## 2 Methodologies
### Vmd  
VMD is an adaptive and fully non-recursive method with modal change and signal processing [44], which decomposes time series data into multiple IMFs. It can not only retain the inherent information of original signals, but also avoids the overlap of variable information. Moreover, VMD has superior performance in sampling and denoising compared to WD, EMD and EEMD [45]. The process of decomposition using VMD can be described as follows:  
Step 1: The analytic signal \(f(t)\) can be transformed by the Hilbert theory into a unilateral frequency spectrum for a given mode, as follows:  
\[\text{Hilbert}[f(t)]=\frac{1}{\pi}\int_{-\infty}^{+\infty}\frac{f(v)}{t-v}dv \tag{1}\]  
Step 2: Each mode is added by an exponential term, and its frequency spectrum is shifted to the baseband as follows:  
\[\bigg{[}\bigg{(}\delta(t)+\frac{j}{\pi t}\bigg{)}u_{k}(t)\bigg{]}e^{-j\omega_{ k}t} \tag{2}\]  
where \(\delta(t)\) is the Dirac distribution, \(u_{k}\) is the \(k\)th mode, \(\omega\) is the angular frequency and \(j\) is an imaginary unit.  
Step 3: The Gaussian smoothness, i.e., the square norm of the gradient, is used to calculate the bandwidth of the signal. Then, the constrained variational expression can be written as follows:  
\[\min_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum_{k}\left\|\partial_{t}\bigg{[} \bigg{(}\delta_{l}+\frac{j}{\pi t}\bigg{)}u_{k}(t)\bigg{]}e^{-j\omega_{k}t} \right\|_{2}^{2}\right\},\;s.t\sum_{k}u_{k}=f \tag{3}\]  
where \(\omega_{k}\) is the center frequency of the \(k\)th mode.  
Step 4: Lagrange multiplier theory is an effective method to enforce constraints. For the convenience of analysis, the constrained variational problem needs to be transformed into an unconstrained one, as shown below:  
\[\mathcal{L}(\{u_{k}\},\{\omega_{k}\},\lambda)\coloneqq\alpha\underset{k}{ \sum}\|\partial_{t}\bigg{[}\bigg{(}\delta_{l}+\frac{j}{\pi t}\bigg{)}u_{k}(t) \bigg{]}e^{-j\omega_{k}t}\underset{2}{\overset{2}{\parallel}}\|f(t)-\underset {k}{\sum}u_{k}(t)\|_{2}^{2}+\langle\lambda(t),f(t)-\underset{k}{\sum}u_{k}(t)\rangle \tag{4}\]  
where \(\alpha\) is a quadratic penalty factor and \(\lambda\) is the Lagrange multiplier.  
Step 5: It is easy to achieve the saddle point of Equation (4) using the alternate direction method of multipliers to further achieve the frequency domain update of each mode. Thus, the expressions of modal component and center frequency are given as follows:  
\[\hat{u}_{k}^{n+1}(\omega)=\frac{\hat{f}(\omega)-\sum_{i\
eq k}\hat{u}_{i}(\omega )+\frac{\hat{\lambda}(\omega)}{2}}{1+2\alpha(\omega-\omega_{k})^{2}} \tag{5}\]  
\[\omega_{k}^{n+1}=\frac{\int_{0}^{\infty}\omega|\hat{u}_{k}(\omega)|^{2}d\omega }{\int_{0}^{\infty}|\hat{u}_{k}(\omega)|^{2}d\omega} \tag{6}\]  
where \(n\) represents the number of iterations. Additionally, Lagrange multiplier \(\lambda\) should be satisfied with the following expression:  
\[\hat{\lambda}^{n+1}(\omega)\leftarrow\hat{\lambda}^{n}(\omega)+\tau\Bigg{(} \hat{f}(\omega)-\sum_{k}\hat{u}_{k}^{n+1}(\omega)\Bigg{)} \tag{7}\]  
Step 6: Solve the saddle point of Equation (2) repeatedly until the modal component satisfies the following convergence criterion:  
\[\sum_{k}\Big{\|}\hat{u}_{k}^{n+1}-u_{k}^{n}\Big{\|}_{2}^{2}/\|u_{k}^{n}\|_{2} ^{2}<\varepsilon \tag{8}\]  
where \(\varepsilon\) is the convergence tolerance that determines the accuracy and iteration times.