## 5 Background Noise Elimination Method
### The Principle of ALO-VMD Method
#### 5.1.2 Ant Lion Optimization Method  
Ant lion optimization (ALO) is one of the swarm intelligence algorithms, which solves the functional optimization problem by imitating the hunting mechanism of ant lions. The principle of the ALO algorithm is introduced as below [44].  
To imitate the random walk of ants inside the search space, a normalized equation is applied in Equation (26).  
\[X_{i}^{Iter}=\frac{(X_{i}^{Iter}-a_{i})\times(d_{i}-c_{i}^{Iter})}{(d_{i}^{Iter }-a_{i})}+c_{i} \tag{26}\]  
where \(a_{i}\) is the minimum of random walk of the \(i\)-th variable, \(d_{i}\) is the maximum of random walk in the \(i\)-th variable, \(c_{i}^{Iter}\) is the minimum of the \(i\)-th variable at the \(Iter\)-th iteration, and \(d_{i}^{Iter}\) indicates the maximum of the \(i\)-th variable at the \(Iter\)-th iteration.  
As mentioned in Reference [33], the random walk of ants is affected by the ant lions' traps. This behavior can be modeled by Equations (27) and (28).  
\[c_{i}^{Iter}=Antlion_{j}^{Iter}+c^{Iter} \tag{27}\]  
\[d_{i}^{Iter}=Antlion_{j}^{Iter}+d^{Iter} \tag{28}\]  
Therefore, \(c^{Iter}\) is the minimum of all the variables at the \(Iter\)-th iteration, \(d^{Iter}\) indicates the vector including the maximum of all variables at the \(Iter\)-th iteration, and \(Antlion_{j}^{Iter}\) shows the position of the selected \(j\)-th ant lion at the \(Iter\)-th iteration.  
According to the roulette wheel, the ALO algorithm selects the fittest ant lion to catch the ants. Once the ant is in the trap, the ant lions throw the sand toward the center of the hole. The process is described mathematically by Equations (29) and (31).  
\[c^{Iter}=\frac{c^{Iter}}{I} \tag{29}\]  
\[d^{Iter}=\frac{d^{Iter}}{I} \tag{30}\]  
\[I=\left\{\begin{array}{ll}1&I\text{er}\leq 0.1 Iter_{\text{max}}\\ 10^{\gamma}\frac{Iter}{Iter_{\text{max}}}&I\text{er}>0.1 Iter_{\text{max}}\\ \end{array}\right. \tag{31}\]  
As such, \(Iter\) is the current iteration, \(Iter_{\text{max}}\) is the maximum number of iterations, and \(\gamma\) is a constant defined based on the current iteration. The value of \(\gamma\) can be found in Reference [44].  
Assumed by the ALO algorithm, the hunting stage is ended when the fitness value of ant is greater than that of the ant lion. To increase the probability of catching new ants, the ant lion is then expected to replace its position to the latest position of the hunted ant, as defined by Equation (32).  
\[Antlion_{j}^{Iter}=Antl_{i}^{Iter} \tag{32}\]  
where \(Antlion_{j}^{Iter}\) shows the position of selected \(j\)-th ant lion at the \(Iter\)-th iteration, and \(Antl_{i}^{Iter}\) indicates the position of \(i\)-th ant at the \(Iter\)-th iteration.  
To maintain the best solution for the optimization process, the fittest ant lion is selected as an elite, which can affect the random walk of all ants. That is, the random walk of each ant is guided by the selected ant lion and the elite ant lion. Therefore, the average of two randomly walks are used to represent the repositioning of a given ant, which can be modeled by Equation (33).  
\[Antl_{i}^{Iter}=\frac{R_{A}^{Iter}+R_{E}^{Iter}}{2}. \tag{33}\]  
where \(R_{A}^{Iter}\) is the random walk around the roulette wheel selected ant lion, and \(R_{E}^{Iter}\) is the random walk around the elite ant lion.