## 3 Calibration
By resorting to the mathematical relationship defining the PCA projection (\(\mathbf{T}=\mathbf{X}\mathbf{P}\), see also Equation (1)), it is possible to express the linear model in Equation (15) directly as a function of \(\mathbf{X}\) and \(\mathbf{Y}\), as:  
\[\mathbf{\hat{Y}}_{\text{PCR}}=\mathbf{T}\boldsymbol{\mathcal{B}}=\mathbf{X} \mathbf{P}\boldsymbol{\mathcal{B}}=\mathbf{X}\mathbf{B}_{\text{PCR}} \tag{16}\]  
where the matrix of PCR coefficients, \(\mathbf{B}_{\text{PCR}}\), is given by:  
\[\mathbf{B}_{\text{PCR}}=\mathbf{P}\boldsymbol{\mathcal{B}} \tag{17}\]  
With respect to the MLR solution reported in Equation (10), PCR is said to be biased, as not all the information originally present in \(\mathbf{X}\) is used to estimate the parameters of the regression model [67; 68]; the calculation of \(\mathbf{B}_{\text{PCR}}\), in fact, relies solely on the variability captured by the PCA scores in \(\mathbf{T}\), and a different model will be returned depending on how many principal components are extracted from the data matrix \(\mathbf{X}\). Accordingly, the number of principal components to retain in a PCR model is a metaparameter which has to be optimized; normally, models with different numbers of principal components are tested and the one resulting in the lowest prediction error (usually computed by cross-validation or other resampling procedures) is selected [69; 70; 71] (see also Section 7.2). PCR can be considered a highly performant statistical approach as it enables the possibility of calculating regression models for ill-conditioned data matrices. Nevertheless, even in similar contingencies it may not necessarily constitute the best option to choose: indeed, the two steps PCR combines (i.e., data compression by PCA and regression by MLR) have different objectives, and it is not always verified that the directions of maximum explained variance in \(\mathbf{X}\) are also those corresponding to the maximum correlation with the response(s) [72]. Conversely, PLS tries to overcome this issue, by actively using the information in \(\mathbf{Y}\) already at the data compression stage, so that the scores extracted from \(\mathbf{X}\) are relevant for describing at the same time the variance in the descriptors and in the properties of interest. PLS rests on the extraction of two sets of scores, one from the independent and one from the dependent data block, having maximum covariance with one another:  
\[\underset{\mathbf{r}_{a},\mathbf{q}_{a}}{\text{argmax}}\ \mathbf{u}_{a}^{\text{T}} \mathbf{t}_{u,\text{PLS}}=\underset{\mathbf{r}_{a},\mathbf{q}_{a}}{\text{argmax}}\ \mathbf{q}_{a}^{\text{T}}\mathbf{Y}^{\text{T}}\mathbf{X}\mathbf{x}_{a} \tag{18}\]where \(\mathbf{t}_{a,\text{PLS}}\) and \(\mathbf{u}_{a}\) are the \(\mathbf{X}\)- and \(\mathbf{Y}\)-scores along the \(a\)-th component, respectively, and \(\mathbf{r}_{a}\) and \(\mathbf{q}_{a}\) represent the vectors of coefficients allowing to retrieve such scores from the corresponding original matrices:  
\[\mathbf{t}_{a,\text{PLS}}=\mathbf{X}\mathbf{r}_{a} \tag{19}\] \[\mathbf{u}_{a}=\mathbf{Y}\mathbf{q}_{a}\]  
The functional relation (usually a linear dependence), also known as inner relationship, between the two data blocks is then implemented at the scores level; component by component, the \(\mathbf{Y}\)-scores are approximated by the \(\mathbf{X}\)-scores through a univariate linear regression model:  
\[\hat{\mathbf{u}}_{a}=\mathbf{t}_{a,\text{PLS}}c_{a} \tag{20}\]  
with \(c_{a}\) being a proportionality constant (_inner regression coefficient_).