## 2 Material and Method
### Ridge Regression Method  
In this study, the ridge regression (RR) model was employed to estimate the frequencies obtained from the FE model more efficiently. The selection of ridge regression as the primary methodology in this study was based on its superior performance in preliminary analyses. When compared to alternative regression methods including linear regression, Lasso regression, and ElasticNet, ridge regression demonstrated the highest prediction accuracy while maintaining computational efficiency. This was particularly crucial given our large dataset of 3240 finite element models. Moreover, ridge regression's ability to effectively handle multicollinearity among geometric parameters and generate practical prediction equations made it the most suitable choice for achieving the study's objectives. RR is a form of linear regression specifically designed to address issues of multicollinearity. High correlations among independent variables due to multicollinearity can compromise the performance and accuracy of traditional linear regression models. Ridge regression mitigates this problem [34] ([PERSON] and [PERSON], 1988). It is regarded as a more reliable and realistic regression method because it reduces the bias associated with the least squares method when estimating regression coefficients. While the least squares method treats variables equally, the relationships among them can vary significantly due to multicollinearity, leading to notable discrepancies between observed and actual values. RR demonstrates greater robustness than the least squares method when dealing with implausible data [35] ([PERSON], 2012). Furthermore, when multicollinearity exists among independent variables, the ordinary least squares (OLS) regression method becomes less effective for calculating parameter estimates. In light of this, [PERSON] introduced the ridge regression estimation in 1962 ([PERSON] et al., 2023).  
Ridge regression (RR) has emerged as a valuable technique for analyzing multiple regression data that exhibit multicollinearity. By introducing a certain degree of bias into the regression estimates, RR effectively reduces standard errors, leading to more accurate estimates of the regression coefficients compared to the ordinary least squares method. The RR parameter is obtained by minimizing the sum of squared errors, which incorporates a constraint on the coefficients, encouraging them to approach zero. This method penalizes features with larger coefficients by adding a penalty that is proportional to the square of the coefficients' magnitudes, thereby lessening the cost function ([PERSON], 2021). The penalty term employed in RR is referred to as the L2 norm, mathematically represented by the following equation 1 ([PERSON] and [PERSON], 1970):  
\[\text{Cost Function}=\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}+\lambda\sum_{j=1}^{ P}\hat{\beta}_{j}^{2} \tag{1}\]  
where \(\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}\), Residual Sum of Squared Errors. \(\lambda\) (lambda): regularization parameter. This parameter controls how much the model is penalized. \(\beta\)_j: They are the coefficients of the independent variables of the model. This formula reveals the main difference in ridge regression: as the regularization parameter \(\lambda\) increases, the coefficients become smaller, which makes the model more generalizable ([PERSON], 2020).