## 6 Other Approaches
### Curve Resolution
\begin{split}\mathbf{C}&=\mathbf{X}\mathbf{S} \mathbf{S}^{\mathrm{T}}\mathbf{S}\mathbf{S}^{\mathrm{-1}}\\ \mathbf{S}^{\mathrm{T}}&=\left(\mathbf{C}^{\mathrm{ T}}\mathbf{C}\right)^{\mathrm{-1}}\mathbf{C}^{\mathrm{T}}\mathbf{X}\end{split} \] (33)
4. Step 3 is repeated until a certain convergence criterion is met. It is evident for the algorithmic procedure outlined before that one of the main features of MCR-ALS is its \"self-modeling\" nature, i.e., in principle, it does not require any specific preliminary information about the data at hand: it is only needed that the bilinear model in Equation (31) is satisfied and that some very generic characteristics of the pure concentration or spectral profiles (e.g., their non-negativity) are known. If further information is also available, though, it can be actively exploited in the form of additional mathematical restrictions. Basically, it is the possibility of implementing in a rather straightforward way a large variety of these restrictions what has made and still makes MCR-ALS very popular compared to other curve resolution strategies. Indeed, they not only reduce the ambiguity associated with the final model (that is to say the range of possible \(\mathbf{C}\)/\(\mathbf{S}^{\mathrm{T}}\) sets returning the same fit when approximating \(\mathbf{X}\)) but, at the same time, they render the resolved profiles more physico-chemically meaningful [230,231]. Examples of possible restrictions are unimodality (if the constrained profiles are expected to exhibit a unique global maximum), closure (if a mass or concentration balance among all or part of the mixture constituents holds), and selectivity (if some of the resolved species are absent in some experiments or do not absorb at specific wavelengths) [222,26,232,233,234,224]. Moreover, as already anticipated, MCR-ALS can easily be applied to multiple data matrices in a multi-set configuration. This provides a different way of analyzing multi-block data with respect to what discussed in Section 5, as here the information shared by the different investigated arrays is resorted to for improving the unmixing of the pure constituents. Concurrently, since some of these constituents can be present only in one or a few matrices and some others may be present in all, multi-set MCR-ALS is one of those multi-block techniques by which information about both common and distinctive components can be obtained [235]. Similarly to ICA, MCR-ALS has mainly been employed in order to retrieve the spectral contributions of particular compounds constituting complex food samples and to determine how their abundance/concentration varies during, e.g., a renenting process (see Table 4). In [236, 237, 238], for instance, the dynamic evolution of various forms of milk (liquid, sol-gel, coagulated) over the progression of a lactic acid fermentation reaction was monitored and their individual FT-NIR fingerprints disentangled. In [239], a similar study was conducted on beer fermentation: here, maltose, maltotriose, fructose, sucrose, dextrins and ethanol spectral and concentration profiles were retrieved for enabling a better understanding of the various phenomena behind their biotransformation.