Alternating Least-Squares

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... 

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

Multivariate Curve Resolution Alternating Least Squares... 

An important group of methods relies on the inherent order of the data, typically time in kinetics or chromatography. These methods are often based on Evolving Factor Analysis and its derivatives. Another well known family of model-free methods is based on the Alternating Least-Squares algorithm that solely relies on restrictions such as positive spectra and concentrations. 

This algorithm has many aspects similar to Iterative Target Transform Factor Analysis, ITTFA, as discussed in Chapter 5.2.2, and Alternating Least-Squares, ALS as introduced later in Chapter 5.4. The main difference is the inclusion of the window information as provided by the EFA plots. 

The method of Alternating Least-Squares, ALS, is very simple and exactly for that reason it can be very powerful. ALS has found widespread applications and it is an important method in the collection of model-free analyses. In contrast to most other model-free analyses, ALS is not based on Factor Analysis. 

P.J. Gemperline and E. Cash, Advantages of soft versus hard constraints in self-modeling curve resolution problems. Alternating Least squares with penalty functions. Anal. Chem., 75, 4236 (2003). 

There are several different iterative algorithms that have been used for SMCR, including alternating least squares (ALS)63 and iterative target transformation factor analysis (ITTFA).64 For more detailed information, the reader is referred to these references. 

This alternating least squares processing is continued until some convergence criteria is met. We continue until the square root of the sum of all the residuals squared in the relation (Residuals = A - CK) changes by <0.0001. 

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 convergence criterion in the alternating least-squares optimization is based on the comparison of the fit obtained in two consecutive iterations. When the relative difference in fit is below a threshold value, the optimization is finished. Sometimes a maximum number of iterative cycles is used as the stop criterion. This method is very flexible and can be adapted to very diverse real examples, as shown in Section 11.7. 

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 ... 

Esteban, M., Anno, C., Dfaz-Cruz, J.M., Dfaz-Cruz, M.S., and Tauler, R., Multivariate curve resolution with alternating least squares optimization a soft-modeling approach to metal complexation studies by voltammetric techniques, Trends Anal. Chem., 19, 49-61, 2000. 

Van Benthem, M.H., Keenan, M.R., and Haaland, D.M., Application of equality constraints on variables during alternating least squares procedures, J. Chemom., 16, 613-622, 2002. 

Bezemer, E. and Rutan, S.C., Study of the hydrolysis of a sulfonylurea herbicide using liquid chromatography with diode array detection and mass spectrometry by three-way multivariate curve resolution-alternating least squares, Anal. Chem., 73, 4403 4409, 2001. 

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. 

Wang, J.H., Hopke, P.K., Hancewicz, T.M., and Zhang, S.L., Application of modified alternating least squares regression to spectroscopic image analysis, Anal. Chim. Acta, 476, 93-109, 2003. 

Alternating least squares (ALS) methods are both slower, due to their numeric intensity, and more flexible than eigenvalue-eigenvector problem-based methods for solving Equation 12.1a and Equation 12.1b. The basic PARAFAC model of Equation... 

PARAFAC refers both to the parallel factorization of the data set R by Equation 12.1a and Equation 12.lb and to an alternating least-squares algorithm for determining X, Y, and Z in the two equations. The ALS algorithm is known as PARAFAC, emanating from the work by Kroonenberg [31], and as CANDECOMP, for canonical decomposition, based on the work of Harshman [32], In either case, the two basic algorithms are practically identical. 

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 ... 

The alternating least-squares procedure in steps 4 and 5 involves the operations C = DS(S S) and = (C C) C D, respectively. The end of the iterative process takes place when the reproduction of the original image from the product of the resolved concentration profiles and spectra has enough quality and there is no significant variation among the results of consecutive cycles. The quality in the data reproduction can be estimated through the lack of fit, expressed as ... 

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