## 2 Theoretical Background of Applied Method
### Multivariate Adaptive Regression Splines  
Multivariate adaptive regression splines (MARS) is a non-parametric regression technique that looks like an extension of linear models. This method was proposed by [PERSON] [47] and extensively discussed in [48; 49; 50; 51; 52; 53]. Endpoint BOS temperature prediction from static data using the MARS model has recently been investigated by [PERSON] et al. [54]. This method aims to find the dependency of variables \(y_{i}\) on one or more independent variables \(\mathbf{x}_{i}\). The relationship between \(y_{i}\) and \(\mathbf{x}_{i}\) (\(i=1,...,N\)) from the data tensor (1) or (2) can be represented as:  
\[y_{i}=f(x_{i}^{1},x_{i}^{2},...,x_{i}^{p})+\varepsilon=f(\mathbf{x}_{i})+\varepsilon, \tag{3}\]  
where \(f\) is an unknown single valued deterministic function that captures the joint predictive relationship of \(y_{i}\) on \((x_{i}^{1},x_{i}^{2},...,x_{i}^{p})\) and \(\varepsilon\) is an error. The additive stochastic component \(\varepsilon\), for which the expected value is determined to be zero, usually reflects the dependence of \(y_{i}\) on values other than \((x_{i}^{1},x_{i}^{2},...,x_{i}^{p})\), that is, neither controlled nor observed. In the multidimensional case, the goal is to form _reflected pairs_ for each input component \(x^{j}\) of the vector \(\mathbf{x}=(x^{1},...,x^{j},...,x^{p})^{T}\) with knots at each observed value \(x_{i}^{j}\) of that input \((i=1,2,...,N;j=1,2,...,p)\). In MARS, the regression function called basis function (BF) is approximated by smoothing splines for a general representation of data in each subgroup. The BF is unique between any two knots and is shifted to another BF at each knot [47]. The MARS model for the prediction of melt temperature and carbon content can be expressed by the following equation [52]:  
\[y_{T/C}=\hat{f}(\mathbf{x})+\varepsilon=c_{0}+\sum_{m=1}^{M}c_{m}B_{m}( \mathbf{x})+\varepsilon, \tag{4}\]  
where \(y_{T/C}\) represents the output variable that is predicted (i.e., temperature or carbon), \(\mathbf{x}\) is the vector of input variables, i.e., observations from the BOS process, \(M\) is the number of BFs in the model (i.e., number of spline functions), \(c_{0}\) is the coefficient of the constant BF \(B_{0}\), and the sum is over the BFs \(B_{m}\) produced by an algorithm that implements the stepwise forward part of the MARS strategy by incorporating the modification to recursive partitioning. The coefficients \(c_{m}\) are estimated by minimizing the residual sum-of-squares. \(B_{m}(\mathbf{x})\) is the \(m\)-th function in a set of constructed BFs, or a product of two or more such functions, and parameter \(\varepsilon\) is the additive stochastic component. This paper compared the piecewise-linear and piecewise-cubic types of the MARS model for BOS modeling and predictions. In this paper, the open-source implementation of MARS proposed by [PERSON] [55] was applied to the BOS process. The piecewise-linear MARS model uses the \(\max(0,\,x-t)\) function, where \(t\) is the univariate knot, which is selected for each of the factor variables \(x\). The \(\max(. )\) function represents the positive part of \((0,x-t)\), which can be formally expressed as the following:  
\[\max(0,x-t)=\left\{ \begin{array}{ll}x-t,&\text{ if }\quad x\geq t\\ 0,&\text{ otherwise}\end{array} \right. \tag{5}\]  
The cubic BF has the following form [56]:  
\[C(x|s=+1,t_{-},t,t_{+})=\left\{ \begin{array}{ll}0,&x\leq t_{-}\\ \alpha_{+}(x-t_{-})^{2}+\beta_{+}(x-t_{-})^{3},&t_{-}<x<t_{+}\\ x-t,&x\geq t_{+}\end{array} \right. \tag{6}\]  
data modeling. In the working settings of SVR, the training data were standardized, i.e., the software centers and scales each column of the predictor data \(\mathbf{x}\) by the weighted column mean and standard deviation, respectively. In addition, the software divides all elements of the predictor matrix \(\mathbf{x}\) by the appropriate scale factor that was found using a heuristic procedure. Sequential minimal optimization (SMO) was used as an algorithm for solving quadratic programming (QP).