Investigation and visualization of resolution theorems in self modeling curve resolution (SMCR) methods

Abstract

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.

Definition and detection of data-based uniqueness in evaluating bilinear (two-way) chemical measurements

“…Based on Manne's theorems, by direct methods like window factor analysis and sub-window factor analysis [20] special local rank patterns have been identified for components to obtain the related component's profiles. Akbari and Abdollahi [21] demonstrated that the implementation of this information as zero-region constraint in the iterative curve resolution methods leads to the same results that can be provided by the mentioned direct methods. Also, different techniques based on some external information about acceptable or desirable shapes of factors, and in particular on assumptions regarding the zero elements in scores and/or loadings were investigated to control rotations of the computed factors [22].…”

“…The constraints such as selectivity [17][18] and local rank [2,[19][20][21] may have drastic effects on the bands of feasible solutions and in favorable cases feasible solutions turn into unique. Manne [17] has provided the full application of the local rank information to acquire unique solution, and he classified his results into several theorems called resolution theorems.…”

Speciation of adsorbates on surface of solids by infrared spectroscopy and chemometrics

“…information, the necessary conditions required to obtain a complete resolution of X surf , are not fulfilled [31,32]. For example, the concentration profile c 1 of the first species appearing (species 1), is always non-zero, hence there is no subwindow where species 2 to m appear without the presence of species 1, which forbids full resolution of c 1 according to resolution theorem 1 of Manne [32].…”

“…local rank on concentration profiles and selectivity on spectral profiles, were extracted from EFA analysis. Local rank constraint, implemented by forcing to zero concentration each species absent in a given submatrix, is well known to be one of the most powerful strategies to reduce rotational ambiguity [30,31]. Hence, only one new pure spectrum and segments of concentration profiles have to be estimated at each MCR-ALS run.…”

On generalized Borgen plots. I: From convex to affine combinations and applications to spectral dataSpectra

“…The genuinely numerical procedures to compute the AFS in [7,8,[19][20][21] Similarly, a regular matrix T 2 R s s is used in order to define the factors…”