## 3 Methodological Procedure  
### Regression Algorithms  
Regression algorithms are employed to establish mathematical connections between variables that interacts in a data set. This can be used to predict the future characteristics of the system from which the data were obtained. The scientific literature [18; 36] features a wide range of regression algorithms that have found success in various applications. These range from the classic linear regression to more sophisticated models that incorporate penalties to reduce variance and enhance prediction reliability, such as the Ridge and LASSO algorithms. Additionally, methods like Bayesian Ridge and Deep Neural Networks offer alternative structures, providing diverse perspectives for evaluating their performance. To estimate the slope of the HFF using the analytical data set \(\mathcal{D}_{set}\), we will utilize various regression algorithms outlined in Items 1-6. In the description of these algorithms, we will denote \(s_{i}\) as the true value of the HFF slope and \(\hat{s}\) as the predicted value. 1. Classical Linear Regression [31]: Their primary objectives are twofold: (i) quantifying the relationship between the dependent variable and one or several independent variables and (ii) predicting the system's evolution concerning a given parameter based on the provided data. The linear regression associated with a first-order equation is \[s=\beta_{0}+\sum_{j=1}^{n}\beta_{j}x_{j}+\epsilon\,,\] (6) and it predicts the behavior of the dependent variable, \(s\), in terms of a set of k-observable variables, \(x_{j}=[x_{1},x_{2},\ldots,x_{k}]\), estimating the k-parameters, \(\beta_{j}=[\beta_{1},\beta_{2},\ldots,\beta_{k}]\). Here, \(\beta_{0}\) denotes the intercept and the error, and \(\epsilon\) corresponds to a random variable not included in the observable. Prediction of the \(s\)-value, based on the \(x_{j}\)-value, is quantified using the equation \[\hat{s}=\hat{\beta_{0}}+\sum_{j=1}^{n}\hat{\beta_{j}}x_{j}\,,\] (7) where the hat symbol is introduced to denote the estimated coefficients. To fit the linear regression model (see Equation (6)), the ordinary least squares method (OLS) is introduced to determine the set of parameters \(\hat{\beta_{j}}\), such that the regression line is the closest line to the data points \((x_{j},s)\). The OLS method estimates coefficients \([\beta_{1},\beta_{1},\ldots,\beta_{n}]\) by minimizing a quantity known as the residual sum of squares (RSS) [37]. This is defined as \[\text{RSS}=\sum_{j}^{n}(s_{i}-\hat{s})^{2}.\] (8)LASSO Regression [38]: Least Absolute Shrinkage and Selection Operator (LASSO) combines a linear regression method, described by Equation (6), implementing a penalty on the sum of the absolute values of the regression coefficients, known as regularization L1, inducing that the coefficients of the predictors tend to zero; as a consequence, LASSO excludes the least relevant predictors in the estimation process, with an additional constraint given by \(\lambda\sum_{j=1}^{n}|\beta_{j}|\). \[\hat{\beta_{L}}=\arg\min_{\beta}\left\{\sum_{j=1}^{n}\left(s_{i}-\left(\beta_{ 0}+\sum_{j=1}^{n}\beta_{j}x_{j}\right)\right)^{2}+\lambda\sum_{j=1}^{n}|\beta_ {j}|\right\}.\] (9)
3. Ridge Regression [38]: Implements a penalty on the sum of the squares of the regression coefficients, called L2. This penalty proportionally reduces the magnitude of all the coefficients of the model, avoiding that the latter tend to zero. The coefficients for the Ridge Regression are obtained by minimizing the RSS (Equation (8)), with an additional constraint given by \(\lambda\sum_{j=1}^{n}\beta_{j}^{2}\). The following equation describes Ridge Regression: \[\hat{\beta_{R}}=\arg\min_{\beta}\left\{\sum_{j=1}^{n}\left(s_{i}-\left(\beta_ {0}+\sum_{j=1}^{n}\beta_{j}x_{j}\right)\right)^{2}+\lambda\sum_{j=1}^{n}\beta_ {j}^{2}\right\}.\] (10)
4.