Multivariate calibrations

The data generated from a NIR or Raman spectrum do not immediately provide the concentrations of the species at any time, so there is no predictive capability. Construction of a calibmtion set requires an independent measure of the property, e.g. by HPLC or by NIR of known mixtures of the components. Two such methods are principal-component regression (PCR) and partial least squares (PLS). As soon as quantitative analysis is considered, the question of noise and reproducibility of the data set becomes important. It is therefore necessary to treat the mw data to remove the drift in baseline etc. which will occur over a long period of spectml acquisition. 

In all quantitative analysis the drift in the spectra is commonly addressed hy normalization of all peaks against an invariant peak. This requires knowledge of the system as well as assignments for the major hands. Where possible, normalization is the first step in pretreatment of the data. If Raman spectm are strongly influenced by fluorescence then this is removed (by a polynomial fit to the broad, underlying profile) prior to normalization (Stellman et al, 1995). 

It is apparent that there has been significant offset of the baseline between samples and hence that the data are unsuitable for quantitative analysis in the raw form. Three different methods of correction of these data have been compared (Shimoyama et al, 1998)  

The second derivative enables the location of the peaks in the spectra and thus assists in assignment, since it will be negative if the original spectrum has a local maximum, positive if it has a local minimum and a point of inflection when it is zero. Again the baseline drift is eliminated (Shimoyama et al, 1998) and either the first or the second derivative may be used in subsequent data analysis by one of the regression methods. In a comparison of the three correction methods specified above, it was found that MSC was best for the discrimination of the EVA copolymers due to their chemical similarity (Shimoyama et al, 1998), while in a comparison of rheological properties (Vedula and Hansen, 1998) the first-derivative spectra were satisfactory. 

In spectral data there will be noise from both multiplicative scatter (MS) and additive scatter (AS). Normalization works well to limit the effects of MS and the derivative spectra remove AS, but only the MSC method can minimize the effects of both (Shimoyama et al, 1997). 

In a study of the simultaneous determination of Mn04, Cr207 and Co , several mixtures of standard solutions of these three chemical species were prepared and analyzed by UV-visible spectrophotometry, at three different wavelengths 530, 440 and 410 nm (Scarminio et al., 1993). The data relative to the permanganate ion are given in Table 5A.3. The fitted linear model is 

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If the task is multivariate calibration, for example, the proper choice of a pre-processing method will essentially aflFect the quality of the resultant model. For more details about the use of these techniques together with PCA and PLS, readers are advised to consider the fundamental monograph by Erikson et al [8]. 

L. Eriksson, J. Trygg, E. Johansson, R. Bro, S. Wold, Orthogonal signal correction, wavelet analysis, and multivariate calibration of complicated process fluorescence data. Anal. Chim. Acta, 2000 420, 181-195. 

H. A. Martens, Multivariate Calibration Ph.D. dissertation. Technical University of Norway, Trondheim, Norway, 1985. 

A solvent free, fast and environmentally friendly near infrared-based methodology was developed for the determination and quality control of 11 pesticides in commercially available formulations. This methodology was based on the direct measurement of the diffuse reflectance spectra of solid samples inside glass vials and a multivariate calibration model to determine the active principle concentration in agrochemicals. The proposed PLS model was made using 11 known commercial and 22 doped samples (11 under and 11 over dosed) for calibration and 22 different formulations as the validation set. For Buprofezin, Chlorsulfuron, Cyromazine, Daminozide, Diuron and Iprodione determination, the information in the spectral range between 1618 and 2630 nm of the reflectance spectra was employed. On the other hand, for Bensulfuron, Fenoxycarb, Metalaxyl, Procymidone and Tricyclazole determination, the first order derivative spectra in the range between 1618 and 2630 nm was used. In both cases, a linear remove correction was applied. Mean accuracy errors between 0.5 and 3.1% were obtained for the validation set. 

Mancozeb is a dithiocarbamate pesticide with a very low solubility in organic and inorganic solvent. In this work we have developed a solvent free, accurate and fast photoacoustic FTIR-based methodology for Mancozeb determination in commercial fungicides. The proposed procedure was based on the direct measurement of the solid samples in the middle infrared region using a photoacoustic detector. A multivariate calibration approach based on the use of partial least squares (PLS) was employed to determine the pesticide content in commercially available formulations. 

Different calibration models, such as classical least squares and multivariate calibration approaches have been considered. 

Beebe, K.R., Kowalski, B.R., "An Introduction to Multivariate Calibration and Analysis", Anal. Chem. 1987 (59) 1007A-1017A. 

Martens, H, Naes, T., Multivariate Calibration, John Wiley and Sons, New York, 1989. 

Capitan-Vallvey, L.F. et al.. Simultaneous determination of the colorants tartrazine, ponceau 4R and sunset yellow FCF in foodstuffs by solid phase spectrophotometry using partial least square multivariate calibration, Talanta, 47, 861, 1998. 

Ni, Y.N., Bai, J.L., and Jin, L., Simultaneous adsorptive voltammetric analysis of mixed colorants by multivariate calibration approach. Anal. Chim. Acta, 329, 65, 1996. 

H. Martens and T. Naes, Multivariate Calibration, Wiley, New York, 1993. 

K, Beebe and B.R. Kowalski, An introduction to multivariate calibration and analysis. Anal. 

Many other subjects are important to achieve successful pattern recognition. To name only two, it should be investigated to what extent outliers are present, because these can have a profound influence on the quality of a model and to what extent clusters occur in a class (e.g. using the index of clustering tendency of Section 30.4.1). When clusters occur, we must wonder whether we should not consider two (or more) classes instead of a single class. These problems also affect multivariate calibration (Chapter 36) and we have discussed them to a somewhat greater extent in that chapter. 

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 question of how many components to include in the final model forms a rather general problem that also occurs with the other techniques discussed in this chapter. We will discuss this important issue in the chapter on multivariate calibration. 

The ultimate goal of multivariate calibration is the indirect determination of a property of interest (y) by measuring predictor variables (X) only. Therefore, an adequate description of the calibration data is not sufficient the model should be generalizable to future observations. The optimum extent to which this is possible has to be assessed carefully when the calibration model chosen is too simple (underfitting) systematic errors are introduced, when it is too complex (oveifitting) large random errors may result (c/. Section 10.3.4). 

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 ... 

There are two points of view to take into account when setting up a trmning set for developing a predictive multivariate calibration model. One viewpoint is that the calibration set should be representative for the population for which future predictions are to be made. This will generally lead to a distribution of objects in experimental space that has a higher density towards the center, tailing out to the boundaries. Another consideration is that it is better to spread the samples more or... 

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 the direct standardization introduced by Wang et al. [42] one finds the transformation needed to transfer spectra from the child instrument to the parent instrument using a multivariate calibration model for the transformation matrix = ZgF. The transformation matrix F (qxq) translates spectra Zg that are actually measured on the child instrument B into spectra Z that appear as if they were measured on instrument A. Predictions are then obtained by applying the old calibration model to these simulated spectra Z ... 

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... 

T. Naes and E. Risvik (Editors), Multivariate Analysis of Data in Sensory Science, Data Handling in Science and Technology Series. Elsevier, Amsterdam, 1996 P.K. Hopke and X.-H. Song, The chemical mass balance as a multivariate calibration problem. Chemom. Intell. Lab. Assist., 37 (1997) 5-14. 

E. de Noord, The influence of data preprocessing on the robustness nd parsimony of multivariate calibration models. Chemom. Intell. Lab. Systems, 23 (1994) 65-70,... 

D. Jouan-Rimbaud, D.L. Massart, R. Leardi, et al.. Genetic algorithms as a tool for wavelength selection in multivariate calibration. Anal. Chem., 67 (1995) 4295 301. 

S. Sekulics, B.R. Kowalski, Z.Y. Wang, et al., Nonlinear multivariate calibration methods in analytical chemistry. Anal. Chem., 65 (1993) A835-A845. 

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