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Two new models have been recently introduced for studying the remaining rotational ambiguity in the bilinear decomposition of matrix data. One of the models is N‐BANDS, which yields two extreme profiles per sample component, corresponding to maximum or minimum signal contribution function or relative component area under its concentration profile. It is highly useful for computing the relative root mean square error due to rotational ambiguity in estimated analyte concentrations (RMSERA), which numerically quantifies the impact of the phenomenon in terms of prediction uncertainty. Since N‐BANDS successfully consider the presence of instrumental noise in the data, it is extremely useful for the analysis of real data sets. The other model is SW‐N‐BANDS, which is similar to N‐BANDS, but is applied in a sensor wise manner, that is, computing the maximum and minimum intensity value at each sensor. It provides the boundaries of the full set of feasible profiles, and helps to better understand the behavior of a given component under the application of several constraints. Both models are described in light of both simulations and experimental data, illustrating their main characteristics of importance to analytical chemistry studies. The new N‐BANDS and SW‐N‐BANDS are described for estimating and interpreting rotational ambiguity effects in analytical chemistry. [ABSTRACT FROM AUTHOR]

A new characteristic term, data point importance (DPI), is introduced in order to quantify the information ranking in multivariate data. DPI defines an easily calculable value corresponding to each row or column of data matrix to reflect its impact for keeping the pattern of the data structure. Usually, a lot of data points have DPIs equal or very close to zero so that they do not carry on useful information about keeping the data pattern. DPI values for some of the data points are significant, and they have been sorted regarding to their importance values. The applicability of the proposed procedure is tested for three typical examples of simulated and experimental data including hyperspectral image data. The possible incorporation of the DPI values for more informative score plots is suggested. [ABSTRACT FROM AUTHOR]

A new algorithm named N‐BANDS has been developed for the estimation of the combined effect of noise and rotational ambiguity in the bilinear decomposition of data matrices using the popular multivariate curve resolution–alternating least‐squares model. It is based on a nonlinear maximization and minimization of a component‐wise signal contribution function (SCF), with a single‐objective function and a separate module for applying a variety of constraints. The algorithm can be applied to multicomponent systems and efficiently estimates the extreme component profiles corresponding to maximum and minimum SCF in the presence of varying amounts of instrumental noise. Simulated systems mimicking multicomponent and multisample analytical calibration protocols have been studied, having uncalibrated interferents in the test samples. Different noise structures (independent and identically distributed, proportional and correlated) were added to the data matrices, with results that indicate an increase in the extension of feasible bands as the noise level increases. N‐BANDS has been developed as a new algorithm for estimating the combined effect of noise and rotational ambiguity in bilinear decompositions using multivariate curve resolution–alternating least‐squares, based on the maximization/minimization of component‐wise signal contribution functions, with a single objective function and a separate module of constraints. Multi component and multi sample simulated systems were studied with uncalibrated interferents in test samples. Noise addition results indicate an increase in the extension of feasible bands as the noise level increases. [ABSTRACT FROM AUTHOR]

Extended bilinear modeling is popular in second‐order multivariate calibration, particularly when the matrix data for each sample are of chromatographic origin. Since elution time profiles vary across samples, in both shape and peak position, it is not possible to process these data in a three‐way trilinear format. In these cases, the most successful model for quantitating analytes in the presence of interferents is multivariate curve resolution‐alternating least squares (MCR‐ALS) in its extended version, ie, processing an augmented data matrix built with the matrices for a test sample and the calibration samples, appended in the direction of the elution time mode. MCR‐ALS starts with certain initial profiles and applies a set of natural constraints during the ALS phase, whose purpose is to reduce the range of feasible solutions or to lead to unique bilinear solutions if possible. In this report, a simulated second‐order three‐component system (two calibrated analytes and one uncalibrated interferent in test samples) is studied regarding the presence of rotational ambiguity in the bilinear solutions, using (a) a grid search methodology to compute the feasible solutions and (b) MCR‐ALS on a large set of test samples to estimate the average prediction errors. Various initialization schemes and constraints are probed, and the results are compared in terms of the extent of rotational ambiguity and global uncertainty in predicted concentrations. [ABSTRACT FROM AUTHOR]

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 factor-based MCR methods from OLS to WLS are also discussed. [ABSTRACT FROM AUTHOR]

Different approaches have been proposed during recent years to improve the solutions obtained by multivariate curve resolution methods, among them studies on circumstances that result in unique answers are of particular importance. Three so-called resolution theorems proposed by Rolf Manne comprehensively discuss and survey these conditions. Despite the importance of these theorems, they have not attracted much attention in the literature. In this work, we have returned to the resolution theorems by using visualization tools for calculation and demonstration the area of feasible solutions in a simulated three component system. The effect of selectivity and local rank constraints on rotational ambiguity in the results of self modeling curve resolution methods is the focus. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Parallel factor analysis (PARAFAC) is believed to be one of the most widely used multiway curve resolution techniques. The superiority of PARAFAC is due to its uniqueness property that is provided under mild assumptions. Unambiguously recovered chemically and physically meaningful profiles by PARAFAC are reliable for quantitative and qualitative application. The essential uniqueness condition of PARAFAC breaks totally or partially in data sets with linearly dependent loadings. PARAFAC uniqueness levels include full uniqueness, partial uniqueness, uni-mode uniqueness, and full non-uniqueness. Although PARAFAC uniqueness conditions are more or less known, they cannot cover the whole of existing uniqueness problems, especially complex rank-deficient systems. The questions investigated here are as follows: Under which conditions is PARAFAC still unique and what is the extent of deterioration of PARAFAC uniqueness by linearly dependent loadings? These are among the most important points in dealing with three-way data arrays with PARAFAC models. Direct visualization of feasible bands using an explicit, well-defined, and trustworthy method is a key feature of our approach to the investigation of uniqueness and recasting partial uniqueness rules in this study. Different simulated examples with different uniqueness levels were designed, and rotational ambiguity was calculated for each case. In this way, the full implication of partial uniqueness was investigated. Direct visualization of feasible bands is beneficial not only for confirmation of previous rules of uniqueness and partial uniqueness but also for development of new theorems. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Rotational ambiguity is a significant concern of chemometricians when using soft modeling methods to resolve bilinear data. Applying different types of constraints appropriate to a system under study can significantly reduce rotational ambiguity in soft models and the resulting range of feasible solutions. The first and most commonly used constraint is non‐negativity and can be applied to concentration and spectral response profiles as appropriate. Unimodality constraints, trilinearity constraints, and physical chemical modeling of concentration profiles are natural constraints that can significantly reduce rotational ambiguity in soft modeling curve resolution methods. Also, functional fitting of spectral response profiles (eg, peak shape modeling) has been used as a constraint for resolving spectroscopic data matrices. In the present work, the use of peak shape model constraints for spectral profiles was investigated using the Voigt function (a combination of Gaussian and Lorentzian functions). Mixture spectra of a simulated two‐component system and a real data set were fitted to several different Voigt models. The results of applying peak shape models on the range of feasible solutions for simulated and real data sets and simultaneously fitting the data to different models show that spectral peak shape constraints are not effective for limiting the range of feasible bands. Indeed, the non‐specific spectral Voigt model does not provide enough information about spectra to be effective for significant reduction of rotational ambiguity. These studies show that in the absence of sufficient information, many sets of peak functions can be well‐fitted to spectra and rotational ambiguity is not improved when resolving bilinear spectroscopic data. The results of applying peak shape models on the range of feasible solutions for simulated and real data sets and simultaneously fitting the data to different models show that the use of the sum of Voigt functions (a combination of Gaussian and Lorentzian functions) as spectral fitting constraints is not effective for limiting the range of feasible bands. In the absence of sufficient information, many sets of peak functions can be well fitted to spectra and rotational ambiguity is not improved. [ABSTRACT FROM AUTHOR]