Mixture spectra

Woodruff and co-workers introduced the expert system PAIRS [67], a program that is able to analyze IR spectra in the same manner as a spectroscopist would. Chalmers and co-workers [68] used an approach for automated interpretation of Fourier Transform Raman spectra of complex polymers. Andreev and Argirov developed the expert system EXPIRS [69] for the interpretation of IR spectra. EXPIRS provides a hierarchical organization of the characteristic groups that are recognized by peak detection in discrete ames. Penchev et al. [70] recently introduced a computer system that performs searches in spectral libraries and systematic analysis of mixture spectra. It is able to classify IR spectra with the aid of linear discriminant analysis, artificial neural networks, and the method of fe-nearest neighbors. 

Spectra at p (=20) wavelengths. Because of the Lambert-Beer law, all measured spectra are linear combinations of the two pure spectra. Together they form a 15x20 data matrix. For example the UV-visible spectra of mixtures of two polycyclic aromatic hydrocarbons (PAH) given in Fig. 34.2 are linear combinations of the pure spectra shown in Fig. 34.3. These mixture spectra define a data matrix X, which can be written as the product of a 15x2 concentration matrix C with the 2x20 matrix of the pure spectra ... 

The rows of X are mixture spectra and the columns are chromatograms at the p = 20 wavelengths. Here, columns as well as rows are linear combinations of pure factors, in this example pure row factors, being the pure spectra, and pure column factors, being the pure elution profiles. 

Fig. 34.5. Score plot (PC score vs PC2 score) of the mixture spectra given in Fig. 34.2.

The boundaries, m and n, define two lines in the (v, Vj) plane the line x, = m%2 and the line Xj = m2- The intersection of these two lines with the line defined by the mixture spectra gives two points A and B which are the estimates of the pure spectra. The intervals (A-A ) and (B-BO define the solution bands between which the pure spectra are situated. 

It should be borne in mind that the two spectra given in Fig. 34.14 are estimates of the pure spectra, which exactly fulfil the constraints. Two other estimates of the pure spectra are the purest mixture spectra A and B. When plotted in the same figure (see Figs. 34.15 and 34.16) a good impression is obtained of the remaining... 

Pure variables are fully selective for one of the factors. This means that only one pure factor contributes to the values of that variable. When the pure variables or selective wavelengths for each factor are known then the pure spectra can be calculated in a straightforward memner from the mixture spectra by solving ... 

Determination of pure variables from mixture spectra... 

Fig. 34.39. Mixture spectra and the corresponding relative standard deviation spectrum.

Once the pure variables have been identified, the data set can be resolved into the pure spectra by solving eq. (34.11). For the mixture spectra in Table 34.4, this gives ... 

A basic assumption of OPA is that the purest spectra are mutually more dissimilar than the corresponding mixture spectra. Therefore, OPA uses a dissimilarity criterion to find the number of components and the corresponding purest spectra. Spectra are sequentially selected, taking into account their dissimilarity. The dissimilarity of spectrum i is defined as the determinant of a dispersion matrix Y,. In general, matrices Y, consist of one or more reference spectra, and the spectrum measured at the /th elution time. 

As mentioned previously, the complex emission spectrum F (l) of samples containing multiple fluorophores is assumed to be the linear sum of individual component spectra Ffl), F2(X), FfX), weighted by their abundance xu x2, x3. Let Fj(X) and F2(X) be the reference emission spectra of pure samples of fluorophore (e.g., Cerulean and Venus). The term reference emission spectra is used because these spectra describe the emission at excitation wavelength /. x of a defined concentration of fluorophore (e.g., 10 /rM) acquired using the same excitation light intensity as was used to acquire an emission spectra of an unknown sample mixture. Under these conditions, the shape and magnitude of the fluorophore mixture spectra will be ... 

Multivariate analysis of single point NIR spectra has become a mainstay for a wide range of pharmaceutical applications. Single point methods are generally based on a relatively small number of individually collected reference spectra, with the quahty of a method dependent on how well the reference spectra model the unknown sample. With a robust method, highly accurate concentration estimates of sample components can be extracted from unknown mixture spectra. 

In contrast w DCLS, the ptire spectra in the indirect approach are not measured direcfly, but are estimated from mixture spectra. One reason for using ICLS is that a is not possible to physically separate die components (e.g., when one cd the components of interest is a gas and future prediction samples are mixtures of the gas dissolved in a liquid). Indirect CLS is also used when the model assumptions do not hold if the pure component is run neat. By preparing mixtures, it is possible to dilute a strongly absorbing component so that the modd assumptions hold. 

To estimate the pure spectra using the ICLS approach, a series of mixture spectra are obtained based on an experimental design with known concentration values [Pg.114]

To estimate 3he pure component spectra, the mixture spectra and concentrations of the calibration samples are supplied to the computer and a regression is performedas described by Equation 5-16. 

CalibraJwn Measurement Residual Plot (Model Diagnostic) After the pure specta are estimated (S), they are used with the original C matrix to generate esti es of the mixture spectra (R CS). These are then used to calculate a caUbration residual matrix which contains the portion of the mixture spectra that are not fit by the estimated pures (Equation 5.18). 

The resKluals shown in Figure 5-36 are randomly distributed about zero, which is expected when the model is correctly specified. The magnitude of the residue is small relative to the size of the features found in the plot of mixture spectra and estimated pures. 

This is what the mixture spectrum is expected to look like if the CIS as-stimptions are obeyed and c is a reasonable estimate of the true concentrations. The residual is the difference between the observed and reconstructed mixture spectra. 

Reference spectra choice is critical when applying supervised pattern recognition methods. The first solution is to use pure compound spectra as references. The drawback is that mixture spectra in data cubes often differ from the reference spectra. Applying the model may therefore give wrong results. The second solution, suitable in a few studies, is to select image pixels where only one compound is present in order to obtain the calibration sets. 

One of the major problems in studying polymers quantitatively is the absence of model compounds for the purpose of calibration. A method of obtaining spectra of the components of a mixture spectra is based on obtaining the ratio of absorbances. This method was first used by Hirschfeld 851 for mixtures of components differing in relative concentration. This approach was later generalized but is limited to a rather... 

The determination of the spectra and the relative concentrations of each component in the mixture spectra are the results that make factor analysis worth the effort. This statement appears almost too good to be true but, with some restrictions to be noted later, it is possible. Let us begin with the covariance matrix... 

The classical approach to the analysis of mixtures by use of infrared spectroscopy consists in identifying specific, strong bands that belong to a suspected component, obtain a pure spectrum of the suspected component, and then remove those in the spectrum of the mixture that are due to the identified compound. The process is repeated for the remaining bands in the mixture spectra. Once the component spectra are known for a mixture, a series of calibration curves is produced. These curves relate concentration to absorbance, using Beer s law. The concentration of the components of the mixture are then obtained by interpolation. The advantage of Fourier-transform, infrared spectroscopy is that components of a mixture may be... 

Windig, W. and Meuzelaar, H.L.C., Numerical Extraction of Components from Mixture Spectra by Multivariate Data Analysis. In Meuzelaar, H.L.C. and Isenhour, T.L. (eds) Computer Enhanced Analytical Spectroscopy Plenum New York, 1987 pp. 81-82. 

There are a series of aids for the interpretation of NMR spectra of equilibrated redistribution mixtures. Spectra of systems involving equilibria in simple compounds, of course, are easier to correlate than those of families of compounds. 

In this section, lowercase boldface type denotes a column vector and uppercase boldface type a matrix, and the superscript T denotes matrix transpose.) Multiple mixture spectra with varying analyte concentrations can be written together in matrix form as... 

The idea behind PCA is that a spectral dataset of mixtures of the same components can be expressed as a linear combination of a small set of spectral representations. This is most easily understood if we consider a set of mixture spectra for three components. Assuming that there is no noise and that the mixture spectra are simply the sum of absorptivity spectra for the pure components, each of the mixture spectra can... 

Principal component analysis makes it possible to find a set of representations for mixture spectra in which noise and interactions are taken into account without knowing anything about the spectra of the pure components or their concentrations. The basic idea is to find a set of representations that can be linearly combined to reproduce the original mixture spectra. In PCA, Equation (4.3) is rewritten as... 

The loading and scores for PCA can be generated by singular value decomposition (SVD). Instead of expressing the matrix containing the mixture spectra, A, as a product of two matrices as in Equation (4.4), SVD expresses it as a product of three matrices... 

The singular values contained in are the square roots of the eigenvalues in 2. The scores in U are obtained by projecting the mixture spectra onto the loading... 

The relative absorbance values obtained by these self-modeling procedures are proportional to concentrations of the components in the mixtures and are used as the first estimates for concentrations. The method of alternating least squares14 is then applied to the data. In this method, the mixture spectra in the absorbance matrix, A, are written in terms of Beer s law as... 

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