## 4 Validation and Comparison
### Optimization and Selection of the Best Neural Network Ensemble  
To ensure reliable model evaluation, we divide the high-precision dataset into three subsets: the training set (80%), the validation set (10%), and the test set (10%). All data is normalized using min-max normalization, mapping input and output variables to the range [0, 1]. The mean and standard deviation of each feature were calculated across all sets to confirm consistent distributions. Additionally, a K-S test was performed, and the \(p\)-values were found to be greater than 0.05, indicating no significant differences between the sets and validating the balanced data split. In addition, K-Fold Cross-Validation is applied to enhance generalization assessment. The training set is split into K non-overlapping subsets, with one subset used for validation and the remaining K-1 subsets for training. This process is repeated K times, and the final evaluation metric is obtained by averaging the results across all iterations. This approach reduces bias from a single dataset split and provides a more robust performance assessment. Figure 8: Flowchart of the WOA-optimized ensemble model method. This study sets K to 5 and selects the model that achieves the best performance across evaluation metrics. During training, key strategies are implemented to ensure efficient and accurate convergence. The optimizer is used with an adaptive learning rate, starting at 0.01, which adjusts automatically to improve convergence and avoid local optima. Early stopping is applied, terminating training if the validation loss does not decrease for five consecutive epochs, preventing overfitting while maintaining computational efficiency. To further enhance generalization, L2 regularization (weight decay) with a coefficient of 0.0001 is introduced to penalize large weights, promoting smoother weight distribution. The choice of activation function plays a crucial role in the model's learning ability, influencing convergence speed and accuracy. Comparative experiments on the proposed model show that while ReLU helps mitigate the vanishing gradient problem and accelerates early convergence, tanh ultimately delivers superior overall performance. Specifically, compared to ReLU, tanh reduces RMSE and MAE by 0.44 dB and 0.43 dB, respectively, lowers MAPE by 0.43%, and improves R\({}^{2}\) by 0.08. Based on these findings, tanh is selected as the activation function for this study. After performing cross-validation, the model is further refined by exploring different network architectures. The experiments compared the performance of models with one, two, and three hidden layers, revealing that while adding a second hidden layer improved performance, further layers led to diminishing returns and overfitting. Therefore, a two-hidden-layer architecture was selected for the final model. Based on empirical guidelines [36], each hidden layer consists of a maximum of 12 neurons. This results in 144 distinct neural network configurations generated through various combinations of hyperparameters. These models are trained and evaluated on the model selection and ensemble optimization set to identify the best-performing architecture. The performance distribution of these 144 models is shown in Figure 9.  
The number of neurons in the hidden layers plays a crucial role in determining the model's complexity and learning capacity. More neurons allow the model to capture intricate features but increase the risk of overfitting, while fewer neurons may lead to insufficient feature learning, affecting prediction accuracy. This trade-off explains the variations in MAPE performance observed in Figure 9.  
Figure 9: MAPE performance of 144 models with varying hidden layer neuron counts. As shown in Table 3, the performance of the neural network ensemble model under different hidden layer configurations is presented in detail. While the number of neurons does affect model performance, increasing the number of neurons does not always lead to improved performance.