## 2 Material and Method
### Model Evaluation Metrics
Additionally, moderate-to-strong positive correlations are evident between 'bid' and 'bod' (r \(\approx\) 0.63), as well as between 'bid' and '\(\prime\)' (r ranging from approximately 0.55 to 0.75, with stronger correlations in the initial modes). These findings suggest that the variables 'bid', 'bod', 'tid', and 'tod' are interrelated. Conversely, an examination of the lower correlations among the variables reveals the following:
* The variable 'elastic' displays correlations that are nearly zero with most other variables (specifically with 'height', 'bid', 'bod', 'tid', and 'tod') across all modes. A very weak positive correlation is noted only with the 'f' values (r \(\approx\) 0.08 to 0.14), indicating that the property 'elastic' is largely linearly independent of the other parameters under consideration. * The 'height' variable also lacks a significant linear relationship with the group comprising 'bid', 'bod', 'tid', and 'tod', as the correlations remain close to zero. While the overall patterns of correlation are predominantly consistent across modes, minor variations in the strength of certain relationships do exist. The most pronounced alteration is seen in the strengthening of the negative correlation between 'height' and 'f' \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **Mean** & **Std.** & **Min** & **25\%** & **50\%** & **75\%** & **Max** \\ \hline Elastic (Gpa) & 33.67 & 2.358 & 30 & 32 & 34 & 36 & 37 \\ \hline Height (m) & 10.92 & 1.14 & 9.3 & 10.11 & 10.92 & 11.73 & 12.54 \\ \hline Bid (m) & 0.164 & 0.044 & 0.094 & 0.131 & 0.156 & 0.2 & 0.25 \\ \hline Tid (m) & 0.089 & 0.032 & 0.041 & 0.064 & 0.082 & 0.11 & 0.179 \\ \hline Bod (m) & 0.082 & 0.014 & 0.062 & 0.075 & 0.087 & 0.1 & 0.1 \\ \hline Tod (m) & 0.044 & 0.014 & 0.021 & 0.033 & 0.045 & 0.054 & 0.072 \\ \hline F1–2 (Hz) & 2.70 & 1.16 & 0.73 & 1.83 & 2.49 & 3.35 & 7.73 \\ \hline F3–4 (Hz) & 11.47 & 3.86 & 4.31 & 8.57 & 10.87 & 13.80 & 25.50 \\ \hline F5 (Hz) & 28.56 & 9.46 & 10.55 & 21.46 & 27.05 & 34.20 & 66.29 \\ \hline F6–7 (Hz) & 53.13 & 16.90 & 19.80 & 40.39 & 50.71 & 63.67 & 116.22 \\ \hline F8 (Hz) & 78.44 & 21.20 & 32.07 & 62.48 & 76.15 & 92.42 & 152.90 \\ \hline F9 (Hz) & 84.52 & 25.23 & 32.07 & 65.19 & 81.72 & 101.51 & 171.17 \\ \hline F10 (Hz) & 102.54 & 25.80 & 47.32 & 83.31 & 99.50 & 119.41 & 198.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Statistical properties of the data. Alongside the analysis results, the performance of the models was assessed using both actual and estimated values. The graphs presented in Figure 10 illustrate that as prediction success increases, the data approaches the red line. Overall, prediction success is found to be high across all modes, with Mode 5 exhibiting the highest level of prediction success. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Ridge Regression (RR)**} & \multicolumn{4}{c}{**Lasso Regression (LR)**} & \multicolumn{4}{c}{**ElasticNet Regression (ENR)**} \\ \hline  & **MAPE (\%)** & **MAE** & **RMSE** & **R\({}^{2}\)** & **MAPE (\%)** & **MAE** & **RMSE** & **R\({}^{2}\)** & **MAPE (\%)** & **MAE** & **RMSE** & **R\({}^{2}\)** \\ \hline F1-2 & 8.42 & 0.1933 & 0.2655 & 0.9496 & 9.19 & 0.2327 & 0.3201 & 0.9266 & 17.03 & 0.4334 & 0.5803 & 0.7590 \\ \hline F3-4 & 5.54 & 0.5692 & 0.7468 & 0.9644 & 5.76 & 0.6062 & 0.7990 & 0.9593 & 9.07 & 1.0359 & 1.3827 & 0.8780 \\ \hline F5 & 5.37 & 1.3694 & 1.7801 & 0.9661 & 5.38 & 1.3833 & 1.7989 & 0.9653 & 8.86 & 2.5077 & 3.3262 & 0.8815 \\ \hline F6–7 & 5.13 & 2.5626 & 3.5479 & 0.9583 & 5.14 & 2.5740 & 3.5555 & 0.9581 & 9.14 & 4.7606 & 6.1872 & 0.8731 \\ \hline F8 & 7.74 & 5.9556 & 7.3659 & 0.8887 & 7.80 & 6.0140 & 7.4239 & 0.8869 & 10.38 & 7.8779 & 9.7891 & 0.8034 \\ \hline F9 & 5.24 & 4.3827 & 6.0895 & 0.9451 & 5.23 & 4.3767 & 6.0984 & 0.9449 & 9.30 & 7.5007 & 9.5521 & 0.8650 \\ \hline F10 & 6.44 & 6.4751 & 7.9178 & 0.9070 & 6.44 & 6.4704 & 7.9165 & 0.9070 & 7.92 & 7.9368 & 10.0789 & 0.8493 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Regression analysis results.