent variables in the block of observed instmment responses. Thus PLS regression can solve the colinearity problem and provide all of the advantages discussed earlier. Principal components analysis coupled with multiple regression, often called Principal Component Regression (PCR), is another cahbration approach that has been compared and contrasted to PLS (52—54). Cahbration problems can also be approached using the Kalman filter as discussed (43). 

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. ... 

We will see that CLS and ILS calibration modelling have limited applicability, especially when dealing with complex situations, such as highly correlated predictors (spectra), presence of chemical or physical interferents (uncontrolled and undesired covariates that affect the measurements), less samples than variables, etc. More recently, methods such as principal components regression (PCR, Section 17.8) and partial least squares regression (PLS, Section 35.7) have been... 

The application of principal components regression (PCR) to multivariate calibration introduces a new element, viz. data compression through the construction of a small set of new orthogonal components or factors. Henceforth, we will mainly use the term factor rather than component in order to avoid confusion with the chemical components of a mixture. The factors play an intermediary role as regressors in the calibration process. In PCR the factors are obtained as the principal components (PCs) from a principal component analysis (PC A) of the predictor data, i.e. the calibration spectra S (nxp). In Chapters 17 and 31 we saw that any data matrix can be decomposed ( factored ) into a product of (object) score vectors T(nxr) and (variable) loadings P(pxr). The number of columns in T and P is equal to the rank r of the matrix S, usually the smaller of n or p. It is customary and advisable to do this factoring on the data after columncentering. This allows one to write the mean-centered spectra Sq as ... 

The method of PCA can be used in QSAR as a preliminary step to Hansch analysis in order to determine the relevant parameters that must be entered into the equation. Principal components are by definition uncorrelated and, hence, do not pose the problem of multicollinearity. Instead of defining a Hansch model in terms of the original physicochemical parameters, it is often more appropriate to use principal components regression (PCR) which has been discussed in Section 35.6. An alternative approach is by means of partial least squares (PLS) regression, which will be more amply covered below (Section 37.4). 

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. 

Galego and Arroyo [14] described a simultaneous spectrophotometric determination of OTC, hydrocortisone, and nystatin in the pharmaceutical preparations by using ratio spectrum-zero crossing derivate method. The calculation was performed by using multivariate methods such as partial least squares (PLS)-l, PLS-2, and principal component regression (PCR). This method can be used to resolve accurately overlapped absorption spectra of those mixtures. 

Principal Component Regression, PCR, and Partial Least Squares, PLS, are the most widely known and applied chemometrics methods. This is particularly the case for PLS, for which there is a tremendous number of applications and a never-ending stream of proposed improvements. The details of these latest modifications are not within the scope of this book and we concentrate on the essential, classical aspects. 

Regression can be performed directly with the values of the variables (ordinary least-squares regression, OLS) but in the most powerful methods, such as principal component regression (PCR) and partial least-squares regression (PLS), it is done via a small set of intermediate linear latent variables (the components). This approach has important advantages ... 

Sivakesava et al. also used Raman (as well as FT-IR and NIR) to perform a simultaneous on-line determination of biomass, glucose, and lactic acid in lactic acid fermentation by Lactobacillus casei.2 Partial least squares (PLS) and principal components regression (PCR) equations were generated after suitable wavelength regions were determined. The best standard errors were found to be glucose, 2.5 g/1 lactic acid, 0.7 g/1 and optical cell density, 0.23. Best numbers were found for FT-IR with NIR and Raman being somewhat less accurate (in this experiment). 

For partial least-squares (PLS) or principal component regression (PCR), the infrared spectra were transferred to a DEC VAX 11/750 computer via the NIC-COM software package from Nicolet. This package also provided utility routines used to put the spectra into files compatible with the PLS and PCR software. The PLS and PCR program with cross-validation was provided by David Haaland of Sandia National Laboratory. A detailed description of the program and the procedures used in it has been given (5). 

Principal Component Regression (PCR) was used by Tuchbreiter and MueUiaupt to determine the composition of a number of random ethane/propene, ethane/1-hexene, and ethane/l-octene copolymers [120]. After polymerization, the polymers were characterized by both Attenuated Total Reflection Fourier Transform Infrared Spectroscopy (ATR-FT-IR) and C NMR and multivariate calibration models using PCR were subsequently developed to estimate the co-monomer content. 

Partial least squares (PLS) and principal component regression (PCR) are the most widely used multivariate calibration methods in chemometrics. Both of these methods make use of the inverse calibration approach, where it i.s... 

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