Multiple-wavelength Linear Regression

This approach is simply an extension of Beer s law to include multiple wavelengths. The need for several wavelengths is dictated by the inherent richness of the IR spectra - it is generally difficult or impossible to find a single absorption corresponding to a particular analyte that is not overlapped by the absorptions of at least one other constituent. The Beer s law relationship shown in Equation (1)  

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This section introduces three of the more common techniques multiple-wavelength linear regression (MLR),... 

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. 

The set of selected wavelengths (i.e. the experimental design) affects the variance-covariance matrix, and thus the precision of the results. For example, the set 22, 24 and 26 (Table 41.5) gives a less precise result than the set 22, 32 and 24 (Table 41.7). The best set of wavelengths can be derived in the same way as for multiple linear regression, i.e. the determinant of the dispersion matrix (h h) which contains the absorptivities, should be maximized. 

Fig. 1.1. Examples for standard curves resulting from multiple determinations of different amounts of BSA. Line with circies protocol according to Lowry et al. Soiid iine. nonlinear regression dotted iine linear regressions wavelength 720 nm. Line with squares BCA protein determination. Soiid iine nonlinear regression dotted iine linear regression wavelength 562 nm). Findings means an example for graphical evaluation...

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. 

When MLR is used to provide a predictive model that uses multiple analyzer signals (e.g. wavelengths) as inputs and one property of interest (e.g. constituent concentration) as output, it can be referred to as an inverse linear regression method. The word inverse arises from the spectroscopic application of MLR because the inverse MLR model represents an inverted form of Beer s Law, where concentration is expressed as a function of absorbance rather than absorbance as a function of concentration 1,38... 

NIR spectroscopy became much more useful when the principle of multiple-wavelength spectroscopy was combined with the deconvolution methods of factor and principal component analysis. In typical applications, partial least squares regression is used to model the relation between composition and the NIR spectra of an appropriately chosen series of calibration samples, and an optimal model is ultimately chosen by a procedure of cross-testing. The performance of the optimal model is then evaluated using the normal analytical performance parameters of accuracy, precision, and linearity. Since its inception, NIR spectroscopy has been viewed primarily as a technique of quantitative analysis and has found major use in the determination of water in many pharmaceutical materials. 

In a semiquantitative method, six wavelengths were used in a Mahalanobis distance calculation, and it was possible to distinguish the ETH extracts at concentrations below 0.05%. For quantitative analysis, multiple linear regression (MLR) was employed. The correlations obtained were r2 = 0.85 (ETH) and r2 = 0.86 (NOR). With low drug concentrations and a short range of values, the SECs were high. 

Some application software programs use methods known as MLR multiple linear regression) that permits the statistical treatment of a large number of data points in order to establish a calibration equation. These chemiometric methods take advantage of all the absorptions measured at different wavelengths, irrespective of their origin analyte to be measured, matrix or artefacts of the instrument. 

If the number of measurements (wavelengths, n) exceeds the number of components (p), then the above system will be over-dimensioned and resolvable by multiple linear regression (MLR). Such a system can be expressed in a matrix form as ... 

On the other hand, atomic emission spectra are inherently well suited for multivariate analysis due to the fact that the intensity data can be easily recorded at multiple wavelengths. The only prerequisite is that the cahbration set encompasses all likely constituents encountered in the real sample matrix. Calibration data are therefore acquired by a suitable experimental design. Not surprisingly, many of the present analytical schemes are based on multivariate calibration techniques such as multiple linear regression (MLR), principal components regression (PCR), and partial least squares regression (PLS), which have emerged as attractive alternatives. 

It is often possible to focus on a specific absorption band in the MIR and create a simple Beer s law equation for one or more species. Several wavelengths may be employed in a procedure called multiple linear regression, wherein a more complex version of Beer s law is developed. In very complex mixtures with severely overlapping absorption bands, PCA and... 

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