## 3 Materials and Methods
### Multivariate Adaptive Regression Splines (MARS)  
In the year 1991, the concept of multivariate adaptive regression splines (MARS), a method for non-parametric regression, was proposed by [PERSON] [48]. This approach aims to model a relationship between a set of predictor variables, represented as \(X\) (\(n\times p\)), and a target variable, denoted as \(y\) (\(n\times 1\)). Where \(n\) is the number of observations or samples, \(p\) is the number of predictor variables or features, \(X\) is the matrix of predictor variables with size \(n\times p\), and \(y\) is the column vector of the target variable with size \(n\times 1\). The MARS model's mathematical representation can be expressed through the equation below:  
\[y=f(X)+e \tag{1}\]  
The term \(f(X)\) represents the prediction or approximation of the target variable \(y\) given the predictor variables \(X\) by the MARS model. The equation also gives us a residual vector, \(e\) (\(n\times 1\)), which forms an integral part of this model. MARS can be considered as a sophisticated extension of the Classification and Regression Trees (CART) [49] technique, but it stands apart from conventional parametric methods. While parametric methods necessitate a predefined functional correlation between predictor and target variables, MARS refrains from such presumptions. Instead, MARS utilizes the predictor variables present in the specified dataset to produce a set of coefficients and piece-wise polynomials that carry the power '\(q\)'. These polynomials are specifically generated to encapsulate the relationship among variables, which may exhibit either a linear or a non-linear nature. These polynomials, referred to as'splines', are seamlessly connected to assemble the complete MARS model. Moreover, the process of fitting these splines involves segmenting the data points of the independent variables into discrete regions, with the knot locations symbolized by '\(t\)'. The power '\(q\)' of these splines essentially determines their linearity or non-linearity. For any variable '\(x\)', the MARS model calculates these splines through applying the particular equations below:  
\[[-(x-t)]_{+}^{q}= \begin{cases}(t-x)^{q},&\text{if $x<t$}\\ 0,&\text{otherwise}\end{cases}  \tag{2}\]  
\[[+(x-t)]_{+}^{q}= \begin{cases}(t-x)^{q},&\text{if $x\geq t$}\\ 0,&\text{otherwise}\end{cases}  \tag{3}\]  
In the given equations, '\(q\)' is not less than zero, and it is a key factor in shaping the smoothness of the ultimate MARS model. In this particular investigation, '\(q\)' is set to 1, signifying the analysis of linear splines. Figure 2 illustrates two splines related to an individual variable '\(x\)', having '\(q\)' set to 1 and a defined knot '\(t\)'. The left spline provides positive outcomes, while the right spline produces zero for '\(x\)' values on the left side of the knot. On the flip side, when '\(x\)' values lie on the right side of the knot, the right spline gives positive results, and the left spline is zero. These splines, also called as the foundation functions for '\(x\)', play a significant role in the MARS technique. The MARS model related to the aim variable '\(y\)' is formed via including a series of '\(M\)' basis functions, as demonstrated in the following equation:  
\[\hat{y}=\hat{f}_{M}(x)=a_{o}+\sum_{m=1}^{M}a_{m}B_{m}(x) \tag{4}\]  
Equation (4) depicts \(\hat{y}\) as the target variable, which the MARS model predicts. The term '\(a_{o}\)' signifies the intercept term, which is a fixed constant. '\(B_{m}(x)\)' denotes the \(m\)th basis function, which could be an individual basis function or a combination of multiple basis functions, while '\(a_{m}\)' symbolizes the coefficient that corresponds to the \(m\)th basis function. The estimation of these coefficients is achieved via the least squares method. The MARS model, as represented by Equation (4), is formulated using a biphasic methodology that encompasses a forward pass and a backward pass.