7 NIR Data Exploration and Regression by Chemometrics—A Primer
145
most works concerning spectroscopy, a nonnegativity constraint in the spectral mode
(S) and in the concentration mode (C) is employed to guide the algorithm, i.e.,
using the knowledge that the NIR spectra only have positive absorbances and that
the concentrations can only be positive. Detailed description of other constraints,
limitations and other aspects of MCR is discussed in the literature [19].
Even in the absence of error, three indeterminacies exist for the MCR solution
[22]: (i) Permutation indeterminacy—there is no defined order of the components in
C and S and no sequential calculation of the components. This is a minor bookkeeping
problem, which should be solved when, for example, repeating the model in crossvalidation scenarios (see validation section 7.6). (ii) Intensity indeterminacy—two
identical spectra, but scaled differently, will provide the same model fit since the
concentrations will be adjusted accordingly. This provides two different solutions.
The problem is easily solved by normalizing the spectral profiles to have the norm
1 or by constraining the concentrations to add up to 1 (closure constraint). (iii)
Rotational indeterminacy—similarly to the intensity indeterminacy, a rotation of the
concentration profiles and consequently of the spectral profiles can in some cases
reproduce the original data with the same fit quality. The closure constraint will not
help solving this problem, but nonnegativity constraint on both concentrations and
spectra will reduce the solution space considerably, often sufficiently to finding the
correct solution.
Even with constraints, MCR does not always provide a unique solution, and the
result will sometimes depend on the initial guess of C or S; therefore, only specialized
chemometric software packages include the MCR. However, by repeating the MCR
model with many different, random initial guesses and subsequently analyzing the
solution space, it is possible to find a unique global solution.
The number of components in the MCR model can be validated by inspecting the
explained variation as a function of the number of MCR components (f ) in the same
way as the number of components in other chemometric algorithms is validated (see
validation section 7.6). However, due to the ambiguity in the solution space the MCR
results should always be validated by a priori knowledge about the chemical system
being investigated. Spectral pre-processing that is focused on “cleaning” the spectra
from scatter and artifacts will normally be an advantage, while other operations such
as centering and autoscaling will “destroy” the pure spectral information sought and
thus the MCR model.
7.3.1 Application of MCR to NIR Spectra
MCR is in general best suited for relative simple and well-behaving systems. It has
nevertheless been applied to numerous NIRS applications such as whey powder [23],
protein denaturation [24], edible oils [25], porcine fat tissue [26], process analytical
technology [27] and many more studies.
Figure 7.14 shows the result of a 3-component MCR model fitted to the Dataset 2.
Figure 7.14a shows the concentration matrix C from the MCR model plotted in the
145
most works concerning spectroscopy, a nonnegativity constraint in the spectral mode
(S) and in the concentration mode (C) is employed to guide the algorithm, i.e.,
using the knowledge that the NIR spectra only have positive absorbances and that
the concentrations can only be positive. Detailed description of other constraints,
limitations and other aspects of MCR is discussed in the literature [19].
Even in the absence of error, three indeterminacies exist for the MCR solution
[22]: (i) Permutation indeterminacy—there is no defined order of the components in
C and S and no sequential calculation of the components. This is a minor bookkeeping
problem, which should be solved when, for example, repeating the model in crossvalidation scenarios (see validation section 7.6). (ii) Intensity indeterminacy—two
identical spectra, but scaled differently, will provide the same model fit since the
concentrations will be adjusted accordingly. This provides two different solutions.
The problem is easily solved by normalizing the spectral profiles to have the norm
1 or by constraining the concentrations to add up to 1 (closure constraint). (iii)
Rotational indeterminacy—similarly to the intensity indeterminacy, a rotation of the
concentration profiles and consequently of the spectral profiles can in some cases
reproduce the original data with the same fit quality. The closure constraint will not
help solving this problem, but nonnegativity constraint on both concentrations and
spectra will reduce the solution space considerably, often sufficiently to finding the
correct solution.
Even with constraints, MCR does not always provide a unique solution, and the
result will sometimes depend on the initial guess of C or S; therefore, only specialized
chemometric software packages include the MCR. However, by repeating the MCR
model with many different, random initial guesses and subsequently analyzing the
solution space, it is possible to find a unique global solution.
The number of components in the MCR model can be validated by inspecting the
explained variation as a function of the number of MCR components (f ) in the same
way as the number of components in other chemometric algorithms is validated (see
validation section 7.6). However, due to the ambiguity in the solution space the MCR
results should always be validated by a priori knowledge about the chemical system
being investigated. Spectral pre-processing that is focused on “cleaning” the spectra
from scatter and artifacts will normally be an advantage, while other operations such
as centering and autoscaling will “destroy” the pure spectral information sought and
thus the MCR model.
7.3.1 Application of MCR to NIR Spectra
MCR is in general best suited for relative simple and well-behaving systems. It has
nevertheless been applied to numerous NIRS applications such as whey powder [23],
protein denaturation [24], edible oils [25], porcine fat tissue [26], process analytical
technology [27] and many more studies.
Figure 7.14 shows the result of a 3-component MCR model fitted to the Dataset 2.
Figure 7.14a shows the concentration matrix C from the MCR model plotted in the