## 3 Data Description and Methods
### Proposed Model  
Despite substantial advances engendered by the deployment of LSTM architectures augmented with attention mechanisms in financial forecasting, extant studies continue to exhibit several critical shortcomings. Predominantly, existing approaches emphasize short-term trend extrapolation [32], while under-representing the modeling of long-range sequential dependencies that are pivotal for capturing macro-scale market dynamics [11; 12]. Although Attention layers mitigate this to an extent by adaptively weighting temporal features, they remain insufficient for discerning multi-scale temporal structures and attenuating high-frequency, noise-induced artifacts inherent in raw financial time series. Moreover, a sizable portion of attention-based frameworks operate on raw, untransformed sequences laden with nonstationary components and stochastic volatilities leading to pronounced overfitting and performance instability under volatile market conditions [33; 34]. Finally, most prior investigations are constrained to univariate or sparsely multivariate models, thereby failing to capture synergistic interactions between critical market indicatorssuch as the S&P 500 and the VIX index [35; 36]. To address these limitations, we advocate for the integration of signal decomposition techniques such as VMD that deconstruct financial series into denoised, frequency-specific modes prior to the application of attention, thereby furnishing models with stable, information-rich feature representations [37; 38]. In this study, we propose the VMD-Cascaded LSTM with Attention model to improve the stability and accuracy of VIX volatility predictions. The overall structure of the model is shown in Figure 4.  
The proposed model consists of two main stages. In the first stage, the S&P 500 close data are decomposed using VMD, generating multi-scale features represented as VMD-extracted S&P 500 features. These features are then fed into the first LSTM layer, which captures the initial temporal patterns of the market. This design leverages the S&P 500 data as a representation of market trends and ensures that key temporal dependencies are learned effectively. In the second stage, the VIX close data are also processed through VMD, producing VMD-extracted VIX features. These data are then passed through a Multi-Head Attention mechanism, which dynamically emphasize critical features within the VMD-extracted VIX features. The enhanced features are subsequently used as inputs to the second LSTM layer, which refines the patterns learned from the first stage and integrates them with the newly emphasized VIX features. This hierarchical structure stabilizes the learning process and ensures more precise predictions of VIX volatility. The detailed architecture of the proposed model is already illustrated in Figure 4, which captures both stages of the VMD-Cascaded LSTM with Attention structure. As shown in the figure, the first stage begins with the decomposition of the S&P 500 data into its VMD modes (sig1, sig2, sig3), which are then summed to form VMD S&P 500 data. These data are input to the first LSTM layer, where primary temporal patterns are captured. Simultaneously, the VIX data are decomposed into its VMD modes and summed to produce VMD VIX data, which serves as the input to the Multi-Head Attention mechanism. The Attention mechanism dynamically highlights the most critical features within the VMD VIX data, which are passed to the second LSTM layer as features for further learning. The dimension transformations in the proposed model play a crucial role in ensuring seamless data flow between different layers. After the first LSTM layer processes the VMD S&P 500 data, it outputs a 2D tensor in the form (batch size, LSTM units). To prepare this output for subsequent integration, it is flattened into a 2D structure. Simultaneously, the VIX data, initially represented as a 3D tensor (batch size, sequence length, 1), are reshaped into a 2D tensor (batch size, sequence length). Figure 4: Overview of the VMD-Cascaded LSTM with Attention model.