## 4 Brief Description of the Learning Models
### Performance Evaluation of the ML Models  
Different performance metrics of the ML models are investigated to validate the prediction of the resonance frequency with respect to the simulated and the measured resonance frequency. In this research work, the mean absolute error (MAE), the mean squared error (MSE), the root mean squared error (RMSE), and the variance score are observed as performance indicators of the ML regression models. Predicting errors are computed, and prediction models are assessed using these metrics. The mean absolute error, mean squared error, and root-mean-squared error are well-known scale-dependent measurements based on absolute and squared values [44,45]. The MAE considers the average amount of error over a group of projections but ignores the direction of the mistake, giving less weight to extreme predictions [46]. In [47], the MAE is mathematically formulated as:  
\[MAE=\frac{1}{n}\sum_{i=1}^{n}\left|Pi-Oi\right| \tag{8}\]  
where \(n\) = number of errors and \(\left|Pi-Oi\right|\) = Absolute error. The MSE is a common statistic used in estimation. The root-mean-squared error (\(RMSE\)) is used in place of MSE by taking the square root. The root-mean-squared error (\(RMSE\)) quantifies how far estimates deviate from reality. The definition of \(RMSE\) is [48]:  
\[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(Pi-Oi\right)^{2}} \tag{9}\]  
Another performance indicator is the explained variance score that reflects the error scatter in a data collection. It can be defined as [49]:  
\[explained\ variance(y,\hat{y})=1-\frac{Var(y-\hat{y})}{Var(y)} \tag{10}\]  
Figure 12 shows that a total of 86 samples were used to predict the resonance frequency of the designed MPA. Among the 86 samples, 68 samples were used to train the models and 18 samples were used to test the models' performance. The number of training samples and the number of sampling data determines the accuracy of the prediction model. The 86 data samples were generated using CST simulation software by varying the different dimensional parameters of the proposed MPA. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & **Simulated** & **Predicted** & **Error** & \multirow{2}{*}{**No.**} & **Simulated** & **Predicted** & **Error** \\  & **Frequency** & **Frequency** & **Percentage** & & **(GHz)** & **Frequency** & **Frequency** & **Percentage** \\  & **(GHz)** & **(GHz)** & **(\%)** & & **(GHz)** & **(GHz)** & **(\%)** \\ \hline
1 & 8.1414 & 8.6086 & 5.73869 & 10 & 9.064 & 9.222 & 1.74353 \\ \hline
2 & 9.0396 & 9.225 & 2.05042 & 11 & 9.0296 & 9.6116 & 6.44494 \\ \hline
3 & 9.1176 & 9.2757 & 1.73436 & 12 & 8 & 8.7789 & 9.7363 \\ \hline
4 & 9.0507 & 9.265 & 2.36726 & 13 & 11.81 & 10.3787 & 12.11948 \\ \hline
5 & 7.2623 & 8.7624 & 20.65657 & 14 & 9.1413 & 9.2286 & 0.95446 \\ \hline
6 & 9.351 & 9.7752 & 4.53594 & 15 & 11.734 & 10.8604 & 7.44519 \\ \hline
7 & 11.988 & 10.3457 & 13.69946 & 16 & 11.649 & 9.7128 & 16.62076 \\ \hline
8 & 8.1487 & 8.8375 & 8.45233 & 17 & 9 & 9.2611 & 2.90101 \\ \hline
9 & 11.865 & 10.2844 & 13.32168 & 18 & 9.0639 & 9.8493 & 8.6656 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Simulated and predicted resonant frequency comparison of the test set using CNN. Figure 13 shows a heat map plot (correlation matrix) which represents each of the ten attributes of antenna design parameters named Lp, Ls, Lp, Wp, h, Ws, Mtp, Mt, S11, and frequency on the y- and x-axis of the plot. These values represent the input dimensions of the proposed antenna structure, as shown in Table 1 and Figure 2 in the previous section. The maximum correlation occurs at the center of the plot with a value of 1.00. As the point moves away from the maximum line (1.00), the correlation decreases accordingly. In this mapping plot, the correlation value ranges from 0.03 to 0.72. Figure 14 shows the simulated vs. predicted frequency of test data samples using DTR and CNN, respectively, for 18 test samples. In the analysis, the frequency tuning ranges from 7 GHz to 12 GHz.