## 6 Other Approaches
### Curve Resolution  
In the previous sections, it was shown how a spectral data matrix **X** can be decomposed according to a bilinear model and how this decomposition can be regarded as its projection onto a subspace of latent (_abstract_) directions. However, under normal experimental conditions, it is well known that spectroscopic data can also be approximated by a bilinear model whose elements are directly interpretable in chemical terms. Indeed, by assuming that **X** contains the spectral intensities or absorbance values for the various analyzed samples at different wavelengths, and, without any loss of generality, that the optical path is unitary, Beer-Lambert-Bouger law can be expressed as:  
\[\textbf{X}=\textbf{CS}^{\text{T}} \tag{32}\]  
where **C** and **S** are the arrays collecting the concentrations of the chromophores constituting the spectral mixture across all the specimens and their individual spectroscopic signatures (molar absorbivities), respectively [221]. Based on the relation described by Equation (31), the aim of curve resolution methods is to perform a data-driven unmixing of the matrix **X**, i.e., to estimate the number of constituents in the aforementioned mixture and obtain their pure concentration and spectral profiles from the information encoded in the recorded data. Among the possible approaches for accomplishing such a task, the most popular (due to its flexibility and to the possibility of being applied also to multi-set or multi-way data) is Multivariate Curve Resolution-Alternating Least Squares (MCR-ALS [222; 223; 224; 225; 226; 227]). As the name suggests, MCR-ALS is an iterative algorithm based on the alternating least squares concept, which involves the following steps: \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Methods**} \\ \hline
**Name of the Method** & **Aim** & **Reference** \\ \hline Hierarchical PLS (H-PLS) & Regression & [178] \\ Joint and Individual Variation Explained (JIVE) & Exploratory & [213] \\ Multiblock PLS serial extension & Regression & [214] \\ Network-Induced Supervised Learning (NI-SL) & Regression & [215] \\ Parallel Orthogonalized Partial Least Squares (PO-PLS) & Regression & [216] \\ Multiblock Redundancy Analysis & Regression & [217] \\ OnPLS & Regression & [218] \\ \hline \multicolumn{1}{c}{**Applications**} \\ \hline
**Aim** & **Multi-Block method** & **Reference** \\ \hline Prediction of bread sensory properties & MB-PLS & [190] \\ Prediction of wine ageing time & MB-PLS/H-PLS/NI-SL/SO-PLS & [192] \\ Quantification of protein and moisture in soybean flour & MB-PLS & [193] \\ Discrimination of lemon essential oils & MB-PLS & [194] \\ Determination of the geographical origin of wine & MB-PLS & [196] \\ Authentication of spirits & SO-PLS/SO-CovSel & [128; 202] \\ Determination of the geographical origin of safron & SO-PLS/SO-CovSel & [205] \\ Path modelling & SO-PLS & [219] \\ Path modelling & SO-PLS & [220] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Additional data fusion methods and application examples in the field of spectroscopy-based food analysis. MB-PLS, SO-PLS, SO-CovSel and OnPLS stand for Multi-Block-Partial Least Squares, Sequential and Orthogonalized-Partial Least Squares, Sequential and Orthogonalized Covariance Selection and n-block Orthogonal Projections to Latent Structures, respectively. 1. The number of components (mixture constituents) is estimated (e.g., according to a priori knowledge of the systems under study or by SVD/PCA). 2. A first guess of either \(\mathbf{C}\) or \(\mathbf{S}^{\mathrm{T}}\) is calculated by methodologies like Evolving Factor Analysis (EFA [228]) or SIMPLE-to-use Interactive Self-Modeling Analysis (SIMPLISMA [229]). 3. \(\mathbf{C}\) and \(\mathbf{S}^{\mathrm{T}}\) are iteratively updated using alternating least squares under appropriate constraints (e.g., non-negativity of the values in \(\mathbf{C}\) and/or \(\mathbf{S}^{\mathrm{T}}\)) as: \[