## 3 Vibration Signal Preprocessing and Fault Feature Extraction
### GWO-VMD-Based Vibration Signal Decomposition
#### 3.1.3 GWO-Based [\(K\), \(\alpha\)] Optimization for VMD  
Grey Wolf Optimizer is a swarm intelligent technique developed by [PERSON] et al. in 2014 [38]. The algorithm mimics the leadership hierarchy of wolves, which are well known for their group hunting social behavior. When compared with the Genetic Algorithm (GA) and other swarm algorithms, such as PSO, GWO is believed to be more effective [39]. Therefore, GWO was chosen in this paper to optimize the two parameters of VMD.  
In the social hierarchy of grey wolves, the fittest is named _alpha_, the second and third fittest are named _beta_ and _delta_, respectively, and the rest of the wolves are called _omega_. The optimization process of GWO is led by _alpha_, _beta_, and _delta_. With the leadership in the hierarchy, the GWO hunting procedure mainly consists of three steps known as encircling prey, hunting, and attacking prey. The mathematical models of these three behaviors were formalized by [PERSON] et al. in Reference [38].  
When GWO is applied to simultaneously hunt the optimal values of [\(K\), \(\alpha\)], the procedure is as follows.  
Step 1. Parameter initialization.  
Initialize the size of the wolf group, the maximum iteration steps, the upper and lower bounds of \(K\) and \(\alpha\), and the values of _alpha_, _beta_, _delta_, and _omega_.  
Step 2. Define and calculate the fitness function.  
In order to ensure that the reconstructed vibration signals after decomposition by GWO-VMD can retain as much fault feature as possible while removing as much noise as possible, a fitness function is constructed for GWO to minimize the reconstructed signal by using permutation entropy [40] and Kurtosis [13] as follows:  
\[fn=-\,kt+H_{p} \tag{6}\]  
where \(fn\) stands for the fitness function, \(kt\) and \(Hp\) represent Kurtosis and permutation entropy, respectively. The mathematical expressions of \(kt\) and \(Hp\) are as follows:  
\[kt=\frac{E(x-\mu)^{4}}{\sigma^{4}} \tag{7}\]  
\[H_{p}=-\frac{1}{ln(m_{1}!)}\sum_{i=1}^{q}p_{i}lnp_{i} \tag{8}\]  
where \(x\) is an arbitrary signal to calculate Kurtosis, \(\mu\) and \(\sigma\) are the mean value and standard deviation of \(x\), respectively, \(p_{i}\) is the probability of each row in the \(m_{1}\times q\) reconstruction matrix of the signal, and \(m_{1}\) is the embedded dimension.  
Step 3. Update the wolf group.  
After calculating the fitness value of each wolf, _alpha_, _beta_, _delta_, and _omega_ are updated with the following rules: if \(fn<alpha\), then \(alpha=fn\); else if \(alpha<fn<beta\), then \(beta=fn\); else if \(beta<fn<omega\), then \(omega=fn\).  
Step 4. Recalculate fitness values of the wolf group.  
Recalculate the fitness value of each wolf, compare the new value with the previous one--if the updated value is smaller than the old one, keep the latest position of the wolf and replace the previous value with the new one; otherwise retain the previous fitness.  
Step 5. Terminating condition judgement.  
Determine whether the maximum iteration is reached; if not, go back to Step 2; otherwise, stop the iteration and return the optimal values of \([K,\alpha]\).