Regression multivariate calibration

Berzas, J.J. Rodriguez, J. Castaneda, G. Determination of ethinylestradiol and gestodene in pharmaceuticals by a partial least-squares and principal component regression multivariate calibration, Anal.ScL, 1997,13, 1029-1032. 

An important aspect of all methods to be discussed concerns the choice of the model complexity, i.e., choosing the right number of factors. This is especially relevant if the relations are developed for predictive purposes. Building validated predictive models for quantitative relations based on multiple predictors is known as multivariate calibration. The latter subject is of such importance in chemo-metrics that it will be treated separately in the next chapter (Chapter 36). The techniques considered in this chapter comprise Procrustes analysis (Section 35.2), canonical correlation analysis (Section 35.3), multivariate linear regression... 

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 offset a, and the multiplication constant bj are estimated by simple linear regression of the ith individual spectrum on the reference spectrum z. For the latter one may take the average of all spectra. The deviation e, from this fit carries the unique information. This deviation, after division by the multiplication constant, is used in the subsequent multivariate calibration. For the above correction it is not mandatory to use the entire spectral region. In fact, it is better to compute the offset and the slope from those parts of the wavelength range that contain no relevant chemical information. However, this requires spectroscopic knowledge that is not always available. 

Several approaches have been investigated recently to achieve this multivariate calibration transfer. All of these require that a small set of transfer samples is measured on all instruments involved. Usually, this is a small subset of the larger calibration set that has been measured on the parent instrument A. Let Z indicate the set of spectra for the transfer set, X the full set of spectra measured on the parent instrument and a suffix Aor B the instrument on which the spectra were obtained. The oldest approach to the calibration transfer problem is to apply the calibration model, b, developed for the parent instrument A using a large calibration set (X ), to the spectra of the transfer set obtained on each instrument, i.e. and Zg. One then regresses the predictions (=Z b ) obtained for the parent instrument on those for the child instrument yg (=Z b ), giving... 

In recent years there has been much activity to devise methods for multivariate calibration that take non-linearities into account. Artificial neural networks (Chapter 44) are well suited for modelling non-linear behaviour and they have been applied with success in the field of multivariate calibration [47,48]. A drawback of neural net models is that interpretation and visualization of the model is difficult. Several non-linear variants of PCR and PLS regression have been proposed. Conceptually, the simplest approach towards introducing non-linearity in the regression model is to augment the set of predictor variables (jt, X2, ) with their respective squared terms (xf,. ..) and, optionally, their possible cross-product... 

On the other hand, when latent variables instead of the original variables are used in inverse calibration then powerful methods of multivariate calibration arise which are frequently used in multispecies analysis and single species analysis in multispecies systems. These so-called soft modeling methods are based, like the P-matrix, on the inverse calibration model by which the analytical values are regressed on the spectral data ... 

Nonlinearity is a subject the specifics of which are not prolifically or extensively discussed as a specific topic in the multivariate calibration literature, to say the least. Textbooks routinely cover the issues of multiple linear regression and nonlinearity, but do not cover the issue with full-spectrum methods such as PCR and PLS. Some discussion does exist relative to multiple linear regression, for example in Chemometrics A Textbook by D.L. Massart et al. [6], see Section 2.1, Linear Regression (pp. 167-175) and Section 2.2, Non-linear Regression, (pp. 175-181). The authors state,... 

Multivariate calibration has the aim to develop mathematical models (latent variables) for an optimal prediction of a property y from the variables xi,..., jcm. Most used method in chemometrics is partial least squares regression, PLS (Section 4.7). An important application is for instance the development of quantitative structure—property/activity relationships (QSPR/QSAR). 

QSPR models have been developed by six multivariate calibration methods as described in the previous sections. We focus on demonstration of the use of these methods but not on GC aspects. Since the number of variables is much larger than the number of observations, OLS and robust regression cannot be applied directly to the original data set. These methods could only be applied to selected variables or to linear combinations of the variables. 

The aim of multivariate calibration methods is to determine the relationships between a response y-variable and several x-variables. In some applications also y is multivariate. In this chapter we discussed many different methods, and their applicability depends on the problem (Table 4.6). For example, if the number m of x-variables is higher than the number n of objects, OLS regression (Section 4.3) or robust regression (Section 4.4) cannot be applied directly, but only to a selection... 

Not just by accident PLS regression is the most used method for multivariate calibration in chemometrics. So, we recommend to start with PLS for single y-variables, using all x-variables, applying CV (leave-one-out for a small number of objects, say for n < 30, 3-7 segments otherwise). The SEPCV (standard deviation of prediction errors obtained from CV) gives a first idea about the relationship between the used x-variables and the modeled y, and hints how to proceed. Great effort should be applied for a reasonable estimation of the prediction performance of calibration models. 

This method can be considered a calibration transfer method that involves a simple instrument-specific postprocessing of the calibration model outputs [108,113]. It requires the analysis of a subset of the calibration standards on the master and all of the slave instmments. A multivariate calibration model built using the data from the complete calibration set obtained from the master instrument is then applied to the data of the subset of samples obtained on the slave instruments. Optimal multiplicative and offset adjustments for each instrument are then calculated using linear regression of the predicted y values obtained from the slave instrument spectra versus the known y values. 

R. Heikka, P. Minkkinen, and V-M Taavitsainen, Comparison of variable selection and regression methods in multivariate calibration of a process analyzer. Process Control and Quality, 6, 47-54 (1994). 

In order to construct a calibration model, the values of the parameters to be determined must be obtained by using a reference method. The optimum choice of reference method will be that providing the highest possible accuracy and precision. The quality of the results obtained with a multivariate calibration model can never exceed that of the method used to obtain the reference values, so the choice should be carefully made as the quality of the model will affect every subsequent prediction. The averaging of random errors inherent in regression methods can help construct models with a higher precision than the reference method. 

P.J. Gemperhne, J.R. Long and V.G. Gregoriov, Nonlinear multivariate calibration using principal components regression and artificial neural networks. Anal Chem., 63, 2313-2323 (1991). 

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... 

The simplest form of multivariate calibration is multiple linear regression (MLR), which is well suitable in X-ray analysis The concentration of a given element is a linear combination of the intensities found at certain wavelengths in X-ray analysis. 

This method is probably the simplest of the software-based standardization approaches.73,74 It is applied to each X-variable separately, and requires the analysis of a calibration set of samples on both master and slave instruments. A multivariate calibration model is built using the spectra obtained from the master instrument, and then this model is applied to the spectra of the same samples obtained from the slave instrument. Then, a linear regression of the predicted Y-values obtained from the slave instrument spectra and the known Y-values is performed, and the parameters obtained from this linear regression fit are used to calculate slope and intercept correction factors. In this... 

Heikka, R., Minkkinen, P. and Taavitsainen, V.-M., Comparison of Variable Selection and Regression Methods in Multivariate Calibration of a Process Analyzer Process Contr. Qual. 1994, 6, 47-54. 

Calibration is the process by which a mathematical model relating the response of the analytical instrument (a spectrophotometer in this case) to specific quantities of the samples is constructed. This can be done by using algorithms (usually based on least squares regression) capable of establishing an appropriate mathematical relation such as single absorbance vs. concentration (univariate calibration) or spectra vs. concentration (multivariate calibration). 

---