Alternating Least-Squares, ALS

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. 

ALS should more correctly be called Alternating Linear Least-Squares as every step in the iterative cycle is a linear least-squares calculation followed by some correction of the results. The main advantage and strength of ALS is the ease with which any conceivable constraint can be implemented its main weakness is the inherent poor convergence. This is a property ALS shares with the very similar methods of Iterative Target Transform Factor Analysis, TTTFA and Iterative Refinement of the Concentration Profiles, discussed in Chapters 5.2.2 and 5.3.3. 

The diagram starts with initial guesses for the concentration profiles C. It is, of course, equally possible to start with initial guesses for the component spectra A and swapping the order of the linear regression/correction steps calculating first C and then A while the structure of the rest is the same. 

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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. 

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. 

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... 

A class of algorithms which is specialized for multilinear problems is known as alternating least-squares (ALS). Multilinear models are all conditionally linear in a function of each of the three or so independent variables for example, spectral intensity is linear in concentration if the other variables are fixed. Each step of an ALS algorithm fixes the vectors for all but one independent variable, then applies linear regression to select the vectors for the one variable to minimize the error sum of squares. The algorithm cycles among the sets of parameters to be estimated, updating each in turn. Most applications of multilinear models use ALS code. ... 

In PARAFAC analysis, the set of matrices A, B and C are usually obtained by iteratively solving alternating least-squares (ALS) problems min - A(C B) over A for fixed... 

Shinzawa, H., Iwahashi, M., Noda, I. Ozaki, Y. (2008b). A Convergence Criterion in Alternating Least Squares (ALS) by Global Phase Angle. Journal of Molecular Structure, Vol. 883-884, No. 30, pp. 73-78... 

Estimation of the parameters A, B, and C is usually carried out by the alternating least squares (ALS) algorithm. As an example for PARAFAC decompositions, we consider the evaluation of the folding states of a protein by means of mass spectrometry... 

Ivanisevic et al. (2009) applied pure curve resolution method (PCRM) and alternative least square (ALS) method to pXRD data of ASD for miscibility studies. The PCRM-based approach follows the analysis of the variance among the measured intensity points of pXRD patterns of ASD with varying drug loadings thus... 

Minimizing this difference, thus leaving the noise out, is a classical least squares problem that can be handled by different algorithms. One of the most popular ones in curve resolution is alternating least squares (ALS). The main benefit of ALS with respect to others is the simplicity of the involved substeps ... 

The SMCR method uses alternating least-squares (ALS) steps to find concentration profiles and pure spectra of the chemical compounds that fit the data as good as possible. The spectra are considered to be a sum of the contributions of each of the chemical compounds having signals in the wavenumber region under consideration. In Figure 48, only contributions of styrene, 1,3-butadiene, and polybutadiene are present and the spectra X that were used can be represented by... 

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

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)... 

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 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 ... 

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 ... 

FIGURE 9.6. Scheme of steps of multivariate curve resolution based on alternating least squares (MCR-ALS procedure). 

An implementation of Multivariate Curve Resolution Alternating Least Squares (MCR-ALS) can be downloaded from http //www.ub.es/gesq/mcr/ mcr.htm.7 MCR is also available in The Unscrambler (Camo Inc., Wood-bridge, New Jersey, USA). 

Nonnegativity is a lower bounded problem, where each parameter is bound to be above or equal to zero. Such a bounded problem can be efficiently solved with an active set algorithm [Gill et al. 1981], How this algorithm can be implemented in an alternating least squares algorithm, e.g., for fitting the PARAFAC model is explained. 

Garrido, M., Rius, F.X., Larrechi, M.S. (2008). Multivariate Curve Resolution-alternating Least Squares (MCR-ALS) Applied to Spectroscopic Data from Monitoring Chemical Reaction Processes, Anal. Bioanal. Chem., Vol.390, No.8,... 

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