## 2 Tools and Methods
### Model Assessment Metrics  
In data analysis, linear regression models are often used in mathematical modelling, as evidenced by numerous applications. However, it is sometimes necessary to apply a non-linear model to a dataset, particularly when the data show that there is some curvature or interaction in the relationship between the response and input variables. To achieve the research objective of building a non-linear regression (mathematical) model for complex data with explanatory variables and a residual (error term), the estimated model can be expressed in the well-known form [17; 18; 19]  
\[Y_{t}=f\big{(}X_{1,t},X_{2,t},\cdots,X_{p,t}\big{)}+\varepsilon_{t}, \tag{1}\]  
The parameters \(p=1,\ 2,\cdots,\ P\) and \(t=1,\ 2,\cdots,\ T\) signify the index range, while \(\varepsilon_{t}\) indicates the residual (error term). The collection \(\big{(}X_{1t},X_{2t},\cdots,X_{pt}\big{)}\in\mathbb{R}\) represents the explanatory variables, while \(Y=(Y_{t})\in\mathbb{R}\) denotes the response. In practice, researchers are required to create a fairly precise mathematical model that describes a dataset; for instance, a regression function \(f:\{X\}\rightarrow\{Y\}\) is formed, and its accurate form \(f(\circ)\) is assessed. Due to the challenge of establishing the appropriate form \(f(\circ)\) of the model [20; 21], an error arises from using an incorrect model form, namely the regression function \(f\). This type of error can be denoted by the term \(\delta_{t}\) in a modified version of the \(f_{m}\) model/regression presented in Equation (1):  
\[Y_{t}=f_{m}(X_{1,t},X_{2,t},\cdots,X_{p,t})+\delta_{t}+\varepsilon_{t}, \tag{2}\]  
In order to obtain a reasonably precise mathematical representation of the observed data, the value of \(\delta_{t}\) must be sufficiently small so that the model \(f_{m}(\circ)\) can closely approximate \(f(\circ)\), with \(f\) being any continuous function in the regression model (1). If \(f\) represents a nonparametric function, we obtain a nonparametric model; conversely, if \(f(\circ)\) denotes a parametric function, we have a parametric model that encompasses traditional linear and non-linear models. The quest for the regression function is determined by Relation (3), where \(\beta_{j},\ j=0,\ 1,\cdots,\ k\) represent the regression coefficients. \[Y=g\big{(}X,\ \beta_{0},\beta_{1},\cdots,\beta_{p}\big{)}=E(Y|X) \tag{3}\]  
According to Formula (3), where \(E(Y|X)\) indicates how the conditional mean values of the random variable Y are dependent on the values of the random variable X, it is clearthat multiple options exist for constructing a regression model (various types, such as linear, quadratic, polynomial, hyperbolic, logarithmic, etc.) aligned with (1) and utilising specific mathematical relations [22; 23]. The least squares method is the most appropriate technique to derive the parameters of the regression curve, based on the premise that the sum of the residual squared errors of the model is minimised when estimating the regression function. In scientific papers, many researchers tend to favour a non-linear regression model that provides an estimate of the dependent variable under study, typically represented by a polynomial equation (of degree n) [6; 7]: \[\widetilde{y}=\beta_{0}x_{0}+\sum_{j=1}^{n}\beta_{j}x_{j}+\sum_{ \begin{array}{ l}u,j=1\\ u\
eq j\end{array} }^{n}\beta_{uj}x_{u}x_{j}+\sum_{ \begin{array}{l}u,j=1\\ u\
eq j\end{array} }^{n}\beta_{uj}x_{u}x_{j}+\cdots+\sum_{ \begin{array}{l}u,j= 1\\ u\
eq j\end{array} }^{n}\beta_{uj}x_{u}x_{j}^{n}+\sum_{j=1}^{n}\beta_{ji}x_{j}^{n}\] (4) The following section will describe the most used metrics for assessing model suitability. * R\({}^{2}\)--The coefficient of determination (R\({}^{2}\)) is frequently utilised to assess the quality of linear fit in regression models (5). R\({}^{2}\) represents the square of the correlation between the real variable and the forecasted variable [24], or the fraction of the forecasted variable that the regression model elucidates [25].