## 3 Data Description and Methods
### The Baseline Methodologies
#### 3.2.1 Vmd  
VMD is a decomposition technique frequently utilized in the analysis of nonstationary signals. It decomposes complex signals with various frequency components into multiple narrow-band modes. This method was proposed to overcome the drawbacks of the traditionally widely used EMD. In EMD, issues such as mode mixing, boundary effects, and sensitivity to noise have been identified during the sequential decomposition of the signal into IMFs. VMD mitigates these problems by adopting a variational optimization approach to simultaneously estimate the modes. Here, the concept of variational optimization means setting an objective function to efficiently identify multiple narrow-band components inherent in the signal and minimizing it through an iterative update process.  
The core idea pursued by VMD is to express the original signal as a sum of multiple modes, each assumed to be a narrow-band component with a central frequency, while simultaneously updating all modes. Mathematically, the given signal \(f(t)\) can be represented as the sum of \(K\) modes \(\{u_{k}(t)\}_{k=1}^{K}\) and their corresponding central frequencies \(\{\omega_{k}\}_{k=1}^{K}\):  
\[f(t)=\sum_{k=1}^{K}u_{k}(t) \tag{1}\]  
VMD is formulated as an optimization problem that minimizes the following objective function to ensure each mode \(u_{k}(t)\) maintains narrow-band characteristics:  
\[\min_{\{u_{k}\},\{\omega_{k}\}}\sum_{k=1}^{K}\biggl{\|}\partial_{t}\biggl{[} \biggl{(}\delta(t)+\frac{j}{\pi t}\biggr{)}*\Bigl{(}u_{k}(t)e^{-j\omega_{k}t} \biggr{)}\biggr{]}\biggr{\|}_{2}^{2}. \tag{2}\]  
The algorithm initializes the central frequencies \(\omega_{k}^{(0)}\) and the initial modes \(u_{k}^{(0)}(t)\). It then iteratively performs mode updates, recalculates the central frequencies, and updates the Lagrange multipliers and penalty parameters. The process continues until the rate of  
Figure 1: Comparison of S&P 500 and VIX movements over time.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **Mean** & **Std. Dev.** & **Skewness** & **Kurtosis** & **J-B.** & **J-B. \(p\)-Value** & **ADF** & \(p\)**-Value** \\ \hline VIX & 21.4152 & 8.2642 & 2.5211 & 11.1301 & 7819.7997 & 0.0000 & \(-\)4.1679 & 0.0007 \\ S\&P 500 & 4258.3165 & 766.3876 & 0.2480 & \(-\)0.0865 & 13.2813 & 0.0013 & \(-\)0.3038 & 0.9250 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary statistics of VIX and S&P 500 data, including normality (J.-B. = Jarque–Bera) and stationarity (ADF) test results.  
change falls below a threshold or a maximum number of iterations is reached, resulting in the convergence to the final modes. To ensure that the sum of the decomposed modes matches the original signal, Lagrange multipliers are introduced to impose this constraint, and penalty terms are used to maintain accuracy.  
Unlike traditional EMD, which extracts one mode at a time, VMD simultaneously optimizes a predetermined number of modes that interact with each other. This simultaneous optimization reduces interference between modes and enhances robustness against noise. In the frequency domain, VMD updates each mode's frequency component and recalculates the central frequency to maintain narrow-band characteristics. When applying VMD, several key parameters must be appropriately set. The most important parameter is the number of modes to decompose, which should be chosen based on the frequency characteristics of the signal to achieve proper decomposition. Additionally, the penalty parameter \(\alpha\) controls the narrow-bandness of the modes and is typically set empirically or determined through cross-validation to find the optimal value.