## 4 Empirical Results
### Key Results  
In this section, we conduct VIX prediction using the proposed model and benchmark models, presenting the results accordingly. First, Figure 7 visualizes the prediction results of the proposed model, further demonstrating its ability to capture market volatility patterns effectively. The figure highlights that the VMD-Cascaded LSTM with Attention model closely follows actual VIX movements, indicating its robustness in capturing complex financial time-series dynamics. The combination of VMD, cascaded LSTM, and Attention allows the model to extract relevant features while mitigating noise, leading to improved predictive accuracy. Furthermore, Figure 8 illustrates the results of hyperparameter tuning for each benchmark model, providing a visual comparison of their predictive performance. The results show that different architectures respond differently to hyperparameter optimization. While some models exhibit noticeable improvements after tuning, others show limited gains. For example, VMD LSTM 1 (2,1) and VMD LSTM 2 (2,1), which utilize additional input features, demonstrate enhanced performance compared to basic LSTM models. However, even after tuning, their prediction errors remain higher than those of the proposed model, reaffirming the effectiveness of combining VMD with Attention in reducing prediction errors. This reduction in prediction error highlights the significance of combining VMD with an advanced sequence-learning mechanism. To evaluate the robustness of this substantial improvement, we conducted three complementary analyses. First, we constructed a bootstrap-based confidence interval for MSE, confirming that the low error value was not a result of statistical outliers. For example, the VMD-Cascaded LSTM with Attention model achieved a 95% confidence interval of [0.0015, 0.0030] for MSE (mean = 0.0023), significantly outperforming the next-best model (VMD-Cascaded LSTM), which recorded [0.0327, 0.1002] (mean = 0.0590). In contrast, basic LSTM baselines such as LSTM1 and LSTM2 yielded much higher intervals--[0.1399, 0.2689]  
Figure 7: Prediction results of the VMD-Cascaded LSTM with Attention model. and [0.4812, 0.7402], respectively--demonstrating the superior predictive stability of the proposed model. Second, a rolling-window out-of-sample test was conducted to verify temporal generalization across different market conditions. The model maintained consistent performance across periods, with only slight degradation during highly volatile intervals. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Model** & **95\% CI(MSE)** & **MSE** & **RMSE** & **MAE** \\ \hline LSTM1 & [0.1399, 0.2689] & 0.2212 & 0.4704 & 0.3929 \\ LSTM2 & [0.4812, 0.7402] & 0.4995 & 0.7068 & 0.5123 \\ VMD LSTM 1 (1,1) & [0.1113, 0.1422] & 0.1290 & 0.3592 & 0.3142 \\ VMD LSTM 1 (2,1) & [0.0545, 0.0714] & 0.0699 & 0.2644 & 0.2274 \\ VMD LSTM 2 (1,1) & [0.0894, 0.1170] & 0.1138 & 0.3373 & 0.2960 \\ VMD LSTM 2 (2,1) & [0.0810, 0.1000] & 0.0843 & 0.2903 & 0.2598 \\ VMD-Cascaded LSTM & [0.0327, 0.1002] & 0.0532 & 0.2306 & 0.1305 \\
**VMD-Cascaded LSTM with Attention** & **[0.0015, 0.0030]** & **0.0018** & **0.0427** & **0.0316** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance comparison of the proposed model and benchmark models (MSE with 95% confidence interval). Figure 8: Comparison of benchmark models based on hyperparameter tuning results. Finally, prediction residuals were visualized both over time and as a distribution. The absence of systematic drift or extreme skew in the residuals indicates that the model does not overfit noise, further validating the robustness of the results. These robustness checks collectively confirm that the significant performance improvements are not due to random chance or data leakage, but instead arise from the architectural advantages of combining VMD with Multi-Head Attention and a Cascaded LSTM framework. The proposed model outperforms all benchmark models due to two key factors.