## 2 Overview of the EPR Technique  
In this paper, an advanced data modeling strategy, based on the evolutionary polynomial regression (EPR) approach, has been applied. EPR is carried out in two primary stages: an evolutionary process and a conventional numerical regression step. The first stage employs a multi-objective Genetic Algorithm (GA) to explore and identify the polynomial structure of the expressions, while the second stage utilizes the Ordinary Least Squares (OLS) method to estimate the optimal values of the model parameters [7; 12]. The general model structure that EPR can manage is given by:  
\[Y=a_{0}+\sum_{j=1}^{m}a_{j}\cdot\left(X_{1}^{ES(j,1)}\cdot\ldots\cdot X_{k}^{ES (j,k)}\right)\cdot f(X_{1}^{ES(j+1)}\cdot\ldots\cdot X_{k}^{ES(j,2k)}) \tag{1}\]  
where \(m\) is the number of additive terms; \(a_{1}\), \(\ldots\), \(a_{m}\) are numerical parameters to be estimated; \(X_{1}\), \(\ldots\), \(X_{k}\) represent the candidate explanatory variables; and \(ES\) (\(j\), \(z\)) (with \(z=1,\ldots\), \(k\)) is the exponent of the \(z\)-th input within the \(j\)-th term. The exponents \(ES\) (\(j\), \(z\)) are selected from a user-defined set of candidate values as well as \(f(X_{1}^{ES(j,k+1)}\cdot\ldots\cdot X_{k}^{ES(j,2k)})\) which is a user-selected function among some possible alternatives (including the option of no function). The product among the candidate variables and the selected function gives the transformed variable:  
\[Z_{j}=\left(X_{1}^{ES(j,1)}\cdot\ldots\cdot X_{k}^{ES(j,k)}\right)\cdot f(X_{1 }^{ES(j,k+1)}\cdot\ldots\cdot X_{k}^{ES(j,2k)}) \tag{2}\]  
The method starts from a number \(n\) of available observations representing the dataset size, by performing a random initialization of the population of explanatory variable exponents. Next, the model parameters are estimated using OLS by solving a linear inverse problem, which involves minimizing the squared errors between the observed and predicted data. This process guarantees a unique correspondence between each model structure and its associated parameters. Furthermore, EPR provides a nonlinear mappingof the data with a limited number of coefficients, thus simplifying the identification of the optimal pseudo-polynomial model. Thus, common over-fitting issues can be prevented [28]. Naturally, model accuracy tends to improve as complexity increases, which is reflected in the number of pseudo-polynomial terms and/or the number of input variables included in the model. Therefore, identifying the optimal model structure involves an optimization process aimed at maximizing or minimizing a predefined Objective Function (OF). The most accurate models are generated by the application of genetic operators: crossover and mutation. Crossover works by randomly recombining the exponent vectors of the best-performing solutions, while mutation introduces randomness by altering one or more exponent values. These operators iteratively produce new populations of exponent vectors, addressing the search for improved model structures. The search is completed when the algorithm returns the required value of the OF. A flowchart summarizing the EPR procedure is depicted in Figure 1.  
Figure 1: Flowchart of the EPR procedure. The database is constituted of \(n\) observations described through \(k\) candidate input variables. The symbolic expression is composed of \(m\) parameters, defined by setting the error distribution. The distribution of the errors between the estimated and observed values follows a given Probability Density Function (pdf) with mean \(\mu\) and standard deviation \(\sigma\). The errors are usually assumed to be normally distributed with a null mean and given variance \(\sigma\). In order to limit the negative effects of extreme experimental values, the EPR model parameters are evaluated by adopting a robust regression approach. It consists of assigning a weight \(\psi_{i}\) to each observation, that is set as inversely proportional to the residual \(\varepsilon_{i}\) obtained from the estimates.