N‐BANDS: A new algorithm for estimating the extension of feasible bands in multivariate curve resolution of multicomponent systems in the presence of noise and rotational ambiguity

Abstract

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.

On the signal contribution function with respect to different norms

“…Because the present experimental data are rather noisy, negative entries occur in the data matrices, and thus, it is important to consider how the available algorithms for computing the band boundaries handle these negative values. The recently published N‐BANDS method was adopted here, because it provides nonnegative component profiles and consistent results for increasing levels of noise 24 . N‐BANDS can be adapted to minimize or maximize any of the three parameters mentioned above.…”

“…The analysis of a typical test sample included the following activities: (1) joining the test sample data matrix with those for the calibration samples along the time decay direction (this is the mode representing component concentrations), building an augmented data matrix of size 12,023 data points, (2) decomposing the augmented matrix using multivariate curve resolution–alternating least‐squares (MCR‐ALS) 25 under the constraints of nonnegativity and sample selectivity (also called species correspondence), and (3) using the retrieved spectral and time decay profiles as starting values for N‐BANDS estimation of band boundaries 24 …”

Noise effects on band boundaries in multivariate curve resolution of three-component systems

“…[1]. It is also possible to approximate the boundaries with two extreme profiles estimated by either the MCR-BANDS [9,10] or N-BANDS [11] algorithms, corresponding to maximum and minimum values of the so-called signal contribution function. In particular, it was recently found that N-BANDS is able to reasonably approximate these boundaries of the feasible solutions in the presence of noise [12].…”

Two new methods for the estimation and interpretation of the range of feasible profiles in multivariate curve resolution and their implications to analytical chemistry

“…8 In view of the above, it is imperative to bridge the gap between the body of theoretical work on rotational ambiguity in bilinear decomposition on one side, and experimental analytical protocols based on second-order multivariate calibration on the other. Especially useful in this regard are two recently introduced models: N-BANDS 9 ("N" stands for noise) and sensor-wise N-BANDS (SW-N-BANDS). 10 The former permits the estimation of the uncertainty in analyte concentration produced by rotational ambiguity, and the latter provides qualitative information for the correct interpretation of the component profiles retrieved by bilinear decomposition.…”