## 3 Statistical Methods--Partial Least Squares  
Statistical methods can help in identifying which variables in a system can be controlled (factors) to explain and predict the behavior of the output variables (responses).  
Different strategies can be used to establish these relationships, namely Multilinear Regression (MLR) [41] and Partial Least Squares (PLS) [42]. MLR is used when the number of factors is small; the factors are not collinear; and the relationship between the factors and responses can be in some way understood a priori [43]. When any of these three conditions is not present, MLR is not an adequate strategy to establish a model relating the factors to the responses. On the other hand, PLS can be an appropriate strategy to predict the dependent variables from a large set of independent variables, which are called, in this case, predictors, even if there is a high degree of collinearity between the independent variables. PLS relates two data matrices: the predictor (input variable) matrix (X variables) and the response matrix (Y variables), making use of a linear multivariate model for that purpose [42; 44].  
PLS is a bilinear calibration method that calculates linear combinations of input variables that have a maximum covariance with the response. These linear combinations (latent variables) are found sequentially. Each combination results from the progressive analysis of a different part of the space until prediction ability stops improving, based on the analysis of the value of the predicted determination coefficient (R\({}^{2}\) (pred)).  
Minitab (version 17.1.0) was the software used to apply the PLS method to the data set in the present study. Minitab uses the Nonlinear Iterative Partial Least Squares (NIPALS) algorithm developed by [PERSON] [45; 46; 47; 48]. This algorithm reduces the number of predictors using a technique similar to principal components analysis by extracting a set of components that describes the maximum correlation between the predictors and the response variables. If the predictors are highly correlated, then the number of components in the model might be much less than the number of predictors. Minitab then performs least squares regression on the uncorrelated components. Additionally, cross-validation is often used to select the components that maximize the model's predictive ability [47; 48].  
To validate the model, the best approach is to use an independent data set or \"test set\" and to check the predictive ability of the model based on the results obtained for the independent data set (external validation). However, when the data set collected is small, it is sometimes impossible, in order to validate the model, to guarantee access to an independent test set. In that case, the alternative is to perform what is called a cross  
Figure 1: Schematic representation of the laboratory procedure.  
validation by creating \"test sets\" within the \"training data set\", i.e., the data set used to create the model itself (internal validation) [49].  
In the case of internal validation, the training data set is divided into blocks; one block is left aside in each run, and a new model is obtained with the remaining blocks, which is used to estimate the response of the removed blocks. The cross-validation prediction errors are calculated for this new situation, and this process is repeated successively, each time removing a new block while the previously removed one is integrated again in the training data set. The overall cross-validation error will be estimated once this process is completed [48; 49; 50]. At the end of the cross-validation process, it is also possible to determine the model dimension, i.e., the number of PLS components (latent variables) the model should have.  
In this study, this last methodology was used due to the lack of additional experimental data to be used as an independent test set.