Rotational ambiguity

It is difficult to generalise, but such ambiguous situations often occur when the concentration windows are overlapping in specific ways. Kinetic investigations are typical examples of rotational ambiguity as a result of very wide concentration windows. 

While the resulting concentration profiles, and in particular the computed spectra, seem to be reasonably close to the true ones, there are significant discrepancies, typical for model-free analyses, (a) The computed concentration profile for the intermediate component reaches zero at the end of the measurement, (b) The initial part of the concentration profile for the final product is wrong it does not start with zero concentration. Both discrepancies are the result of rotational ambiguity. The minimal ssq, reached after relatively few iterations, reflects the noise of the data and not a misfit between CA and Y. ssq does not improve if the correct matrices C and A are used. 

Implementation of additional constraints can help remove, or at least reduce, rotational ambiguity. A possibility to narrow down the range is to utilize a known component spectrum. The simplest way of implementing known component spectra is to replace the appropriate spectrum within the iterations with the correct, known one. See 

The incorrect concentration profile for the intermediate B indicates that there is still a reduced level of rotational ambiguity. In fact, the solution is still not unique there is still a discrepancy in the concentration profile of the intermediate and small errors in the spectra. With the introduction of the one correct spectrum, the range of rotational ambiguity has been reduced but not totally removed. 

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The spectrum of B is fairly well defined while the spectrum of A is not, there is a long extrapolation required until the its spectrum turns negative at 400nm. This exemplifies the limitation of model-free methods they rely on very simple constraints but in certain cases the range of feasible answers can be very wide, sometimes too wide to be useful. This will be discussed later in Chapter 5.4.3, Rotational Ambiguity. 

The improvement resulting from the incorporation of the correct spectrum for the intermediate is subtle but significant. The all resulting spectra are improved, with the intermediate spectrum, of course, correct. The new concentration profiles for the starting material A and product C are now correct, while the profile for the intermediate B is untouched. Compared to Figure 5-56, the minimal ssq after the incorporation of the correct spectrum, is not improved and this is a clear indication for rotational ambiguity. 

The quality of the resolution results can be evaluated from a comparison between the actual spectral and time profiles of species with those recovered by MCR-ALS. Actual profiles can be found experimentally by recording CE runs of pure standards of components. The concordance between true and calculated profiles can be measured with correlation coefficients. Values close to 1 suggest that results are not affected by rotational ambiguities. Conversely, values significantly lower than 1 indicate that ambiguities still persist (43). 

Depending on the quality of data and the method selected, constraints on the parameters to be estimated may be required in order to get a chemically meaningful solution. In the case of multivariate curve resolution (MCR) (see Section 3.2) performed on one 2D NMR spectrum, application of constraints is mandatory. If constraints are not applied, it can be shown that there is an infinity of equally well-fitting solutions and hence the true underlying parameters (spectra, concentrations) cannot be estimated directly. This is known as the rotational ambiguity of two-way low-rank models. 

Due to this rotational ambiguity, it is customary to choose a solution in which the component matrices are orthonormal ... 

Once a Tucker3 solution is computed, rotational ambiguity is still present, as already discussed. Therefore, it is still possible to transform the solution so... 

The popularity of PARAFAC stems from its ability to mathematically separate the spectra of overlapping fluorescence components. Hence, PARAFAC is similar to MCR mentioned earlier (actually PARAFAC can be considered the three-way version of the two-way MCR) with the important difference being that PARAFAC does not have rotational ambiguity as does MCR. This means that if the model is correct, it will give chemically meaningful results. For a mathematical explanation of PARAFAC including tutorials on its application we refer the reader to other references (Bro, 1997 Andersen and Bro, 2003). Briefly stated, PARAFAC of a three-way data set decomposes the data signal into a set of trilinear terms and a residual array ... 

Jaumot, J. and Tauler, R. (2010). MCR-BANDS A user friendly MATLAB program for the evaluation of rotation ambiguities in multivariate curve resolution. Chemometr. Intell. Lab. Syst., 103, 96-107. 

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