## 6 Discussion
### Comparative Analysis of the Effectiveness of VMD in Magnitude Prediction  
This paper uses PA, FAR, and MR as criteria for evaluating the optimal model. When applying VMD to earthquake prediction, we found that this method significantly improves the prediction trends and evaluation metrics for earthquakes of magnitude _M_s 5.0 and above compared to direct prediction without VMD, and is better at learning the patterns between magnitudes.  
In addition, we compared the overall error between methods using VMD and those not using VMD. The experimental results (see Tables 4 and 5) show that although traditional error metrics such as MSE, RMSE, and MAE did not show significant improvement with VMD compared to direct prediction, the VMD method exhibits a clear advantage in prediction accuracy.  
Observations of predictions without VMD reveal that the predicted values fluctuate within a narrow range (see Figure 15), indicating that the model has limited learning capabilities regarding the data. In contrast, predictions made after VMD show a wider range. Although there was no significant improvement in evaluation metrics, VMD better predicts the trends in magnitude changes and is closer to the actual magnitude variations. This suggests that VMD is effective in reducing noise and extracting key features, enabling the model to better capture the inherent patterns in the data, thereby improving the reliability and stability of the predictions.  
Figure 15: Prediction result chart without VMD.  
In addition, traditional error metrics have certain limitations when applied to earthquake magnitude prediction:  
1. Sensitivity to Outliers: MSE and RMSE are very sensitive to outliers. The distribution of earthquake magnitudes is often highly irregular and volatile, with a few large-magnitude events potentially causing significant increases in these error metrics. For example, if a predicted value deviates greatly from the actual value, it will substantially increase the overall error.  
2. MAE Limitations: MAE applies linear penalties to all error values, which may not fully reflect the model's performance in handling large-magnitude events [47; 48].  
In the prediction results of this study, when the time window size was 12, the ATT-BiLSTM model achieved the best prediction performance. However, its MSE, RMSE, and MAE were not the lowest. This is because, without VMD, the model's learning ability was poor, leading to predictions fluctuating within a narrow range, which made the errors relatively averaged. Our model, on the other hand, learned some of the underlying patterns, which was initially reflected in its predictions. Additionally, we observed that adjacent data in Mode 1 sometimes had strong correlations. When there is a large and infrequent difference between adjacent earthquake magnitudes, the model struggles to learn these patterns, leading to a certain lag in predictions and thus increased numerical errors. Such data points act as \"outliers\" in the calculation of these metrics, explaining why there was no significant improvement in MSE, RMSE, and MAE. We also attempted methods such as differencing, moving averages, and filtering to mitigate this effect, but did not achieve better results.