## 3 Calibration
Since more than a single component is usually needed to summarize the relevant information in \(\mathbf{X}\) and \(\mathbf{Y}\), Equations (19) and (20) can be rewritten as:  
\[\mathbf{T}_{\text{PLS}}=\mathbf{X}\mathbf{R} \tag{21}\] \[\mathbf{U}=\mathbf{Y}\mathbf{Q}\Longrightarrow\hat{\mathbf{Y}}= \mathbf{U}\mathbf{Q}^{\text{T}}\] \[\hat{\mathbf{U}}=\mathbf{T}_{\text{PLS}}\mathbf{C}\]  
where the columns of the scores matrices \(\mathbf{T}_{\text{PLS}}\) and \(\mathbf{U}\) and those of the coefficient matrices \(\mathbf{R}\) (\(\mathbf{X}\)-weights) and \(\mathbf{Q}\) (\(\mathbf{Y}\)-loadings) are associated with the individual retrieved components (see also Equation (19)), and \(\mathbf{C}\) is a diagonal array whose non-zero elements are the coefficients \(c_{a}\). The overall regression model can then be expressed as:  
\[\mathbf{\hat{Y}}_{\text{PCR}}=\mathbf{T}\boldsymbol{\mathcal{B}}=\mathbf{X} \mathbf{P}\boldsymbol{\mathcal{B}}=\mathbf{X}\mathbf{B}_{\text{PCR}} \tag{16}\]0  
Equation (22) shows that, despite the regression model be calculated at the level of the scores, due to the linearity of the projection, it is possible to express it directly in terms of the original variables (similarly to PCR, see also Equation (16)). At the same time, since only a part of the information in the original data is used for the regression, and, in particular, for the estimation of the regression coefficients:  
\[\mathbf{\hat{Y}}_{\text{PCR}}=\mathbf{T}\boldsymbol{\mathcal{B}}=\mathbf{X} \mathbf{P}\boldsymbol{\mathcal{B}}=\mathbf{X}\mathbf{B}_{\text{PCR}} \tag{16}\]1  
PLS constitutes another case of so-called biased regression [67; 68], requiring the determination of the optimal number of components to be retained in the model (which, once again, is carried out by selecting the value leading to the minimum prediction error, usually in cross-validation). A fundamental aspect of PLS regression is that, contrarily to MLR and, therefore, to PCR, when the problem involves the calibration of multiple responses, individual models yield different outcomes compared to a single global model. Since PLS is also based on the extraction of latent variables from the \(\mathbf{Y}\)-block, in fact, the calculation of a single model for calibrating all the responses at the same time entails their sharing part of their systematic variability, i.e., that they will exhibit a certain degree of intercorrelation that is not due to chance or subsampling. Only in this case is it recommended to build a single model for predicting all the responses; if this condition is not met, then it is suggested that each response be calibrated separately [73]. Altogether, MLR, PCR and PLS represent the large majority of regression methods which are used in scenarios demanding calibration based on spectroscopic data. However, all of them return linear models, while circumstances where the complexity of the problem at hand may call for non-linear functional relations might also occur. Given the nature of spectral data, a relatively popular way of implementing a non-linear functional relation--if necessary--is to use locally linear models [74; 75; 76]. The concept behind local regression is that a problem which is globally non-linear may be approximated by a combination of locally linear regressions, each one covering a relatively small portion of the variable space. Other options are non-linear extensions of the PLS algorithm [77; 78; 79; 80; 81; 82], support vector machines encompassing non-linear kernel transformations [83; 84; 85; 86], and artificial neural networks [87; 88; 89; 90] (which,nonetheless, have more stringent requirements regarding the number of training objects needed for constructing a reliable model). Throughout the years, regression analysis has played and still plays a crucial role in the domain of food characterization.