Resolution methods, multivariate curve

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... 

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]. 

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. 

The field of curve resolution was bom in response to the need for a tool to analyze multivariate experimental data from multicomponent dynamic systems. The common goal of all curve-resolution methods is to mathematically decompose the global instrumental response into the pure-component profiles of each of the components in the system. The use of these methods has become a valuable aid for resolving complex systems, especially when obtaining selective signals for individual species is not experimentally possible, too complex, or too time consuming. 

Different approaches have been proposed during recent years to improve the solutions obtained by curve-resolution methods, and some of them are summarized in the next sections. The field is already mature and, as it has been recently pointed out [26], multivariate curve resolution can be considered as a sleeping giant of chemometrics, with a slow but persistent growth. 

Whenever the goals of curve resolution are achieved, the understanding of a chemical system is dramatically increased and facilitated, avoiding the use of enhanced and much more costly experimental techniques. Through multivariate-resolution methods, the ubiquitous mixture analysis problem in chemistry (and other scientific fields) is solved directly by mathematical and software tools instead of using costly analytical chemistry and instrumental tools, for example, as in sophisticated hyphenated mass spectrometry-chromatographic methods. 

The next subsection deals first with aspects common to all resolution methods. These include (1) issues related to the initial estimates, i.e., how to obtain the profiles used as the starting point in the iterative optimization, and (2) issues related to the use of mathematical and chemical information available about the data set in the form of so-called constraints. The last part of this section describes two of the most widely used iterative methods iterative target transformation factor analysis (ITTFA) and multivariate curve resolution-alternating least squares (MCR-ALS). 

Multivariate curve resolution-alternating least squares (MCR-ALS) uses an alternative approach to iteratively find the matrices of concentration profiles and instrumental responses. In this method, neither the C nor the ST matrix have priority over each other, and both are optimized at each iterative cycle [7, 21, 42], The general operating procedure of MCR-ALS includes the following steps ... 

The examples that are given in the following subsections show the power of multivariate curve resolution to resolve very diverse chemical problems. Different strategies adapted to the chemical and mathematical features of the data sets are chosen, and resolution of two-way or three-way data sets is carried out according to the information that has to be recovered. Because MCR-ALS has proved to be a very versatile resolution method, able to deal with two-way and three-way data sets, this is the method used in all of the following examples. 

In summary, using either principal component analysis or multivariate curve-resolution methods, the main contamination sources of semivolatile organic compounds present in the surface waters of Portugal were identified, and their geographical... 

Jaumot, J., Gargallo, R., and Tauler, R., Noise propagation and error estimations in multivariate curve-resolution alternating least squares using resampling methods, J. Chemom., 18, 324-340, 2004. 

The methods in Section 3.3.1 are concerned primarily with removing noise. Most methods leave peakwidths either unchanged or increased, equivalent to blurring. In signal analysis an important separate need is to increase resolution. In Section 3.5.2 we will discuss the use of filters combined with Fourier transformation. In Chapter 6 we will discuss how to improve resolution when there is an extra dimension to the data (multivariate curve resolution). However, a simple and frequently used approach is to calculate derivatives. The principle is that inflection points in close peaks become turning points in the derivatives. The first and second derivatives of a pure Gaussian are presented in Figure 3.10. 

Multivariate curve resolution-alternating least squares (MCR-ALS) is an algorithm that fits the requirements for image resolution [71, 73-75]. MCR-ALS is an iterative method that performs the decomposition into the bilinear model D = CS by means of an alternating least squares optimization of the matrices C and according to the following steps ... 

Depending on the quality of data and the method selected, constraints on the parameters to be estimated may be required in order to get a chemically meaningful solution. In the case of multivariate curve resolution (MCR) (see Section 3.2) performed on one 2D NMR spectrum, application of constraints is mandatory. If constraints are not applied, it can be shown that there is an infinity of equally well-fitting solutions and hence the true underlying parameters (spectra, concentrations) cannot be estimated directly. This is known as the rotational ambiguity of two-way low-rank models. 

Saurina J, Hernandez-Cassou S, Tauler R, Multivariate curve resolution and trilinear decomposition methods in the analysis of stopped-flow kinetic data for binary amino acid mixtures, Analytical Chemistry, 1997, 69, 2329-2336. 

The factorial methods in this chapter are also called second-order transformations, because only two moments, mean and covariance, are needed to describe the Gaussian distribution of the variables. Other second-order transformations are FA, independent component analysis (ICA), and multivariate curve resolution (MCR). 

Mas, S., Fonrodona, G., Tauler, R., and Barbosa, J., Determinahon of phenohc acids in strawberry samples by means of fast liquid chromatography and multivariate curve resolution methods. Talanta, 71,... 

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