Curve resolution

A number of the techniques discussed above utilize decomposition of the data matrix to reduce the complexity of the techniques. It is common to consider the 

Decomposition of a matrix [14] is accomplished through the use of a transformation matrix, T  

Multivariate curve resolution (MCR) [14] is the latest of the methods that seek to constrain the rotation of the matrices by forcing the C and S matrices to obey certain restrictions. One such restriction is that all absorbances in the spectra must be greater than or equal to zero. This restriction was also imposed in SAO-ITTFA [21]. In MCR it can be imposed that none of the concentrations will be less than zero. Other restrictions can result from chemical mass balance equations to determine composition in a dynamic system, or knowledge of some of the pure component spectra can be fed into the method algorithm to resolve other pure component spectra. One major advantage of MCR (and SAO-ITTFA) is to determine concentration profiles from kinetic systems as well as to determine the spectra of transient species. 

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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. 

An example of a spectrum with a chemical shift is that of the tin 3d peaks in Eig. 2.8. A thin layer of oxide on the metallic tin surface enables photoelectrons from both the underlying metal and the oxide to appear together. Resolution of the doublet 3 ds/2, 3 dii2 into the components from the metal (Sn ) and from the oxide Sn " is shown in Eig. 2.8 B. The shift in this instance is 1.6-1.7 eV. Curve resolution is an operation that can be performed routinely by data processing systems associated with photoelectron spectrometers. 

Buback et a A9 1 11,532 applied FTIR to follow the course of the initiation of S polymerization by AIBN and to determine initiator efficiency. Contributions to the IR signal due lo cyanoisopropyl end groups, AIBN, and the kelenimine can be separated using curve resolution techniques. 

In previous methods no pre-knowledge of the factors was used to estimate the pure factors. However, in many situations such pre-knowledge is available. For instance, all factors are non-negative and all rows of the data matrix are nonnegative linear combinations of the pure factors. These properties can be exploited to estimate the pure factors. One of the earliest approaches is curve resolution, developed by Lawton and Sylvestre [7], which was applied on two-component systems. Later on, several adaptations have been proposed to solve more complex systems [8-10]. 

In their fundamental paper on curve resolution of two-component systems, Lawton and Sylvestre [7] studied a data matrix of spectra recorded during the elution of two constituents. One can decide either to estimate the pure spectra (and derive from them the concentration profiles) or the pure elution profiles (and derive from them the spectra) by factor analysis. Curve resolution, as developed by Lawton and Sylvestre, is based on the evaluation of the scores in the PC-space. Because the scores of the spectra in the PC-space defined by the wavelengths have a clearer structure (e.g. a line or a curve) than the scores of the elution profiles in the PC-space defined by the elution times, curve resolution usually estimates pure spectra. Thereafter, the pure elution profiles are estimated from the estimated pure spectra. Because no information on the specific order of the spectra is used, curve resolution is also applicable when the sequence of the spectra is not in a specific order. 

In Section 34.2 we explained that factor analysis consists of a rotation of the principal components of the data matrix under certain constraints. When the objects in the data matrix are ordered, i.e. the compounds are present in certain row-windows, then the rotation matrix can be calculated in a straightforward way. For non-ordered spectra with three or less components, solution bands for the pure factors are obtained by curve resolution, which starts with looking for the purest spectra (i.e. rows) in the data matrix. In this section we discuss the VARDIA [27,28] technique which yields clusters of pure variables (columns), for a certain pure factor. 

W.H. Lawton and E A. Sylvestre, Self modeling curve resolution. Technometrics, 13 (1971) 617-633. 

B.G.M. Vandeginste, R. Essers, T, Bosman, J. Reijnen and G. Kateman, Three-component curve resolution in HPLC with multi wavelength diode array detection. Anal. Chem., 57 (1985) 971-985. 

B.G.M. Vandeginste, F. Leyten, M. Gerritsen, J.W. Noor, G. Kateman and J. Frank, Evaluation of curve resolution and iterative target transformation factor analysis in quantitative analysis by liquid chromatography. J. Chemom., 1 (1987) 57-71. 

B.G.M. Vandeginste, W.Derks andG. Kateman, Multicomponent self modelling curve resolution in high performance liquid chromatography by iterative target transformation analysis. Anal. Chim. Acta, 173 (1985) 253-264. 

R. Tauler, A.K. Smilde and B.R. Kowalski, Selectivity, local rank, three-way data analysis and ambiguity in multivariate curve resolution. J. Chemom., 9 (1995) 31-58. 

DOSY is a technique that may prove successful in the determination of additives in mixtures [279]. Using different field gradients it is possible to distinguish components in a mixture on the basis of their diffusion coefficients. Morris and Johnson [271] have developed diffusion-ordered 2D NMR experiments for the analysis of mixtures. PFG-NMR can thus be used to identify those components in a mixture that have similar (or overlapping) chemical shifts but different diffusional properties. Multivariate curve resolution (MCR) analysis of DOSY data allows generation of pure spectra of the individual components for identification. The pure spin-echo diffusion decays that are obtained for the individual components may be used to determine the diffusion coefficient/distribution [281]. Mixtures of molecules of very similar sizes can readily be analysed by DOSY. Diffusion-ordered spectroscopy [273,282], which does not require prior separation, is a viable competitor for techniques such as HPLC-NMR that are based on chemical separation. 

Tauler R., Smilde A.K., Hemshaw J.M., Burgess L.W., Kowalski B.R., Multicomponent Determination of Chlorinated Hydrocarbons Using a Reaction-based Chemical Sensor. Part 2. Chemical Speciation Using Multivariate Curve Resolution, Anal. Chem. 1994 66 3337-3344. 

Terrado M, Barcelo D, Tauler R (2009) Quality assessment of the multivariate curve resolution alternating least squares method for the investigation of environmental pollution patterns in surface water. Environ Sci Technol 43 5321-5326... 

Terrado M, Barcelo D, Tauler R (2010) Multivariate curve resolution of organic pollution patterns in the Ebro River surface water-groundwater-sediment-soil system. Anal Chim Acta 657 19-27... 

Keywords Chemometrics, Contamination sources, Ebro River, Multivariate curve resolution, Principal component analysis... 

MCR-ALS Multivariate curve resolution alternating least squares... 

Multivariate Curve Resolution Alternating Least Squares... 

Multivariate curve resolution methods (MCR [17]) describe a family of chemometric procedures used to identify and solve the contributions existing in a data set. These procedures have been traditionally applied for the resolution of multiple chemical components in mixtures investigated by spectroscopic analysis techniques [18]. 

Tauler R, Maeder M, De Juan A (2009) Multiset data analysis extended multivariate curve resolution. In Brown S, Tauler R, Walczak R (eds) Comprehensive chemometrics, vol 2. Elsevier, Oxford, pp 473-505... 

Multivariate curve resolution, 6 54—56 Multivariate linear regression, 6 32—35 Multivariate optical elements (MOE), 6 68 Multiwalled carbon nanotubes (MWCNTs), 77 48, 49 22 720 26 737. See also Carbon nanotubes (CNTs) Multiwall nanotubes (MWNTs) synthesis of, 26 806 Multiwall fullerenes, 12 231 Multiwall nanotubes (MWNTs), 12 232 Multiwall paper bags, 78 11 Multiway analysis, 6 57-63 Multiyear profitability analysis, 9 535-537 Multiyear venture analysis, 0 537-544 sample, 9 542-S44 Mummification, 5 749 Mumps vaccine, 25 490 491 Mumps virus, 3 137 Municipal biosolids, as biomass, 3 684 Municipal distribution, potential for saline water use in, 26 55-56 Municipal effluents, disposal of, 26 54 Municipal landfill leachate, chemicals found in, 25 876t... 

Literature examples include one-, two-, and three-step hydrogenation reactions, hi each case, the reactant and product concentrations were monitored, as were intermediates or side reaction products as necessary. Simple peak height or area measurements were sufficient to generate reaction profiles for the reduction of cyclohexene [116] and of l-chloro-2-nitrobenzene [117]. However, for more spectrally complex systems, such as the reduction of carvone and of 2-(4-hydroxyphenyl) propionate, multivariate curve resolution (MCR) was required [117]. 

Earlier, it was mentioned that due to the orthogonality constraints of scores and loadings, as well as the variance-based criteria for their determination, it is rare that PCs and LVs obtained from a PC A or PLS model correspond to pure chemical or physical phenomena. However, if one can impose specihc constraints on the properties of the scores and or loadings, they can be rotated to a more physically meaningful form. The multivariate curve resolution (MCR) method attempts to do this for spectral data. 

Figure 12.23 Example of multivariate curve resolution (MCR). (A) A series of 210 FTIR. spectra obtained during the course of a chemical reaction (B) the pure spectral profiles obtained using MCR. and (C) the corresponding pure concentration profiles obtained using MCR.

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