Classical Multivariate Calibration

The classical multivariate calibration represents the transition of common single component analysis from one dependent variable y (measured value) to m dependent variables (e.g., wavelengths or sensors) which can be simultaneously included in the calibration model. The classical linear calibration (Danzer and Currie [1998] Danzer et al. [2004]) is therefore represented by the generalized matrix relation 

Direct calibration can be applied when the calibration coefficients are known, otherwise - in case of indirect calibration - the calibration coefficients are computed by means of experimentally estimated spectra-concentrations relations. 

Classical calibration procedure can only be applied when all the species that contribute to the form of the spectra are known and can be included into the calibration. Additionally, there is the constraint that no interactions between the analytes and other species (e.g. solvent) or effects (e.g. of temperature) should occur. 

In case of baseline shift, the sensitivity matrix in Eqs. (6.74) and (6.75) must be complemented by a vector 1  

Instead of the addition of the 1-vector the calibration data may be centered (y — y and x, —3c, respectively). Even if the spectra of the pure species cannot be measured directly then the A-matrix can be estimated indirectly from the spectra provided that all components of the analytical system are known  

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

A comprehensive two-volume Handbook of Chemometrics and Qualimetrics has been published by D. L. Massart et al. (1997) and B. G. M. Vandeginste et al. (1998) predecessors of this work and historically interesting are Chemometrics A Textbook (Massart et al. 1988), Evaluation and Optimization of Laboratory Methods and Analytical Procedures (Massart et al. 1978), and The Interpretation of Analytical Chemical Data by the Use of Cluster Analysis (Massart and Kaufmann 1983). A classical reference is still Multivariate Calibration (Martens and Naes 1989). A dictionary with extensive explanations containing about 1700 entries is The Data Analysis Handbook (Frank and Todeschini 1994). 

Multivariate calibration tools are used to construct models for predicting some characteristic of future samples. Chapter 5 begins with a discussion of the reasons for choosing multivariate over univariate calibration methods. The most widely used multivariate calibration tools are then presented in two categories classical and inverse methods. 

For all the mentioned reasons, there is an ongoing tendency in spectroscopic studies to manipulate samples less and perform fewer experiments but to obtain more data in each of them and use more sophisticated mathematical techniques than simple univariate calibration. Hence multivariate calibration methods are being increasingly used in laboratories where instruments providing multivariate responses are of general use. Sometimes, these models may give less precise or less accurate results than those given by the traditional method of (univariate) analysis, but they are much quicker and cheaper than classical approaches. 

D. M. Haaland, W. B. Chambers, M. R. Keenan and D. K. Melgaard, Multi-window classical least-squares multivariate calibration methods for quantitative ICP-AES analyses, Appl. Spectrosc., 54(9), 2000, 1291— 1302. 

Most chemometricians prefer inverse methods, but most traditional analytical chemistry texts introduce the classical approach to calibration. It is important to recognise that there are substantial differences in terminology in the literature, the most common problem being the distinction between V and y variables. In many areas of analytical chemistry, concentration is denoted by V, the response (such as a spectroscopic peak height) by y However, most workers in the area of multivariate calibration have first been introduced to regression methods via spectroscopy or chromatography whereby the experimental data matrix is denoted as 6X , and the concentrations or predicted variables by y In this paper we indicate the experimentally observed responses by V such as spectroscopic absorbances of chromatographic peak areas, but do not use 6y in order to avoid confusion. 

Multivariate calibration is a very popular area, and the much reprinted classic by Martens and Nses [24] is possibly the most cited book in chemometrics. Much of the text is based around NIR spectroscopy which was one of the major success stories in applied chemometrics in the 1980s and 1990s, but the clear mathematical descriptions of algorithms are particularly useful for a wider audience. 

As a supplement to the more than 80 references given in the three parts an overwiew of books is presented here that are relevant to chemometrics [1] and its applications in spectroscopy. Two comprehensive standard books on chemometrics have been published by D. L. Massart et al. [2], and B. G. M. Vandeginste et al [3]. The predecessor of these books probably was the most used volume in chemometrics for many years [4]. Introductory and smaller books are from M.J. Adams (focus on analytical spectroscopy) [5], K. R. Beebe et al. (almost no mathematics) [6], R.G. Brereton [7], R. Kramer (focus on multivariate calibration) [8], and M. Otto [9]. The classical book on multivariate calibration in chemistry is from H. Mar-... 

Fluorimetry was considered in the 1950s as the natural detector for pharmaceuticals, due to its improved selectivity and sensitivity compared with UV-Vis absorption. Recent FIA applications include the determination of diazepam, nitrazepam, and oxazepam in pharmaceutical formulations using acidic hydrolysis and fluorimetric detection. Oxidation with Ce(IV) and measurement of the fluorescence from the released Ce(III), which can be considered as a classical strategy, is an appropriate technique for mixtures of amoxycillin and clavulanic acid where kinetic data are used in combination with partial least-squares multivariate calibration. 

Determination of a singie species in a mixture Simuitaneous kinetic-based determinations Classical differential kinetic methods Logarithmic-extrapolation method Proportional-equation method Multipoint methods Curve-fitting methods Kalman filter algorithm Artificial neural networks Multivariate calibration methods... 

There are two paradigms of multivariate calibration. Classical least squares (CLS) models the instrumental response as a function of analyte concentration. 

Chapter 4 retrieves the basic ideas of classical univariate calibration as the standpoint from which the natural and intuitive extension of multiple linear regression (MLR), arises. Unfortunately, this generalization is not suited to many laboratory tasks and, therefore, the problems associated with its use are explained in some detail. Such problems justify the use of other more advanced techniques. The explanation of what the multivariate space looks like and how principal components analysis can tackle it is the next step forward. This constitutes the root of the regression methodology presented in the following chapter. 

Typically, NIR spectra contain bands that arise from OH, CH, and NH groups in a sample. NIR bands of individual analytes are generally broader and heavily overlapped than bands are in MIR spectra. As a result, it is difficult to determine an analyte concentration from only one or a few NIR absorbances. Nonetheless, full-spectrum multivariate calibration methods, such as partial least squares (PLSs), classical least squares (CLS), and principal components regression (PCR) have been used to accurately determine analyte concentrations from NIR spectra. 

In the MCR framework, there are few cases in which the quantitative analysis is based on the acquisition of a single spectrum per sample, as is the case for classical first-order multivariate calibration methods, such as partial least squares (PLS), seen in other chapters of this book. There are some instances in which quantitation of compounds in a sample by MCR can be based on a single spectrum, that is, a row of the D matrix and the related row of the C matrix. Sometimes, this is feasible when the compounds to be determined provide a very high signal compared with the rest of the substances in the food sample, for example colouring additives in drinks determined by ultraviolet—visible (UV-vis) spectroscopy [26,27]. Recently, these examples have increased due to the incorporation of a new cmistraint in MCR, the so-caUed correlation constraint [27,46,47], which introduces an internal calihratimi step in the calculation of the elements of the concentradmi profiles in the matrix C related to the analytes to be quantified. This calibration step helps to obtain real concentration values and to separate in a more efficient way the information of the analytes to be quantified from that of the interferences. 

The inverse calibration regresses the analytical values (concentrations), x, on the measured values, y. Although with it a prerequisite of the GAussian least squares minimization is violated because the y-values are not error-free, it has been proved that predictions with inverse calibration are more precise than those with the classical calibration (Centner et al. [1998]). This holds true particularly for multivariate inverse calibration. 

While in classical statistics (univariate methods) modelling regards only quantitative problems (calibration), in multivariate analysis also qualitative models can be created in this case classification is performed. 

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