## 3 Calibration
Mathematically, this would result in setting up a calibration equation analogous to Equation (8) for each of the \(L\) responses to be predicted:  
\[\mathbf{y}_{l}=\hat{\mathbf{y}}_{l}+\mathbf{e}_{l}=\mathbf{X}\mathbf{b}_{l}+ \mathbf{e}_{l}\ \ \ \ \ l=1,\ \ldots,\ L \tag{12}\]  
In the case of MLR, it is possible to demonstrate that there is no difference (in terms of estimated values of the regression coefficients \(\mathbf{b}_{l}\) and, subsequently, of the predicted responses \(\hat{\mathbf{y}}_{l}\)) between building separate models for all the single properties of interest and constructing a unique global calibration [50] that can be expressed as:  
\[\mathbf{Y}=\hat{\mathbf{Y}}+\mathbf{E}=\mathbf{X}\mathbf{B}+\mathbf{E} \tag{13}\]  
where each of the \(L\) columns of the matrices \(\mathbf{Y}\), \(\hat{\mathbf{Y}}\), \(\mathbf{B}\) and \(\mathbf{E}\) contains the values of the corresponding elements in Equation (12) for a particular response:  
\[\mathbf{Y} =\left[ \begin{array}{ccccc}\mathbf{y}_{1}&\cdots&\mathbf{y}_{l }&\cdots&\mathbf{y}_{L}\end{array} \right] \tag{14}\] \[\hat{\mathbf{Y}} =\left[ \begin{array}{ccccc}\hat{\mathbf{y}}_{1}&\cdots&\hat{ \mathbf{y}}_{l}&\cdots&\hat{\mathbf{y}}_{L}\end{array} \right]\] \[\mathbf{B} =\left[ \begin{array}{ccccc}\mathbf{b}_{1}&\cdots&\mathbf{b}_{l }&\cdots&\mathbf{b}_{L}\end{array} \right]\] \[\mathbf{E} =\left[ \begin{array}{ccccc}\mathbf{e}_{1}&\cdots&\mathbf{e}_{l }&\cdots&\mathbf{e}_{L}\end{array} \right]\]  
MLR has the advantages of a clear loss-function, which directly generalizes the least squares criterion to the multivariate case, and of not requiring the tuning of any metaparameter (that is to say, once the \(\mathbf{X}\)-\(\mathbf{Y}\) training pairs are identified, the solution is univocal) [45]. However, due to the mathematical structure of Equation (10), this method cannot be applied in situations where the matrix \(\mathbf{X}\) is ill-conditioned, i.e., when the number of training samples is lower than the number of recorded variables (wavelengths) and/or when such variables are highly correlated [51]. Unfortunately, both these conditions are commonly met by spectroscopic data, and thus, MLR is seldom applied if spectral profiles are concerned, unless some variable selection strategy is adopted [52; 53; 54; 55]. On the other hand, a solution to the problem of having to deal with ill-conditioned matrices of descriptors is provided by the use of bilinear models based on the extraction of principal components or _latent variables_[56]. Indeed, as already outlined in Section 2--where the use of principal components to summarize the relevant information encoded in the data was discussed--an effective dimensionality reduction can be achieved by identifying a reduced number of highly informative orthogonal directions in the multivariate space of the original variables and projecting the data onto the subspace spanned by them, obtaining a set of new coordinates or scores, \(\mathbf{T}\). Using \(\mathbf{T}\) instead of \(\mathbf{X}\) in a calibration framework makes it possible to overcome all the issues related to ill-conditioning, since the number of relevant components is usually lower than the number of samples and because, additionally, they are completely uncorrelated due to their orthogonality. In light of these aspects, various bilinear regression techniques have been proposed in the literature, the most commonly used of which are Principal Component Regression (PCR [48; 57; 58]) and Partial Least Squares (PLS [59; 60; 61; 62; 63; 64; 65; 66]) regression. PCR, as the name suggests, is a two-step method which is based on the sequential use of PCA and MLR, with the PCA scores, \(\mathbf{T}\), exploited as predictors:  
\[\mathbf{Y}=\mathbf{\hat{Y}}_{\text{PCR}}+\mathbf{E}=\mathbf{T}\boldsymbol{ \mathcal{B}}+\mathbf{E} \tag{15}\]  
Here, \(\boldsymbol{\mathcal{B}}\) is the matrix of regression coefficients relating the scores to the predicted responses through the MLR model.