## 5 Results  
### Optimal Parameters for MCDO NNs  
A grid search was conducted to determine the optimal configurations for MCDO neural networks, evaluating the number of hidden layers, activation functions, dropout rates, epochs, and batch sizes.  
For Model 1 (constant amplitude with trend), the best-performing neural network consists of five hidden layers with 240, 180, 120, 60, and 30 nodes, using the ReLU activation function and dropout rates of 0.6, 0.5, 0.4, 0.3, and 0.2. It was trained for 50 epochs with abatch size of 16. This configuration achieved an RMSE of 1.5049 and an \(R^{2}\) of 0.6331, as shown in Figure 3.  
For Model 2 (varying amplitude with trend), the top-performing architecture follows the same structure as Model 1. This configuration resulted in an RMSE of 2.0204 and an \(R^{2}\) of 0.9528, demonstrating its ability to capture the increasing amplitude variations effectively. The results for Model 2 are depicted in Figure 4.  
For Model 3 (mixed frequency with trend), the optimal configuration features three hidden layers with 120, 60, and 30 nodes, trained for 100 epochs with dropout rates of 0.4, 0.3, and 0.2. The batch size is set to 16, and this setup achieves an RMSE of 1.8352 and an \(R^{2}\) of 0.5340, indicating a reasonable performance in capturing multiple overlapping periodic components. The results for this configuration are presented in Figure 5.  
Figure 4: Lowest 15 RMSEs for MCDO configurations for Model 2.  
Figure 5: Lowest 15 RMSEs for MCDO configurations for Model 3.  
Figure 3: Lowest 15 RMSEs for MCDO configurations for Model 1.  
For Model 5 (nonlinear modulation with trend), the network shares the same architecture as Model 3 but undergoes training for 50 epochs instead of 100. This model achieves an RMSE of 1.7906 and an \(R^{2}\) of 0.8562, indicating improved accuracy in forecasting time series with evolving phase shifts. The results are shown in Figure 6.  
For Model 5 (nonlinear modulation with trend), the best-performing neural network consists of four hidden layers, utilizing the ReLU activation function and trained for 100 epochs with a batch size of 64. This configuration achieves an RMSE of 1.4387 and an \(R^{2}\) of 0.8512, suggesting strong predictive performance in scenarios with nonlinear amplitude modulations. The results for this setup are illustrated in Figure 7.  
For Model 6 (noise variability with trend), the optimal neural network consists of two hidden layers with 60 and 30 nodes, trained for 100 epochs with a batch size of 16, using the ReLU activation function. This model attains an RMSE of 1.8831 and an \(R^{2}\) of 0.5969, reflecting moderate predictive accuracy given the added uncertainty from noise fluctuations. The results for this model are displayed in Figure 8.  
The range of optimal configurations shows how well MCDO neural networks can adapt to different data characteristics. The depth and size of the networks, along with choices like dropout rates, batch sizes, and training epochs, are fine-tuned to handle varying levels of trend complexity, amplitude changes, frequency shifts, phase adjustments, nonlinearity, and noise. This flexibility ensures the networks are capable of capturing the unique patterns in each dataset. The grid search process is key here, as it identifies the best setup for each model.  
Figure 6: Lowest 15 RMSEs for MCDO configurations for Model 4.  
Figure 7: Lowest 15 RMSEs for MCDO configurations for Model 5.