Data matrices alternating least squares

The results presented below were obtained by multivariate curve resolution-alternating least squares (MCR-ALS). MCR-ALS was selected because of its flexibility in the application of constraints and its ability to handle either one data matrix (two-way data sets) or several data matrices together (three-way data sets). MCR-ALS has been applied to the folding process monitored using only one spectroscopic technique and to a row-wise augmented matrix, obtained by appending spectroscopic measurements from several different techniques. 

Principal component analysis (PCA) and multivariate curve resolution-alternating least squares (MCR-ALS) were applied to the augmented columnwise data matrix D1"1", as shown in Figure 11.16. In both cases, a linear mixture model was assumed to explain the observed data variance using a reduced number of contamination sources. The bilinear data matrix decomposition used in both cases can be written by Equation 11.19 ... 

Llamas et al. developed other spectrophotometric methods for the determination of Amaranth, Sunset Yellow, and Tartrazine in beverages [32]. The spectra of the samples (simply filtered) were recorded between 359 and 600 nm, and mixtures of pure dyes, in concentrations between 0.01 and 1.8 mg/L for Amaranth, 0.08 and 4.4 mg/L for Sunset Yellow, and 0.04 and 1.8 mg/L for Tartrazine, were disposed in a column-wise augmented data matrix. This kind of data structure, analyzed by multivariate curve resolution-alternating least squares (MCR-ALS), makes it possible to exploit the so-called second-order advantage. The MCR-ALS algorithm was applied to the experimental data under the nonnegativity and equality constraints. As a result, the concentration of each dye in the sample and their corresponding pure spectra were obtained. 

MCR-ALS is an iterative algorithm that performs the decomposition ofthe data matrix D into the bilinear model D = CS by means of a constrained alternating least squares optimization ofthe matrices C and S. The sequence of application of the algorithm involves three main steps ... 

This equation can be combined with either Equation (4.27) or (4.28), and in each procedure alternative values can be computed for AH ,(j y and for A//uipala-An additional method makes use of all three equations simultaneously to take advantage of all data available. In such an overdetermined set of data, one can use the method of least squares (see Appendix A) for more than one independent variable and matrix methods to solve the resulting equations [9]. 

Thus, each row of matrix A, each element of vector y (see Eq. 5.32) and each column of the transpose matrix (see Eq. 5.37) is changed by the multiplier that is inversely proportional to the square root of the experimental error in the corresponding experimental data point. Alternatively, the weighted least squares solution may be expressed as follows... 

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. 

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