## 3 The Proposed IAVMD Method
### Parameter Optimization of VMD Using GWO
From the chart we know that the proposed WSK indicator has the smallest change when the outlier of the signal increases gradually. This implies that the proposed WSK indicator is more reliable and robust to the outlier of the signal compared with the single indicator and WK. Hence, through the combination ofthree indicators (i.e., the sparseness, kurtosis, and correlation coefficient), the WSK indicator is more comprehensive and has more advantages in measuring vibration features. Obviously, a larger WSK implies a richer impact feature and a better signal sparseness and decomposition results. Here, to further ensure a global optimal solution, we deem the mean of the WSK of modes acquired by VMD as an objective function of the optimization procedure. Concretely, the goal of the parameter optimization process of VMD using GWO is visually interpreted as an effective search for the minimum value of the opposite of the objective function, as shown in Equation (12):  
\[\left\{ \begin{array}{c}\mathit{fitness}=\min_{\gamma=(K,\hat{\alpha})}\{- mean(WSK_{i})\}\\ \mathit{s.t.}\ K=2,3,\cdots,8\ \mathit{and}\ \hat{\alpha}=[100,2000]\end{array} \right. \tag{12}\]  
where \(K\) denotes the mode number, \(\hat{\alpha}\) denotes the penalty factor, \(mean(WSK_{i})\) is the mean value of WSK, and \(WSK_{i}\) is the WSK corresponding to the \(i\)-th mode. GWO has a strict population hierarchy, which is mainly divided into four layers (see Figure 5), where the first layer is \(\alpha\) (i.e., leaders/the optimal solution), the second layer is \(\beta\) (i.e., subordinate wolves-advisors/the suboptimal solution), the third layer is \(\delta\) (i.e., scouts, sentinels, elders, hunters, and caretakers/the third optimal solution), and the last layer is \(\omega\) (i.e., lowest ranking wolves-scapegoats/other solutions). Moreover, GWO is applied for parameter optimization by implementing several steps (e.g., hunting, searching for prey, encircling prey, and attacking prey). Figure 6 shows the flowchart of the parameter optimization of VMD using the GWO method. The general procedure of VMD parameter optimization is elaborated in detail below:  
(1) Initialize the grey wolf population \(X_{i}(i=1,2,\cdots,m)\), and set the fitness function and the parameters of the GWO algorithm. Generally speaking, the higher the number of wolves and iterations is, the better the optimization results are, but the longer the computation time is; thus, according to \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Signal** & **Kurtosis** & **Entropy** & **WK** & **WSK** \\ \hline No outlier & 3.19 & 6.89 & 3.19 & 9.18 \\ Single outlier & 7.82 & 6.58 & 6.35 & 9.35 \\ Multiple outlier & 16.41 & 6.32 & 2.87 & 9.29 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The comparisons among different indicators. Figure 4: The time domain waveform of the simulation signal: (**a**) No outlier, (**b**) single outlier, and (**c**) multiple outlier. Ref. [42], we define the maximum number of wolves \(m=30\) and maximum iterations \(T=10\), which is aimed at achieving a tradeoff between the optimization results and computational efficiency. Besides, due to the pre-optimization, the objective only involves two variables (\(\hat{\alpha}\), \(K\)), and thus each wolf is expressed as \(X_{i}=(x_{K},x_{alpha})\), \(i=1,2,\cdots,m\), where \(x_{K}\) and \(x_{alpha}\) represent the penalty factor and mode number, respectively. Due to fault feature information of the practical signal usually spread over the first few high-frequency components obtained by VMD, the first two to eight components are usually selected and analyzed. Besides, the feature information of the main components obtained by VMD is distributed over an appropriate bandwidth, which means that the parameter \(\hat{\alpha}\) is supposed to stay within an empirical range and is neither too big nor too small compared to the default of VMD. For all those reasons, here the search ranges of the two parameters \(\hat{\alpha}\) and \(K\) are empirically set as [2; 8] and [100, 2000], respectively.