## 2 Material and Method
### Model Evaluation Metrics
Figure 9: Correlation matrices of data according to modes. Figure 10: Actual and predicted values graph for each mode. Figure 11 displays the prediction errors. For optimal prediction success, it is essential that prediction errors remain minimal. Therefore, values close to zero indicate that predictions are made with minimal error. The figure reveals that prediction errors are notably high in Modes 8-9 and Mode 10, while errors in the other modes are relatively lower. The equations derived for estimating frequencies as a result of the RR analyses (Equations (6)-(11)) are presented below. It is important to consider the success and error rates of the estimations made for each equation in relation to the values mentioned previously. These equations represent regression models that predict the value of the dependent variable (f) across various modes, based on a set of independent variables (elastic, height, bid, tid, bod, and tod). Each equation delineates the quantitative relationship between these variables for a specific mode or group of modes. \[ \begin{array}{c}f_{1,2}=2.47+(0.16*elastic)+(-0.86*height)+(1.50*\text{ bid})\\ +(-0.70*\text{tid})+(0.01*\text{bod})+(0.23*\text{tod})\end{array}  \tag{6}\]  
\[ \begin{array}{c}f_{3,4}=10.66+(0.68*elastic)+(-3.72*height*)+(3.54*\text{ bid})\\ +(0.30*\text{tid})+(0.20*\text{bod})+(0.83*\text{tod})\end{array}  \tag{7}\]  
\[ \begin{array}{c}f_{5}=26.69+(1.69*elastic)+(-9.26*height)+(6.29*\text{ bid})+(3.60*\text{tid})\\ +(0.69*\text{bod})+(1.78*\text{tod})\end{array}  \tag{8}\]  
\[ \begin{array}{c}f_{6,7}=49.79+(3.15*elastic)+(-16.57*height)+(10.88*\text{ bid})\\ +(6.04*\text{tid})+(0.56*\text{bod})+(4.10*\text{tod})\end{array}  \tag{9}\]  
\[ \begin{array}{c}f_{8}=74.96+(4.53*elastic)+(-21.23*height)+(17.79*\text{ bid})\\ +(-4.85*\text{tid})+(-0.39*\text{bod})+(8.28*\text{tod})\end{array}  \tag{10}\]  
\[ \begin{array}{c}f_{9}=142.75+(1.26*elastic)+(-15.58*height)+(229.84*\text{ bid})\\ +(161.20*\text{tid})+(23.92*\text{bod})+(349.26*\text{tod})\end{array}  \tag{11}\]  
\[ \begin{array}{c}f_{10}=98.03+(6.15*elastic)+(-27.27*height)+(20.67*\text{ bid})\\ +(0.19*\text{tid})+(3.73*\text{bod})+(0.17*\text{tod})\end{array}  \tag{12}\]  
Figure 11: _Cont._  
Figure 11: Prediction errors plot for each mode. When these equations are evaluated, the following is considered:  
**Variable Height:** The coefficient for this variable is negative across all modes, with its absolute value increasing significantly as the mode number rises (from \(-0.86\) to \(-27.27\)). This indicates that the function consistently decreases with an increase in height, and this diminishing effect becomes increasingly pronounced in higher modes. **Variable Elastic:** This variable has a positive coefficient in every mode, with the coefficient rising notably as the mode number increases (from 0.16 to 6.15). This suggests that an increase in the elastic parameter leads to an increase in, with this effect becoming more substantial in higher modes. **Variable Bid:** Likewise, the variable bid features positive coefficients in all modes, with these coefficients escalating as the modes progress (from 1.50 to 20.67). This illustrates that a larger bid size results in an increase in, and the effect intensifies in higher modes. **Variables Tid and Bod:** The effects of these variables are more nuanced. The coefficient for tid is negative in certain modes (for example, \(-0.70\) in Equation (6) and \(-4.85\) in Equation (10)) but positive in others (such as 3.60 in Equation (8) and 6.04 in Equation (9)). This variability indicates that the influence of tid on can shift direction depending on the active mode. Similarly, the coefficient of bod can change sign (e.g., \(-0.39\) in Equation (10) and \(+3.73\) in Equation (11)), and its magnitude also varies. **Variable Tod:** Generally, rod tends to have a positive coefficient, with its magnitude increasing as the modes progress (for instance, from 0.23 in Equation (6) to 8.28 in Equation (10)). However, it diminishes again in Equation (11) (to 0.17).