Chromatographic data matrix

Consider, as an example, an HPLC-UV/Vis chromatographic data matrix A, where two overlapped peaks elute. Let A be an n x m matrix (n rows of spectra recorded at m wavelengths) of mixture spectra. The data matrix A can be expressed as a product of k vectors representing digitized pure-component chromatograms and k vectors representing digitized pure-component spectra, as shown in Equation 4.1. 

A method of resolution that makes a very few a priori assumptions is based on principal components analysis. The various forms of this approach are based on the self-modeling curve resolution developed in 1971 (55). The method requites a data matrix comprised of spectroscopic scans obtained from a two-component system in which the concentrations of the components are varying over the sample set. Such a data matrix could be obtained, for example, from a chromatographic analysis where spectroscopic scans are obtained at several points in time as an overlapped peak elutes from the column. 

PAHs introduced in Section 34.1. A PCA applied on the transpose of this data matrix yields abstract chromatograms which are not the pure elution profiles. These PCs are not simple as they show several minima and/or maxima coinciding with the positions of the pure elution profiles (see Fig. 34.6). By a varimax rotation it is possible to transform these PCs into vectors with a larger simplicity (grouped variables and other variables near to zero). When the chromatographic resolution is fairly good, these simple vectors coincide with the pure factors, here the elution profiles of the species in the mixture (see Fig. 34.9). Several variants of the varimax rotation, which differ in the way the rotated vectors are normalized, have been reviewed by Forina et al. [2]. 

Matching to form a data matrix. Each value of the reference vector is compared with the retention time values for the fractions in every chromatographic run. Only a small variation in the values is tolerated thus no doublex matching should occur. 

Loadings Plot (Model and Sample Diag io tiL) The iouding.s can he used to help determine the optimal number of factors to consider for the model. For spectroscopic and chromatographic data, the point at which the loading displays random behavior can indicate the maximum number to consider. Numerical evaluation of the randomness of the loadings has been proposed as a method for determination of the rank of a data matrix for spectroscopic data... 

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. 

You undoubtedly recognize that we seldom start out knowing the pure spectra and chromatographic profiles of a data set like the one described in Figure 4.1. If we knew them, there would not be any point in applying the technique of PCA. Fortunately, with PCA it is possible to compute unique sets of basis vectors that span the significant space of a data matrix like A without prior knowledge. 

Principal component analysis is ideally suited for the analysis of bilinear data matrices produced by hyphenated chromatographic-spectroscopic techniques. The principle component models are easy to construct, even when large or complicated data sets are analyzed. The basis vectors so produced provide the fundamental starting point for subsequent computations. Additionally, PCA is well suited for determining the number of chromatographic and spectroscopically unique components in bilinear data matrices. For this task, it offers superior sensitivity because it makes use of all available data points in a data matrix. 

In Chapters 4 and 5, we discussed a number of mediods for multivariate data analysis, but the methods described did not take into account the sequential nature of the information. When performing PCA on a data matrix, die order of die rows and columns is irrelevant. Figure 6.1 represents three cross-sections through a data matrix. The first could correspond to a chromatographic peak, the others not. However, since PCA and most other classical methods for pattern recognition would not distinguish these sequences, clearly other approaches are useful. 

The values of the coefficients used are summarized in Table 1 and the results obtained from the calculations for the model considered using the program described above are presented in Table 10, showing good agreement with the experimental chromatographic data. It should be noted that the contribution of the inorganic silicate matrix amounts to about 30% of the value of the total differential heat at the same time, the 5% variations in the values of the alkane-silicate interaction coefficients and do not... 

Now suppose that the data are indeed organized in a matrix form. These can be, for instance, spectroscopic or chromatographic data, which contain a small number of objects and variables, a high number of objects but a small number of variables, a small number of objects but a high number of variables, or high numbers of both, objects and variables (see Fig. 1). 

Table 3. Chromatographic data from different separations on columns on basis of the same silica matrix comments, see text.

Table 5.18 Intra-day accuracy and precision data for the determination of four diarrhetic shellfish poisons using LC-MS and the method of standard additions. Reprinted from J. Chromatogr., A, 943, Matrix effect and correction by standard addition in quantitative liquid chromatographic-mass spectrometric analysis of diarrhetic shellfish poisoning toxins , Ito, S. and Tsukada, K., 39-46, Copyright (2002), with permission from Elsevier Science...

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