## 5 Concluding Remarks  
In this study, we proposed an innovative approach to financial time-series forecasting by integrating VMD-Cascaded LSTM with Attention model. The primary objective was to enhance the predictive accuracy of the VIX, a critical indicator of market uncertainty, using advanced signal processing and deep learning techniques. The proposed model incorporates VMD as a preprocessing step to decompose financial time-series data into multiple IMFs, which effectively separate short-term fluctuations from long-term trends. By leveraging these decomposed features, the model captures essential patterns in the data. Furthermore, the Cascaded LSTM with Attention architec  
Figure 10: Histogram of prediction residuals. Figure 9: Prediction residuals over time. ture enables the model to focus on key temporal dependencies, significantly improving predictive accuracy. The experimental results, using VIX and S&P 500 data from January 2020 to December 2024, demonstrated the model's superiority over seven benchmark models. The proposed model achieved highly competitive error metrics, establishing it as a state-of-the-art approach for financial time-series forecasting. These results highlight the potential of combining signal decomposition techniques and attention mechanisms for addressing the complexities of market volatility. This research contributes to the field of financial time-series forecasting in three key aspects. First, it introduces a novel combination of VMD and Attention-based LSTM to effectively address the complexities of market volatility modeling. Second, the model's performance outperforms benchmark models, achieving a mean squared error (MSE) of 0.0018, which represents a 96.6% improvement compared to the best benchmark model, **VMD LSTM 2 (2,1)**. Additionally, the inclusion of the Attention mechanism significantly enhances the model's performance, as evidenced by a 99.7% reduction in MSE, an 81.5% reduction in RMSE, and a 75.8% reduction in MAE compared to **VMD-Cascaded LSTM** (without Attention). These findings underscore the importance of integrating Attention to capture critical temporal dependencies and improve prediction accuracy. The proposed VMD-Cascaded LSTM with Attention model enhances financial time-series forecasting by integrating advanced signal decomposition and deep learning techniques. VMD decomposes financial data into IMFs, reducing noise and isolating meaningful patterns. This enables the model to extract essential features while minimizing the impact of irrelevant fluctuations. Cascaded LSTM refines temporal dependencies, improving long-term forecasting accuracy by capturing hierarchical representations of sequential data. Additionally, the incorporation of Multi-Head Attention highlights critical time steps, allowing the model to focus on key market movements more effectively than traditional LSTM models. Compared to benchmark models, our approach demonstrates superior performance in multiple aspects. VMD enhances noise handling, providing cleaner input signals for predictive modeling. The cascaded structure of LSTM layers ensures stronger pattern recognition by sequentially refining extracted features. Furthermore, the Attention mechanism improves interpretability by dynamically emphasizing influential time points, leading to better predictive accuracy and more explainable results. Overall, this study highlights the advantages of combining VMD, LSTM, and Attention mechanisms for financial volatility forecasting. The proposed approach improves accuracy, enhances model robustness, and increases interpretability, making it a valuable tool for financial modeling and risk management applications. This study proposes a VMD-Cascaded LSTM with Attention model aimed at improving the accuracy of market volatility forecasts. By doing so, it is anticipated to aid financial institutions, asset management firms, and individual investors in developing more sophisticated and stable risk management and investment strategies.