## 2 Theory and Methodology
### Artificial Neural Network (ANN)
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**ANN Model Parameter** & **Model 1** & **Model 2** & **Model 3** \\ \hline Number of layers & 3 & 3 & 3 \\ Number of neurons (per layer) & 200 & 200 & 200 \\ Activation function & ReLU & ReLU & ReLU \\ Optimizer & Adam * & [PERSON] & [PERSON] \\ Loss function & MSE ** & MSE & MSE \\ Input variables & \(V\), \(\beta\), \(S_{e}(\omega_{e})\) & \(\ast\ast\), \(V\), \(\beta\), \(S_{e}(\omega_{e})\) & \(V\), \(\beta\), \(S_{e}(\omega_{e})\) \\ Output variables & \(S(\omega)\) & \(S(\omega)\) & \(S(\omega)\) \\ Total dataset & 21,060 **** & 231,660 & 231,660 \\ Remark & \(\gamma=2\) (fixed) & \(\gamma=\) range of 1–3 & \(\gamma=\) range of 1–3 \\ \hline \hline \end{tabular}  
* Learning rate: 0.001. ** Mean Squared Error. *** It is assumed that _Hs_ is known with \(Hs=4\sqrt{m_{0}}\), where \(m_{0}\) is the zeroth moment of the encounter-frequency spectrum. *** The amount of data for training is less than those of Models 2 and 3 as the peak enhancement factor is fixed for initial testing.  
\end{table} 
Table 2: Artificial neural network (ANN) model parameters. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model** & **R2 Score** & **NRMSE Mean * & **Layer** & **Nodes** & **Optimizer** & **Loss Function** & **Epochs** \\ \hline v.0 & 0.984 & 0.71\% & 3 & 50 & adam & MSE & 25 \\ v.0.1 & 0.985 & 0.70\% & 3 & 100 & adam & MSE & 25 \\ v.0.1.1 & 0.989 & 0.54\% & 3 & 100 & adam & MSE & 50 \\ v.0.2 & 0.989 & 0.52\% & 3 & 150 & adam & MSE & 25 \\ v.0.2.1 & 0.987 & 0.64\% & 3 & 150 & adam & MSE & 50 \\ v.0.2.2 & 0.988 & 0. The model errors for each estimated spectrum \(\hat{S}\) are defined by  
\[MAE\big{(}S,\hat{S}\big{)}=\frac{1}{n}{\sum_{i=0}^{n-1}}\big{|}S_{i}-\hat{S}_{i} \big{|}\ (\text{Mean Absolute Error}), \tag{12}\]  
\[MSE\big{(}S,\hat{S}\big{)}=\frac{1}{n}{\sum_{i=0}^{n-1}}\big{(}S_{i}-\hat{S}_{i }\big{)}^{2}\ (\text{Mean Squared Error}), \tag{13}\]  
\[RMSE=\sqrt{MSE\big{(}S,\hat{S}\big{)}}\ (\text{Root}-\text{Mean}-\text{Squared Error}), \tag{14}\]  
where \(n\) represents the number of discretization points in the frequency domain. The NRMSE is normalized with the maximum spectral density of the original wave:  
\[NRMSE=\frac{RMSE}{\text{max}(S_{i})} \tag{15}\]  
The R2 score, or coefficient of determination, is a common metric used to evaluate the overall performance of the ML models and is defined as  
\[R^{2}=1-\frac{\sum_{i=1}^{m}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}^{m}(y_{i}- \overline{y_{i}})^{2}}, \tag{16}\]  
where \(y_{i}\) is the actual value, \(\hat{y_{i}}\) is the predicted value, \(\overline{y_{i}}\) is the mean of actual values, and \(m\) is the number of data points. The v.0.3.1 hyperparameter configuration, consisting of three hidden layers with 200 nodes each, was selected based on its highest R2 score and lowest mean NRMSE among the tested variations, while also offering a relatively simple structure to reduce training time. Additionally, feature-wise data normalization was applied across all models to improve training stability and enhance predictive performance. Min-max normalization was applied to both input and output features using _MinMaxScaler_, scaling values to the range [0,1]. This helped improve training stability and ensured consistent feature scaling across all models.