Measuring and comparing the resolution performance and the extent of rotation ambiguities of some bilinear modeling methods

doi:10.1016/j.chemolab.2015.08.005

Chemometrics and Intelligent Laboratory Systems

Volume 147, 15 October 2015, Pages 47-57

https://doi.org/10.1016/j.chemolab.2015.08.005 Get rights and content

Highlights

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Results from different bilinear resolution methods are compared.
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Rotation ambiguities are ubiquitously present and they should be considered.
•
Extent of rotation ambiguities and AFS are calculated for different datasets.
•
The results estimated by the MCR-BANDS and FAC-PACK are in agreement

Abstract

Bilinear models are often used in the analysis of datasets from spectroscopy and chromatography. Whenever bilinear soft modeling approaches are applied, rotation ambiguities are ubiquitously present and they should be considered. In this work, results obtained by the application of different methods like independent component analysis (ICA), principal component analysis (PCA), and minimum volume simplex analysis (MVSA) are compared with those obtained by multivariate curve resolution (MCR). In order to do this comparison, mutual information (MI), Amari index (AI), and lack-of-fit (lof) parameters are used for the evaluation of the different methods, and the corresponding areas or regions of feasible solutions (AFS) and their boundaries are investigated in each case. The results obtained by the MCR-BANDS method in the calculation of the extension of rotation ambiguities are discussed and compared with those obtained by the FAC-PACK method, which has been recently proposed for the estimation of the whole range of feasible solutions.

Introduction

Chemometric methods provide powerful tools to analyze multi- and megavariate data from modern analytical instruments. Some of these chemometric methods, in particular multivariate curve resolution (MCR) methods, have been proposed for the resolution of chemical data obtained from chromatography [1], spectroscopy [2], nuclear magnetic resonance [3], hyperspectral imaging [4], voltammetry [5], omics microarray [6], and LC-MS [7]data, among others [8], [9]. MCR methods are a group of methods based on the fulfillment of a bilinear model which attempt the extraction of the true underlying sources of chemical variation using a minimum amount of prior assumptions about the process under investigation. For the analysis of complex multi-component mixture systems, they offer the possibility of resolution, identification, and also quantification [10] of the different components present in an unknown mixture, without needing their previous chemical and physical separation.

MCR chemometric methods have their intrinsic drawbacks, especially that they cannot assure encountering a unique solution to explain the measured experimental variation in the data and that a range of feasible solutions may be obtained by their application. Ambiguities appear because different linear combinations of the component profiles fulfilling the constraints of the system fit equally well the data [11]. Unfortunately, the presence of rotation ambiguities and of non-unique solutions decreases the reliability of MCR methods and makes their assessment more difficult. The only way to reduce the extent of rotation ambiguities and to obtain solutions closer to true ones is by the application of additional constraints (soft or hard) which implies using more knowledge about the data system, or also moving from bilinear modeling to multilinear modeling [12].

Bilinear modeling methods like minimum volume simplex analysis (MVSA), independent component analysis (ICA), principal component analysis (PCA), and multivariate curve resolution-function minimization (MCR-FMIN) have already been compared in previously published papers [13], [14]. MVSA initially was developed for satellite imaging individual component (end member) resolution, and more recently, it has been also proposed in analytical chemistry [15]. PCA considers the information between the different components to be orthogonal or linearly uncorrelated [16]. ICA assumes that the components are mutually statistically independent [17]. These assumptions are statistically different (the latter is more restricted than the first), and therefore, the results are different. In particular, ICA and PCA can be used for different purposes like data preprocessing, exploration, classification, regression, and resolution. All these methods have been proposed for analytical chemistry purposes, and some authors have investigated whether one method is better than the other. Different from these approaches, based on statistical assumptions, multivariate curve resolution methods, especially those based in alternating least squares (MCR-ALS), use more natural and physically and chemically meaningful assumptions by means of constraints, like non-negativity, unimodality, closure, selectivity, or local rank, and by means of other constraints related to the data structure (like trilinearity or multilinearity) and find an optimum solution from a least squares fitting convergence criterium [18]. MCR-FMIN has been also proposed as a different way for multivariate curve resolution and it is based on non-linear optimization algorithms using non-linear constraints [19]. MCR-FMIN uses PCA scores and loadings to define the subspace of MCR solutions and rotates them to fulfill the constraints of the system. Therefore, it is also interesting to compare their solutions with those obtained by PCA and MCR-ALS.

In order to evaluate the effect of rotation ambiguities associated to a particular MCR solution and to measure its extent, Lawton and Sylvestre [20] already proposed a first algorithm for determining the area of feasible solutions (AFS) in two-component systems under the assumption of non-negative spectra and concentration profiles. Borgen et al. [21] extended Lawton and Sylvestre's method to three component systems and proposed a linear programming optimization method to calculate the permitted ranges of pure component spectra using tangent and simplex rotation algorithms. Rajkó and István [22] revised Borgen's study and used computational geometry tools to draw Borgen plots of three-component systems. Leger and Wentzell developed a dynamic Monte Carlo SMCR method [23] which seeks to define the boundaries of allowable pure component profiles. For the calculation of the whole range of feasible solutions, a systematic grid search method based on species-based particle swarm optimization has been proposed for three-component systems by H. Abdollahi et al [24], [25]. A. Golshan et al. have also developed a method that finds the simplex volume containing all feasible solutions and facilitate the determination and visualization of rotational ambiguities of four-component mixture [25].

R. Tauler developed the MCR-BANDS method [26] based on a previous idea of P. Gemperline [27], for the calculation of the extension of rotation ambiguities, based on the fast maximization and minimization of a function defined by the relative signal component contribution (SCCF) of each component [11], [23]. This method has no limitation for the number of components and it uses the same constraints as those applied to find out the MCR solution. It gives a simple evaluation of the extent of rotation ambiguity from the difference between the maximum and minimum values of the SCCF function. Recently, Sawall et al. suggested a fast accurate algorithm to find the AFS for two- and three-component systems based on the use of a polygon inflation algorithm [28]. FAC-PACK is an interactive MATLAB toolbox for the computation of non-negative multi-component factorizations and for the numerical approximation of the area of feasible solutions using the inflation polygon algorithm [28].

In this work, FAC-PACK results are compared to those obtained by MCR-BANDS, and with the solutions obtained by different bilinear model methods such as PCA, ICA, MVSA, MCR-FMIN, and MCR-ALS. The aim of this work is to get a deeper understanding of MCR methods and evaluate their performance under different constraints. In addition, the extension of rotation ambiguities associated to MCR solutions is investigated by the MCR-BANDS and FAC-PACK methods. The comparison of results obtained by these two methods can help to evaluate the reliability of their results and to get a deeper understanding of their principles.

Section snippets

Theory

Second-order data (a data matrix) generally can be decomposed by bilinear model-based methods according to Eq. (1). $D = C S^{T} + E = D^{*} + E$ where D (I,J) is the experimental data matrix corresponding to a bilinear system with I different samples and J different variables., C (I,N) is the contribution of the N components in each sample, S (J,N) is the pure response matrix of the N components, E (I,J)is the matrix associated to noise or experimental error. Given the data matrix D, the aim of bilinear model is

Principal component analysis (PCA)

PCA provides a mathematical and very efficient way to solve the bilinear model and perform the matrix decomposition given in Eq. (1). PCA decomposes the measurement matrix D into the scores C_PCA ܂and loadings S_PCA orthogonal factor matrices, and a reduced number of components are selected which explain maximum data variance. The aim of the method is to maximize the explained variance in the data with a minimum number of components. Due to the applied constraints during the PCA bilinear

Independent component analysis (ICA)

The aim of ICA is the decomposition of the measured multivariate signals into statistically independent component contributions with a minimum loss of information. ICA assumes that the mixing vectors in C are linearly independent and that the components in S are mutually statistically independent, as well as independent of noise components. This goal is equivalent to finding an unmixing matrix W that satisfies $W X = \hat{S^{T}}$ where Ŝ is the estimation of the S. The main task of ICA is to find out the

Minimum volume simplex analysis (MVSA) method

MVSA also considers that the underlying mixing model is bilinear, i.e. that the measured spectral vectors are a linear combination of signatures (spectra) of pure components. MVSA is a method that finds the pure components (end members) by fitting the data to a minimum volume simplex, under some constraints, such as having for every pixel no less than zero abundance fractions (non-negativity constraint) and that their sum should be equal to one (closure). The MVSA method starts with an estimate

Multivariate curve resolution-alternating least squares (MCR-ALS)

MCR-ALS solves Eq. (1) iteratively using an ALS algorithm, which optimally fits the experimental data matrix D, and resolves the ‘true’ pure response profiles, in concentration C and pure spectra S^T matrices. This optimization is carried out for a proposed number of components using initial estimates of either C or S^T. These initial estimates of C or S^T can be extracted using procedures for purest variables selection, such as SIMPLISMA [32]. During the ALS optimization, several constraints can

Multivariate curve resolution-objective function minimization (MCR-FMIN)

MCR-FMIN is based on the minimization of an objective function defined directly from the non-fulfillment of constraints and being always in the subspace spanned by PCA solutions. In other words, an appropriate rotation of the PCA solutions is performed searching for physically meaningful solutions (like non-negative). The function can be defined as $f (T) = c_{norm} (T) + c_{n o n - n e g} (T)$ where f(T) is the objective scalar function to be minimized and c_norm(T), and c_{non − neg}(T) are scalar functions for the

MCR-BANDS: Calculation of the extent of rotation ambiguities

In order to evaluate the extent of rotation ambiguities associated to a particular MCR solution under a set of constraints, the MCR-BANDS procedure has been proposed [11]. In this paper, it is applied to evaluate the extent of rotation ambiguities associated to MCR-ALS solutions, and as extension, also to those obtained by MCR-FMIN, PCA, ICA, and MVSA methods previously explained. In addition, MCR-BAND results are compared to those obtained by the FAC-PACK method (see below). MCR-BANDS method

FAC-PACK: Evaluation of the area of feasible solutions (AFS)

The area of feasible solutions (AFS) is a subset of the two-dimensional plane consisting of all pairs of solutions which represent non-negative spectra (case of the spectral AFS) or non-negative concentration profiles (case of the concentration AFS). In FAC-PACK toolbox for MATLAB, AFS is computed using a polygon inflation algorithm and its more recent implementation, the inverse polygon inflation algorithm. The polygon inflation method approximates the border of each AFS segment by a sequence

Mutual information (MI) evaluation

To estimate the degree of independence between component profiles, mutual information (MI) values are used. MI values are defined as proposed in previous works as a natural measure of the mutual independence between two variables [36]. MI between two variables can be expressed using the joint probability density function p(x₁, x₂) and the marginal probability density function p(x₁) and p(x₂). $I (x_{1}, x_{2}) = {d x}_{1} {d x}_{2} p (x_{1}, x_{2}) log [\frac{p (x_{1}, x_{2})}{p (x_{1}) p (x_{2})}]$

Krasakov et al. have proposed an efficient method of estimating MI values.

Concentration profiles recovery evaluation using Amari index

To estimate the reliability of the results obtained by the different methods applied here, concentration profiles were compared to the correct ones (if available) using the Amari index [13]. It shows the reliability between the compared concentration profiles, and it is defined as follows: $P = \frac{1}{2 N} \sum_{I, J = 1}^{N} (\frac{|p_{i j}|}{{max}_{k} |p_{i k}|} + \frac{|p_{i j}|}{{max}_{k} |p_{k j}|}) - 1$ where p_ij = (Ĉ⁺C)_i,j, C are the true concentration profiles and Ĉ⁺ is the pseudoinverse of calculated ones using the considered method. The Amari index is equal to zero when

Data fitting: Calculation of lack-of-fit values

Lack-of-fit values are defined form the difference between input data D values and their reproduced values obtained using a particular method. To evaluate the quality of the data fitting finally achieved after application of the different bilinear method, the percentages of lack of fit (lof) are calculated according to the following equation: $l o f (%) = 100 \times \sqrt{\frac{\sum_{i j} {(d_{i j} - {\hat{d}}_{i j})}^{2}}{\sum_{i j} d_{i j}^{2}}}$ where d_ij are the elements of the data matrix D, and ${\hat{d}}_{i j}$ are the corresponding elements recalculated by the considered method,

Datasets

In this work, similar synthetic datasets to those described in a previous work [13] were used. Both reaction and chromatographic type of profiles of two, three, and four components were used for the purposes of this work. The different sets of concentration and spectra profiles used as examples are given in Fig. 1.

Pure spectra were normalized to unit area and relative concentration profiles were always scaled between 0 and 1 in all the cases (white Gaussian noise with zero-mean and constant

Software

All calculations were performed in MATLAB R2013a (Mathworks Inc., Natick, MA, USA) for windows.

MF-ICA [29] methods were obtained from the ICA MATLAB toolbox V3 (ICA Toolbox Homepage (http://isp.imm.dtu.dk/toolbox/ica). MVSA MATLAB codes were downloaded from http://www.lx.it.pt/~bioucas/code.htm. MCR-ALS and MCR-BANDS algorithm code and GUI for MATLAB are freely available from the home page of MCR at http://www.mcrals.info/. FAC-PACK GUI for MATLAB is freely available from webpage //www.math.uni-rostock.de/FAC-PACK/

Results and discussion

Table 1 shows MI values of all spectra profiles, as well as the Amari index (AI) for all concentration profiles and the lack-of-fit values obtained for the different tested methods (MCR-ALS, ICA, PCA, MCR-FMIN, and MVSA) in the analysis of all datasets previously described. Data values in bold font in Table 1 correspond to lowest lack of fit, mutual information, and Amari index, respectively. Fig. 2(a–c) gives the graphical representation of lack-of-fit, MI, and Amari index values in Table 1.

Evaluation of the extension of rotation ambiguities in MCR and other methods

MCR solutions have in general a certain degree of ambiguity, and this can be evaluated by methods like MCR-BANDS. MCR-BANDS is based on the optimization of the function SCCF defined by the relative signal contribution of every component in relation to all the other components in the mixture (see Eq. (2)). In this work, relative signal contributions of all the components measured by SCCF using the different investigated methods such as MCR-ALS, MVSA, PCA, ICA, and MCR-FMIN are evaluated and

Conclusions

When multivariate curve resolution methods are applied to two-way data and local rank/selectivity conditions are not present in the data, there is no way in general to ascertain whether the solutions provided by these methods are correct (equal to the true expected one). Instead of a unique solution, a range of feasible solutions is encountered, all of them equally good from a data fitting point of view and fulfilling the required constraints. Different curve resolution methods may give

Conflict of interest

There is no conflict of interest.

Acknowledgements

Authors acknowledge research grant CTQ2012-3816 from Ministerio de Economía y competividad, Spain. Xin Zhang thanks the grant from China Scholarship Committee (2011637007).

References (38)

V. Boeris et al.
Determination of five pesticides in juice, fruit and vegetable samples by means of liquid chromatography combined with multivariate curve resolution
Anal. Chim. Acta
(2014)
M. Esteban et al.
Multivariate curve resolution with alternating least squares optimisation: a soft-modelling approach to metal complexation studies by voltammetric techniques, TrAC
Trends Anal. Chem.
(2000)
C. Ruckebusch et al.
Multivariate curve resolution: a review of advanced and tailored applications and challenges
Anal. Chim. Acta
(2013)
X. Zhang et al.
Distribution of dissolved organic matter in freshwaters using excitation emission fluorescence and multivariate curve resolution
Chemosphere
(2014)
T. Azzouz et al.
Application of multivariate curve resolution alternating least squares (MCR-ALS) to the quantitative analysis of pharmaceutical and agricultural samples
Talanta
(2008)
J. Jaumot et al.
MCR-BANDS: a user friendly MATLAB program for the evaluation of rotation ambiguities in Multivariate Curve Resolution
Chemom. Intell. Lab. Syst.
(2010)
A. Malik et al.
Performance and validation of MCR-ALS with quadrilinear constraint in the analysis of noisy datasets
Chemom. Intell. Lab. Syst.
(2014)
H. Parastar et al.
Is independent component analysis appropriate for multivariate resolution in analytical chemistry?
TrAC Trends Anal. Chem.
(2012)
H. Abdollahi et al.
Uniqueness and rotation ambiguities in multivariate curve resolution methods
Chemom. Intell. Lab. Syst.
(2011)
S. Wold et al.
Principal component analysis
Chemom. Intell. Lab. Syst.
(1987)

J. Jaumot et al.

A graphical user-friendly interface for MCR-ALS: a new tool for multivariate curve resolution in MATLAB

Chemom. Intell. Lab. Syst.

(2005)

R. Tauler

Application of non-linear optimization methods to the estimation of multivariate curve resolution solutions and of their feasible band boundaries in the investigation of two chemical and environmental simulated data sets

Anal. Chim. Acta

(2007)

O.S. Borgen et al.

An extension of the multivariate component-resolution method to three components

Anal. Chim. Acta

(1985)

M.N. Leger et al.

Dynamic Monte Carlo self-modeling curve resolution method for multicomponent mixtures

Chemom. Intell. Lab. Syst.

(2002)

A. Golshan et al.

Determination and visualization of rotational ambiguity in four-component systems

Anal. Chim. Acta

(2013)

A. Barducci et al.

Aerospace wetland monitoring by hyperspectral imaging sensors: a case study in the coastal zone of San Rossore Natural Park

J. Environ. Manag.

(2009)

R. Tauler

Multivariate curve resolution applied to second order data

Chemom. Intell. Lab. Syst.

(1995)

A. Hyvärinen et al.

Independent component analysis: algorithms and applications

Neural Netw.

(2000)

S. Beyramysoltan et al.

Investigation of the equality constraint effect on the reduction of the rotational ambiguity in three-component system using a novel grid search method

Anal. Chim. Acta

(2013)

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MCR decomposition methods are very suitable for solving the linear unmixing problems in chemistry, however, they are confronted with rotational ambiguities [14] which means that, in practice, a range of feasible solutions is obtained instead of a unique solution, and this range of solutions cannot be estimated in the general case. However, for problems of limited complexity, different methods exist to estimate the extent of rotational ambiguities [15,16] in MCR such as Lawton-Sylvester method [17] and Borgen method [18] to name a few. Nevertheless, trying to resolve more independent component profiles is not an optimal approach for the resolution of “component profiles”, since non-negative chemical profiles do not, in general, conform to the stipulated condition of independence in ICA methods.
In the present contribution, a new approach based on mutual information (MI) is proposed for exploring the independence of feasible solutions in two component systems. Investigating how independent are different feasible solutions can be a way to bridge the gap between independent component analysis (ICA) and multivariate curve resolution (MCR) approaches and, to the best of our knowledge, has not been investigated before. For this purpose, different chromatographic and hyperspectral imaging (HSI) datasets were simulated, considering different noise levels and different degrees of overlap for two-component systems. Feasible solutions were then calculated by both grid search (GS) and Lawton-Sylvester (LS) plots. MI map which is the plot of MI vs. rotation matrix elements was used to estimate the degree of independence between different solutions. Inspection of the results showed that the different solutions in the feasible bands correspond to different MI values and that those values are lower for spectral profiles (more independent) than for concentration profiles (more dependent) as expected from the duality concept and the opposite is true. In addition, component profiles are found near more dependent solutions for concentration profiles and near less dependent solutions for spectral profiles which is due to the fact that “independence” constraint was applied to the spectral profiles in ICA algorithms. The performance of three well-known ICA algorithms (mean-field independent component analysis (MF-ICA), mutual information-based least dependent component analysis (MILCA) and joint approximate diagonalization of eigenmatrices (JADE)) as well as MCR-alternating least squares (MCR-ALS) were investigated. MI maps showed that the solutions of MF-ICA and MCR-ALS are in the feasible bands but the MILCA and JADE solutions which are just based on the independence maximization are outside the MI maps.
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The boundaries of the set of feasible bilinear solutions for three-component systems have been estimated in the presence of noise, as a sequel of a previous report on two-component data sets. Simulations involving a significant degree of overlap among component profiles in both matrix modes have been analyzed using FACPACK and the recently described sensor-wise N-BANDS model. The latter computes the upper and lower envelopes of the set of profiles showing maximum and minimum intensity at each sensor. For three and more components, the graphical representation of the extreme profiles corresponding to maximum and minimum signal contribution function provided by methods such as MCR-BANDS and N-BANDS do not enclose, in general, the full set of feasible profiles. The present results were obtained under non-negativity as the only constraint on the profiles, and were successfully compared with those obtained using a noise addition procedure. A three-component experimental system was also analyzed regarding the combined effect of rotational ambiguity and instrumental noise.
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In fact, for two-component systems it has been proved that the extreme profiles obtained from both N-BANDS and MCR-BANDS can be compared with those obtained by calculating the AFS. For more than two components, the extreme profiles obtained by N-BANDS and MCR-BANDS cannot geometrically enclose the full set of feasible profiles, and thus the comparison with AFS estimations cannot be directly performed in the same way [10–12]. MCR-BANDS and N-BANDS are based on computing the maximum and minimum values of a signal contribution function (see below), whereas the AFS describes the whole space of feasible solutions.
Three models are compared regarding the estimation of the boundaries of the range of feasible bilinear solutions brought about by rotational ambiguity in the presence of noise. A two-component system having significant overlapping between the component profiles in both data modes has been studied using FACPACK, based on the so-called polygon inflation algorithm, and two procedures which estimate the extreme profiles corresponding to maximum and minimum signal contribution function: MCR-BANDS and N-BANDS. These latter two models operate by applying different constraints during the optimization phase. Non-negativity was applied as the only constraint on the retrieved profiles. The results were compared with a simulation based on the addition of identically and independently distributed noise at two levels. An experimental system was also analyzed regarding the combined effect of rotational ambiguity and instrumental noise.
A down-to-earth analyst view of rotational ambiguity in second-order calibration with multivariate curve resolution − a tutorial
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The newly developed N-BANDS uses the area-related parameter α for maximization/minimization in the presence of instrumental noise [46], and has been incorporated in the latest version of the graphical user interface MVC2 for second-order multivariate calibration, which is freely available at www.iquir-conicet.gov.ar (MATLAB version) and at https://www.dropbox.com/sh/nruf3lp0ge1gbww/AAAj6r97UBMIhgQmukRGYFPKa?dl=0 (stand-alone compiled version) [47]. Applications of the latter concepts can be found in refs. [48,49]. A literature discussion concerns the question of whether the extreme feasible solutions, computed in any of the three manners discussed above, are able to enclose the full set of feasible solutions, and/or to measure the extent of rotational ambiguity.
Rotational ambiguity is a phenomenon with the potential of generating an uncertainty in the estimation of analyte concentrations in protocols based on matrix instrumental data processed by multivariate curve resolution - alternating least-squares (MCR-ALS). This is particularly relevant when the second-order advantage is to be achieved, i.e., when selected analytes are determined in unknown samples having unexpected constituents, not considered in the calibration set of samples. It is therefore imperative that analytical chemists developing second-order multivariate calibration methods using MCR-ALS acknowledge the relevance of this issue, and more importantly, have access to the required tools to size the relative impact of this potential source of uncertainty on the estimated analyte concentrations. The purpose of this tutorial is to provide a down-to-earth view of rotational ambiguity, by studying in detail a synthetic example mimicking a typical chromatographic-spectral experiment, where a set of calibration samples is joined with an unknown sample having an uncalibrated interference. After explaining the background information needed to understand the origin of the phenomenon, the available tools for the estimation of the feasible MCR-ALS solutions and the derived uncertainty on analyte predictions will be discussed. A multi-component experimental system will also be discussed, stressing the fact that rotational ambiguity uncertainties, however small, should always be estimated and reported.
Does the signal contribution function attain its extrema on the boundary of the area of feasible solutions?
2020, Chemometrics and Intelligent Laboratory Systems
The signal contribution function (SCF) was introduced by Gemperline in 1999 and Tauler in 2001 in order to study band boundaries of multivariate curve resolution (MCR) methods. In 2010 Rajkó pointed out that the extremal profiles of the SCF reproduce the limiting profiles of the Lawton-Sylvestre plots for the case of noise-free two-component systems.
This paper mathematically investigates two-component systems and includes a self-contained proof of the SCF-boundary property for two-component systems. It also answers the question if a comparable behavior of the SCF still holds for chemical systems with three components or even more components with respect to their area of feasible solutions. A negative answer is given by presenting a noise-free three-component system for which one of the profiles maximizing the SCF is represented by a point in the interior of the associated area of feasible solutions.
Evaluation of the extension of rotation ambiguity associated to multivariate curve resolution solutions by the application of the MCR-BANDS method
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Multivariate Curve Resolution (MCR) methods have been widely used to resolve the spectra (instrumental responses) and concentration profiles of the unknown constituents of chemical mixtures especially when no prior information is available about the nature and composition of these mixtures. Based on the fulfillment of a bilinear model, like the multivariate extension of Beer's law, MCR solutions are affected by rotation ambiguity, which means that a range of feasible solutions can explain the observed data equally well fulfilling the same constraints. The MCR-BANDS method has been proposed to provide a measure of the extension of rotation ambiguity associated to a particular MCR feasible solution. In this work, the two extreme (maximum and minimum) feasible solutions obtained by the MCR-BANDS method are investigated and projected on to the area of feasible solution (AFS) obtained by other methods like the FACPACK method, and compared under the application of different constraints. In contrast to other methods that estimate the whole set of feasible solutions (i.e. the AFS), MCR-BANDS provides a simpler and flexible way to give an estimation of the extension of rotation ambiguity associated to a particular MCR solution (for instance using the MCR-ALS method) of systems with any number of components and under any type of constraints, in the concentration and spectral domains.

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Measuring and comparing the resolution performance and the extent of rotation ambiguities of some bilinear modeling methods

Highlights

Abstract

Introduction

Section snippets

Theory

Principal component analysis (PCA)

Independent component analysis (ICA)

Minimum volume simplex analysis (MVSA) method

Multivariate curve resolution-alternating least squares (MCR-ALS)

Multivariate curve resolution-objective function minimization (MCR-FMIN)

MCR-BANDS: Calculation of the extent of rotation ambiguities

FAC-PACK: Evaluation of the area of feasible solutions (AFS)

Mutual information (MI) evaluation

Concentration profiles recovery evaluation using Amari index

Data fitting: Calculation of lack-of-fit values

Datasets

Software

Results and discussion

Evaluation of the extension of rotation ambiguities in MCR and other methods

Conclusions

Conflict of interest

Acknowledgements

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