## 3 Procedure for Estimation of Optimal Analysis Parameters
### Multiple Nonlinear Regression  
A multiple nonlinear regression model is constructed to mathematically explain the relationship between analysis parameters and output data such as melt pool size obtained from thermal analysis results. A linear regression model including only the first-order  
Figure 5: Procedure for estimation of optimal thermal analysis parameters.  
Figure 6: Box–Behnken design.  
terms has limited accuracy in generating a response surface for complex systems such as the thermal analysis model for LPBF process simulation. Therefore, in this paper, the accuracy of the regression model is increased by adding cross-terms and quadratic terms. Response variables (output variables) \(Y_{1},\ Y_{2},\ \ldots,\ Y_{n}\) and regressors (input variables) \(X_{1},\ X_{2},\ \ldots,\ X_{p}\) have the following expression of relation.  
\[\mathbf{y}=\mathbf{\beta}\mathbf{X}+\mathbf{\epsilon} \tag{9}\]  
\[\mathbf{y}=\left\{\begin{array}{c}Y_{1}\\ Y_{2}\\ \vdots\\ Y_{n}\end{array}\right\}\text{, }\mathbf{\beta}=\left\{\begin{array}{c}\beta_{1} ^{\prime}\\ \beta_{2}^{\prime}\\ \vdots\\ \beta_{n}^{\prime}\end{array}\right\}=\left\{\begin{array}{c}\beta_{10}& \beta_{11}&\beta_{12}&\ldots&\beta_{1,p+1}&\beta_{1,p+2}&\ldots&\beta_{12,p+ \frac{2}{p(p-1)}}\\ \beta_{20}&\beta_{21}&\beta_{22}&\ldots&\beta_{2,p+1}&\beta_{2,p+2}&\ldots& \beta_{2,2p+\frac{2}{p(p-1)}}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ \beta_{n0}&\beta_{n1}&\beta_{n2}&\ldots&\beta_{n,p+1}&\beta_{n,p+2}&\ldots& \beta_{n2,p+\frac{2}{p(p-1)}}\end{array}\right\}\text{, }\mathbf{X}=\left\{\begin{array}{c}1\\ X_{1}\\ X_{2}\\ \vdots\\ X_{2}^{\prime}\\ \vdots\\ X_{n}\end{array}\right\}\text{, }\mathbf{\epsilon}=\left\{\begin{array}{c} \epsilon_{1}\\ \epsilon_{2}\\ \vdots\\ \epsilon_{n}\end{array}\right\}\]  
In this research, \(\mathbf{\beta}\) represents the sensitivity of the thermal analysis parameters (input variables) influencing output variables such as the melt pool size. As shown in Equation (10), to determine \(\mathbf{\beta}\), the method of least squares is used, and LASSO (Least Absolute Shrinkage and Selection Operator) regularization is applied to solve the overfitting problem [34]. However, since underfitting may occur with an inappropriate, the method of least squares is used, and LASSO (Least Absolute Shrinkage and Selection Operator) regularization is applied to solve the overfitting problem [34]. However, since underfitting may occur with an inappropriate \(\lambda\), k-fold cross-validation is performed to set an appropriate \(\lambda\)[34].  
\[\mathbf{S}=\sum_{i=1}^{n}\mathbf{e}_{i}^{2}=\mathbf{e}^{\prime}\mathbf{\epsilon}=(\mathbf{y}-\mathbf{ \beta}\mathbf{X})^{\prime}(\mathbf{y}-\mathbf{\beta}\mathbf{X})+\lambda\left\|\mathbf{\beta} \right\|_{1} \tag{10}\]  
In this way, a regression model is constructed using the thermal analysis data for the parameter cases obtained from the DOE, and by analyzing the sensitivity \(\mathbf{\beta}\), it is possible to determine which thermal analysis parameters have a significant effect on the melt pool size and how large the effect is. The details of the regression model constructed in this study are described in Appendix A.