# MoCoformer: Quantifying Temporal Irregularities in Solar Wind for Long-Term Sequence Prediction
## 2 Prior Technologies  
### Variational Mode Decomposition  
The VMD algorithm [53] is an adaptive time series decomposition method that effectively resolves the modal aliasing problem and produces precise signal separation. In order to efficiently split the time series into low-frequency and high-frequency modes, VMD employs an iterative process to find the variational model's optimal solution. This allows it to ascertain the center frequency and bandwidth of each IMF. In particular, the objective function of the VMD algorithm is to minimize the sum of the estimated bandwidths of the K IMFs and to satisfy the constraints to ensure that the sum of the K IMFs is equal to the input signal f. The mathematical expression for this process is given below:  
\[\min_{u_{k},w_{k}}=\sum_{k=1}^{K}\left\|\partial_{t}[(\delta(t)+\tfrac{j}{\pi t })*u_{k}(t)]e^{-jw_{k}t}\right\|_{2}^{2} \tag{1}\]  
\[s.t.\sum_{k=1}^{K}u_{k}=f\]  
where \(\partial_{t}\) represents an impulse function, \(u_{k}\) represents the k-th IMF obtained from the signal decomposition, \(w_{k}\) denotes the center frequency of the k-th mode, and \(j\) refers to the imaginary unit. The objective of this formulation is to minimize the bandwidth of each \(u_{k}\).  
To cope with this problem, the quadratic penalty parameter \(\alpha\) and the Lagrange multiplier operator \(\lambda(t)\) are introduced to subtly transform the original constrained variational problem into an unconstrained variational problem. Among them, the expression of the augmented Lagrangian function is given as follows:  
\[\begin{split} L=&\alpha\sum_{k=1}^{K}\left\| \partial_{t}[(\delta(t)+\tfrac{j}{\pi t})*u_{k}(t)]e^{-jw_{k}t}\right\|_{2}^{2 }\\ +&\left\|f(t)-\sum_{k=1}^{K}u_{k}\right\|_{2}^{2}+ \left\langle\lambda(t),f(t)-\sum_{k=1}^{K}u_{k}\right\rangle\end{split} \tag{2}\]  
where \(\alpha\) represents a weight parameter, \(\left\langle\cdot\right\rangle\) is inner product operation, \(\partial_{t}\) denotes the differentiation with respect to time, \(\lambda\) represents the Lagrange multiplier, and \(\|\cdot\|_{2}\) represents the L2 regularization to avoid over-fitting of the results.  
For the solution of this complex problem, the IMF decomposition was conducted by transferring it from the time domain to the frequency domain, resulting in the following findings:  
\[\hat{u}_{k}^{n+1}(w)=\frac{\hat{f}(w)-\sum_{i\
eq k}\hat{u}_{i}(w)+\frac{ \lambda(t)}{2}}{1+2\alpha{(w-w_{k})}^{2}} \tag{3}\]  
\[w_{k}^{n+1}=\frac{\int_{0}^{\infty}w|\hat{u}_{k}(w)|^{2}}{\int_{0}^{\infty}| \hat{u}_{k}(w)|^{2}} \tag{4}\]  
When the number of decomposed IMFs is K, the sequence of operations of the VMD algorithm is as follows:  
1. Firstly, initialization is carried out by setting \(u_{k}^{1}\), \(w_{k}^{1}\), \(\hat{\lambda}^{1}\), and the number of iterations n.
2. By applying Equations (3) and (4), update \(u_{k}\) and \(w_{k}\) until k reaches K.
3. Update the operator \(\lambda\) using \(\hat{\lambda}^{n+1}(w)=\hat{\lambda}^{n}(w)+\tau\left(\hat{f}(w)-\sum_{i\
eq k }\hat{u}_{i}(w)\right)\).
4. Keep iterating until the convergence condition \(\sum_{k}\left\|\hat{u}_{k}^{n+1}-\hat{u}_{k}^{n}\right\|_{2}^{2}/\left\|\hat {u}_{k}^{n}\right\|_{2}^{2}<e\) is reached, or until the maximum number of iterations n is reached, or else end the iteration and then return to step (2).  
As previously demonstrated, a finite set of intrinsic modal functions \(u_{k}\) with specific sparse properties can be obtained in a non-recursive manner. It is worth noting that the VMD algorithm exhibits increased robustness in suppressing noise due to its integration of a Wiener filter to dynamically update these modes.