## 3 The Proposed IAVMD Method
### Parameter Optimization of VMD Using GWO  
Two important variables (i.e., the penalty factor \(\hat{a}\) and mode number \(K\)) are required to define in advance when we employ VMD to a process complex compound signal. Some previous studies have shown that the two variables (\(\hat{a}\) and \(K\)) have a great influence on the decomposition performance of VMD. On the one hand, the bigger \(\hat{a}\) represents the smaller bandwidth of modes. Otherwise, the smaller \(\hat{a}\) represents the greater bandwidth of modes. On the other hand, if the mode number \(K\) is too big, it easily leads to the over decomposition phenomenon (i.e., information redundancy), while this will cause the under-decomposition phenomenon (i.e., information loss) when the mode number \(K\) is too small. Hence, how to pick out the appropriate variables (\(\hat{a},K\)), is a challenge when applying VMD to deal with complex compound signals. To address this issue, a new nature-inspired algorithm called GWO is introduced to determine and select automatically the combination parameters (\(\hat{a},K\)) of VMD in this subsection due to its excellent global optimization performance. Obviously, the objective function should be constructed when we use GWO to optimize the parameters of VMD. At present, there are many indicators (e.g., smoothness index, entropy, sparseness, kurtosis, and correlation coefficient) in the analysis of the mechanical vibration signal, but it is not optimal to use any of their single indicators to measure vibration signal characteristics. This indicates that the combination of two or more of their single indicators is valuable for practical vibration detection. Zhang et al. (2017) presented a new combination index named weighted kurtosis (WK) as the objective function of their method to optimize the key parameters of VMD, but the proposed weighted kurtosis does not consider some abnormal impacts (i.e., the outlier) with disperse distributions. Hence, to solve this problem, a comprehensive evaluation indicator named weighted sparseness kurtosis (WSK) is formulated in this paper to work as the objective function of the parameter optimization process of VMD, which is defined by:  
\[WSK=S\cdot Kur\cdot\big{|}\rho\big{|} \tag{8}\]  
\[S=\frac{\sqrt{\frac{1}{N}\sum_{n=1}^{N}x(n)^{2}}}{\frac{1}{N}\sum_{n=1}^{N} \big{|}x(n)\big{|}} \tag{9}\]  
\[Kur=\frac{\frac{1}{N}\sum_{n=0}^{N-1}x^{4}(n)}{\big{(}\frac{1}{N}\sum_{n=1}^{N- 1}x^{2}(n)\big{)}^{2}} \tag{10}\]  
\[\rho=\frac{E[(x-\overline{x})(y-\overline{y})]}{E[(x-\overline{x})^{2}]E[(y- \overline{y})^{2}]} \tag{11}\]  
where \(S\) is the sparseness of the signal \(x(n)\), \(N\) is the length of the signal \(x(n)\), \(\rho\) represents the Pearson correlation coefficient between two signals (i.e., \(x\) and \(y\)) and satisfies \(\big{|}\rho\big{|}\leq 1\), and \(E[\cdot]\) denotes the expectation operator. This indicates that the correlation coefficient \(\rho\) amounts to the weight of the sparseness kurtosis indicator. That is, Equation (8) is reputed essentially as the weighted sparseness kurtosis indicator in this paper. Note that in Equation (8) the sparseness is a statistical indicator reflecting the amplitude distribution of the signal, especially for an abnormal impact amplitude with discrete distribution, while kurtosis is very sensitive to the periodic impulse series of vibration signals. Besides, the correlation coefficient can measure the similarity between two vibration signals. To prove the efficacy of the presented WSK indicator, comparisons between four indicators (i.e., kurtosis, entropy, WK, and WSK) are conducted by analyzing a simulation signal. The simulation signal shown in Figure 4a is composed of the periodic impulse, harmonic signal, and random noise. Figure 4b,c shows the simulation signal containing a single outlier and multiple outlier, respectively. Table 1 gives the comparison results of different indicators for the simulation signal.