## 4 Empirical Results  
### Experimental Design  
This study uses VIX close data and S&P 500 close data obtained from Yahoo Finance, covering the period from 1 January 2020 to 31 December 2024. The dataset contains daily closing prices, which were divided into a training set (80%, 2020-2023) and a test set (20%, 2024). All data were normalized to ensure consistent feature scaling. The raw closing prices of VIX and S&P 500 were first normalized using Z-score standardization (mean 0, standard deviation 1). Missing values were handled by simple row deletion, as they were sparse and did not significantly affect the time-series continuity. The dataset was split chronologically to prevent future information leakage. The last year (2024) was used exclusively as the out-of-sample test set to ensure realistic evaluation. To evaluate the model's robustness across different time periods, a rolling-window backtest was additionally conducted. This test design allows the model to be exposed to various market conditions, including periods of high volatility. The relatively consistent MSE trends suggest strong generalization capabilities. In this approach, the training and testing windows were shifted across the time series using a fixed-size scheme (500-day training, 50-day testing). At each step, the model was retrained and evaluated on new out-of-sample data. This method enables a more comprehensive understanding of temporal generalization performance. Figure 5 presents the MSE values for each step of the rolling-window evaluation. The model maintains stable predictive accuracy across most periods, though slight performance degradation is observed during highly volatile intervals (e.g., mid-2021 and mid-2023), consistent with expectations in financial time-series forecasting. The proposed model consists of two primary stages. First, the S&P 500 and VIX data are decomposed using VMD, generating multiple IMFs that capture different frequency components of the original signals. These IMFs provide multi-scale features that enhance the model's ability to learn temporal dependencies effectively. To generate these IMFs, appropriate configuration of the VMD parameters is crucial. In this study, the number of modes \(K\) and the penalty parameter \(\alpha\) for VMD were set to 3 and 2000, respectively. The value of \(K=3\) was determined empirically through preliminary experiments, as it offered a favorable trade-off between decomposition granularity and predictive performance in our dataset. Meanwhile, \(\alpha=2000\) was selected based on prior literature; notably, [PERSON] et al. [39] employed the same value in a two-stage VMD framework for streamflow forecasting, where it proved effective in achieving stable mode separation without introducing excessive smoothing. Maintaining these fixed parameter values across all experiments ensured consistency when comparing different model configurations. A more detailed sensitivity analysis on \(K\) and \(\alpha\) will be considered in future work to further examine the robustness of these choices. Figure 6 illustrates the decomposed IMFs of the VIX and S&P 500 data. The left column represents the IMFs of VIX, while the right column shows those of the S&P 500. The first IMF captures high-frequency fluctuations, while the subsequent IMFs reveal lower-frequency trends and long-term patterns. By leveraging these multi-resolution representations, the model ensures that both short-term volatility and long-term market trends are incorporated into the forecasting process. In the first processing stage, the aggregated VMD components of the S&P 500 (VMD S&P 500) are passed through an LSTM layer, which captures sequential dependencies and extracts essential trend information. Meanwhile, the aggregated VMD components of the VIX (VMD VIX) are processed separately using a Multi-Head Attention mechanism, enhancing the model's ability to focus on critical patterns in market volatility.