## 5 Data Fusion
### Multi-Block Regression and Classification  
The chemometric literature is rich in multi-block methods for regression and classification. One of the most commonly used is undoubtedly Multi-Block-PLS (MB-PLS). MB-PLS constitutes the direct extension of PLS to the data fusion field. Different variants of its algorithm have been proposed all over the years [179], the most recent being developed by [PERSON] et al. in 2001 [188]. The basic idea behind this latest formulation is that multiple predictor blocks can be concatenated and modelled simultaneously by PLS. Although extremely intuitive and yielding accurate predictions, this strategy does not provide information about the single datasets. Conversely, the solution implemented by [PERSON] and [PERSON] can be exploited for the interpretation of block scores and loadings [179]. In this regard, it is important to stress that MB-PLS requires the datasets at hand to be scaled separately (by, e.g., their individual sum-of-squares or their Frobenius' norm), to prevent structures with greater variance from preferentially driving the model and blurring the information carried by the others [179]. Due to its remarkable flexibility, MB-PLS is nowadays probably the most widely applied multi-block regression approach. It is also often used in combination with LDA for addressing classification problems--MB-PLS is here known as MB-PLS-LDA or MB-PLSDA. So far, many applications of MB-PLS have been reported in the field of food analysis (see Table 3); it has been resorted to (i) for the investigation of sensory parameters of different nature and of their relationships with technological properties of cheese and bread samples [189; 190], (ii) for the prediction of meat spoilage time, wine ageing time and crude protein and moisture content in soybean flour by MIR and NIR spectroscopy [191; 192; 193], (iii) for the discrimination of botanical varieties of extra virgin olive oil, lemon essential oils and wines of different geographical origin by MIR, NIR and Raman spectroscopy [194; 195; 196], and (iv) for distinguishing added-value from low-quality products [132]. Despite all the aforementioned advantages MB-PLS offers, usually redundant information and/or the presence of categorical regressors may jeopardize the predictive capability of MB-PLS. In these situations, techniques like the recent Sequential and Orthogonalized-PLS (SO-PLS [197; 198]) could represent feasible options to overcome such limitations. SO-PLS enables the sequential modelling of each regressor set. The keystone of this methodology is a preliminary orthogonalization step performed to filter out redundancies among blocks. Briefly, taking into consideration the case where a generic response \(\mathbf{Y}\) is to be estimated from two predictor matrices, say \(\mathbf{X}_{1}\) and \(\mathbf{X}_{2}\), the SO-PLS algorithmic scheme encompasses the following four steps (readers are addressed to [140; 141] for a more detailed description of the SO-PLS algorithm):  
1. \(\mathbf{Y}\) is regressed onto \(\mathbf{X}_{1}\) by PLS;
2. \(\mathbf{X}_{2}\) is orthogonalized with respect to the \(\mathbf{X}_{1}\)-scores calculated in step 1, yielding the array \(\mathbf{X}_{2,\text{orth}}\). This ensures the common information shared by \(\mathbf{X}_{1}\) and \(\mathbf{X}_{2}\) is removed from the latter;
3. The PLS residuals resulting from step 1 are regressed onto \(\mathbf{X}_{2,\text{orth}}\) by PLS;
4. The SO-PLS predictive model is expressed by combining the outcomes of steps 1 and 3 as: \[\mathbf{Y}=\mathbf{X}_{1}\mathbf{B}_{1}+\mathbf{X}_{2}\mathbf{B}_{2}+\mathbf{ E}_{\text{SO-PLS}}\] (31) where \(\mathbf{B}_{1}\) and \(\mathbf{B}_{2}\) contain the regression coefficients for \(\mathbf{X}_{1}\) and \(\mathbf{X}_{2}\), respectively, and \(\mathbf{E}_{\text{SO-PLS}}\) carry the final \(\mathbf{Y}\)-residuals. As a consequence of the deflation of the common information shared by the different datasets under study, SO-PLS provides useful insights into the unique and distinctive sources of variation within each one of them.