# Landslide Displacement Prediction Based on Variational Mode Decomposition and GA-Elman Model
## 2 Theory and Method  
### Variational Mode Decomposition  
Variational modal decomposition was first proposed by [PERSON] and [PERSON] in 2014 [40]. It is an adaptive, completely non-recursive approach to modal variational problem and signal processing. The basis of this method is to construct a variational problem and then solve the variational problem by decomposing a deterministic real-valued signal \(Y\) into a discrete number of modes \(Y_{k}(t)\), \(k=1,2,3,\cdots,K\). In this process, it is assumed that each decomposed mode fluctuates around a central pulsation. The biggest advantage of VMD compared with EMD is that it can determine the number of modal decompositions by itself. It overcomes the problems of endpoint effects and modal component aliasing in EMD, and it has a solid theoretical foundation. It can be used to deal with nonlinear sequences with poor regularity and high complexity, and to decompose nonlinear sequences into relatively stable subsequences.  
The process whereby VMD decomposes displacement data is as follows:  
Step 1: Construct the variational problem, decompose the signal \(Y\) into \(K\) components, determine the penalty parameter \(\alpha\) and the rising step \(\tau\), and set the constraint condition that the sum of all modes is equal to the original signal. The corresponding constraint variational expression is  
\[\left\{\begin{array}{l}\min\limits_{\{u_{k}\}\{\omega_{k}\}}\left\{\sum \limits_{k=1}^{K}\left\|\partial_{t}\right[\left(\sigma(t)+\frac{j}{\pi t} \right)u_{k}(t)\right]^{-j\omega_{k}t}\right\|_{2}^{2}\right\}\\ s.t.\sum\limits_{k=1}^{K}u_{k}=f(t)\end{array}\right. \tag{1}\]where \(\{u_{k}\}\) and \(\{\omega_{k}\}\) are the decomposed modal component and the center frequency of the modal component, respectively, \(\left(\sigma(t)+\frac{j}{\pi t}\right)u_{k}(t)\) is the analytical signal of the modal component, and \(f(t)\) is the original signal.  
Step 2: By introducing the Lagrange penalty operator \(\lambda\), the constrained problem is transformed into an unconstrained problem, and Equation (2) is obtained.  
\[\begin{split} L(\{u_{k}\},\{\omega_{k}\},\lambda)=\alpha\underset {k}{\overset{}{\sum}}\Big{\|}\partial_{t}\Big{[}\left(\sigma_{t}+\frac{j}{\pi t }\right)u_{k}(t)\Big{]}e^{-i\omega_{k}t}\Big{\|}_{2}^{2}\\ +\bigg{\|}f(t)-\underset{k}{\overset{}{\sum}}u_{k}(t)\Big{\|}_{2 }^{2}+\left\langle\lambda(t),f(t)-\underset{k}{\overset{}{\sum}}u_{k}(t) \right\rangle\end{split} \tag{2}\]  
where \(\alpha\) is the penalty factor.  
Step 3: Substitute the saddle point obtained in the second step into the first step, and the value obtained is the solution of the model. The detailed process of alternating the iterative optimization of \(\{u_{k}\}\), \(\{\omega_{k}\}\), and \(\lambda\) when solving with VMD is shown in Equations (3)-(5).  
\[\begin{split}\hat{a}_{k}^{n+1}(\omega)\leftarrow\frac{\hat{f}( \omega)-\underset{i\
eq k}{\overset{}{\sum}}\hat{a}_{i}(\omega)+\hat{\lambda} (\omega)/2}{1+2\alpha(\omega-\omega_{k})^{2}}\end{split} \tag{3}\]  
\[\begin{split}\omega_{k}^{n+1}\leftarrow\frac{\int_{0}^{\infty} \omega\Big{|}\hat{a}_{k}^{n+1}(\omega)\Big{|}^{2}d\omega}{\int_{0}^{\infty} \Big{|}\hat{a}_{k}^{n+1}(\omega)\Big{|}^{2}d\omega}\end{split} \tag{4}\]  
\[\begin{split}\hat{\lambda}^{n+1}(\omega)\leftarrow\hat{\lambda} ^{n}(\omega)+\gamma\Bigg{(}f(\omega)-\underset{k}{\overset{}{\sum}}\hat{a}_{ k}^{n+1}(\omega)\Bigg{)}\end{split} \tag{5}\]  
The \(K\) IMF components of the original signal \(f(t)\) can be obtained through the above formula.