Application of maximum likelihood multivariate curve resolution to noisy data sets

Self Cite

Self-modeling curve resolution methods have continuously been improved during recent years. Many efforts have been made on curve resolution methods to reduce the rotational ambiguity by means of different types of constraints. Choosing proper constraints and cost functions is critically important for the reduction of the rotational ambiguity because the constraints have a direct influence on the accuracy of the area of feasible solution (AFS). In this work, we introduce a new improved cost function, which serves to apply nonnegativity, unimodality, equality, and closure constraints. We also investigate the reduction of the AFS under hard and soft constraints. Another point of this work is to evaluate the accuracy and precision of the reduced AFS in the presence of noise and perturbations, under hard and soft implementation of nonnegativity, unimodality, equality, and closure constraints. A comparison is given between the reduced AFS with soft constraints (small deviations from constraints are accepted) and the reduced AFS under hard constraints (restrictedly forced constraints). A graphical visualization of this comparison is presented for various model problems. The resultsshow that an AFS computation with soft constraints provides more reliable results, especially in the presence of noise. The test problems substantiate significant advantages of soft constraints over hard constraints because the obtained profiles are closer to the potentially true noisy profiles, which contain small deviations from ideal responses. Using tunable parameters ", , !, ı is one of the advantages of soft constrained cost function that allows the small deviations from ideal responses. Ultimately, soft constraints can help to reduce the lack-of-fit, and they are a proper instrument to handle the effect of noise on the AFS.

Section: Numerical Results For Model Datamentioning

confidence: 99%

Investigating the effect of flexible constraints on the accuracy of self‐modeling curve resolution methods in the presence of perturbations

Mojdehi

Sawall

Neymeyr

et al. 2016

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“…Positive matrix factorization finds candidate bilinear solution with nonnegativity constraints and can be directly used for MCR. The power of MCR‐WALS, PMF, and other WLS‐based methods was recently demonstrated in many studies and reviews …”

Section: Introductionmentioning

confidence: 94%

“…Here we focused on the latter choice. The main frame of this choice is well known: we should use weighted least squares (WLS) instead of OLS. Weighted least squares minimizes the weighted residues:

\begin{matrix} center \\ center E = X - {boldC}_{normalb} {boldS}_{normalb}^{normalT} & center {(‖‖,, E)}_{boldW}^{2} = \sum_{i, j} w_{ij}^{2} e_{ij}^{2} = min \end{matrix}

…”

Section: Introductionmentioning

confidence: 99%

Multivariate resolution of flattened absorption spectra with weighted least squares; weight selection and rotational ambiguity

Skvortsov

2016

Multivariate curve resolution (MCR) of absorption spectra is now a ubiquitously used tool. However, MCR methods, which use ordinary least squares (OLS) approach, assume that the measurement uncertainties are unbiased and homoscedastic. This is not true for absorption measurements, in which uncertainty variance and bias both increase as the true absorbance increases. The bias produces a well-known flattening/saturation of the peaks at high optical densities, which makes the data nonlinear and unsuitable for OLS-based MCR analysis. This problem can be reduced by using weighted least squares (WLS). In the present paper, the ability of WLS-based MCR to handle simulated and real datasets with realistic optical noise and flattening was assessed. Three weighting schemes were tested: OLS (unity weights), weights based on the maximum likelihood principle (MLP) and the physics of absorption measurement, and weights based on empirical cutoff (zero weights for saturated data points). The abilities of MCR to recover the true profiles and to evaluate rotational ambiguity of the solutions were compared for the 3 weighting schemes. MLP-and cutoff-based WLS-MCR produced better resolution of flattened data than OLS, but the success of the extension to strongly flattened spectra depended on data structure. MLP-based MCR was general and stable, while cutoff-based MCR was more sensitive to the data but could recover unbiased profiles. Generally, the use of WLS can expand MCR functionality to the analysis of flattened spectra. The specifics of finding WLS bilinear solutions and approaches to migrate factorbased MCR methods from OLS to WLS are also discussed. KEYWORDS absorption spectroscopy, multivariate curve resolution, peak flattening, rotational ambiguity, weighted least squares | INTRODUCTIONA combination of optical spectroscopy with multivariate curve resolution (MCR) is a mature and well-developed chemometric approach. 1-5 The rapidity and inexpensiveness of optical spectroscopy and its adaptability for many scientific and practical tasks are naturally complemented by the abilities of MCR to extract contributions of individual sources of spectral variance from strongly overlapped and correlated data. Multivariate curve resolution has been widely used for resolving peaks in chromatography, studying kinetics and equilibriums, etc. 2-5 Recently, adaptations of MCR for solving calibration tasks have also been developed. 6 These methods possess a second-order advantage, ie, the possibility to quantify components in the presence of unknown interferences.Generally, MCR assumes that the measured response x is a bilinear combination of several properties. In 2-way MCR the data matrix X can be written in compact matrix form:where C 0 is an m × k left factor matrix, S 0 is an n × k right factor matrix, and k is the number of factors. In practical

“…Indeed, it has been used for data measured with various techniques (among others: UV, IR, and Raman spectroscopy, gene expression, electrochemical data, and hyphenated techniques) on systems as diverse as hyperspectral images, chromatograms, reactions, environmental monitoring, and others . It has been able to resolve systems with hundreds of components, trace compounds, and challenging noise structures . The other method considered here, BTEM, reconstructs the spectral profiles of the pure components by performing Singular Value Decomposition (SVD) on the initial data and then finding linear combinations of the first few singular vectors that satisfy the conditions of minimum spectral entropy, nonnegativity, and presence of certain spectral features of interest, see Section 2.3.…”

Section: Introductionmentioning

confidence: 99%

“…4,[13][14][15] It has been able to resolve systems with hundreds of components, 16 trace compounds, 17 and challenging noise structures. 18,19 The other method considered here, BTEM, reconstructs the spectral profiles of the pure components by performing Singular Value Decomposition (SVD) on the initial data and then finding linear combinations of the first few singular vectors that satisfy the conditions of minimum spectral entropy, nonnegativity, and presence of certain spectral features of interest, see Section 2.3. The concentrations profiles are obtained in a subsequent step via least-squares fit.…”

Section: Introductionmentioning

confidence: 99%

Systematic comparison and potential combination between multivariate curve resolution–alternating least squares (MCR‐ALS) and band‐target entropy minimization (BTEM)

Bertinetto

Juan

2018

This work does a systematic comparative evaluation of 2 methods originating from different fields, both dedicated to the problem of curve resolution/unmixing: multivariate curve resolution–alternating least squares (MCR‐ALS) and band‐target entropy minimization (BTEM). The MCR‐ALS factorizes the data matrix into spectral and concentration profiles that satisfy constraints expressing physicochemical knowledge on the analyzed system. The BTEM reconstructs the pure components' spectral profiles as linear combinations of singular vectors that minimize the spectral entropy and contain specific peaks. Both methods were applied to 40 simulated and one real data set. The simulated data were generated from real spectral and concentration profiles that include different types of spectroscopy (mass spectrometry, Raman, and UV‐visible), data structures (random mixtures, images, and reaction system), and noise levels; the real data set was a Raman image of kidney calculus. For most data sets, both methods yielded accurate solutions, with a correlation between reference and resolved profiles >0.99. However, MCR‐ALS (here used with nonnegativity constraint only) was affected by rotational ambiguity in the recovery of spectral profiles coming from systems with high correlation or overlap in the concentration direction, whereas BTEM tended to distort UV‐visible spectra, a kind of measurement far in nature from low entropy conditions. MCR‐ALS solutions were more stable than BTEM to the increase in noise level. This work also explores the possibility of combining the 2 methods by performing them in sequence. The results show that this combination can significantly improve the outcome as compared to either method applied alone.