## 2 VMD with Kurtosis Mean (KMVMD)  
[PERSON] introduced the algorithm called variational mode decomposition, which mainly uses signal processing tools of Hilbert transform, Wiener filtering, and frequency mixing. The signal is decomposed into different IMF by VMD [45]. The IMF is written as follows.  
\[u_{k}(t)=A_{k}(t)\cos(\phi_{k}(t)) \tag{1}\]  
The VMD requires to predefine the number of modes \(k\). Improper \(k\) will cause over decomposition or under decomposition. With the occurrence and development of bearing faults, the kurtosis value increases. The kurtosis value is higher, bearing fault is more serious. So, it is often applied to fix the number of mode \(k\). After decomposition, the mode with the largest value is often selected. Its mathematical formula is shown as follows.  
\[kur=\frac{E(x-\mu)^{4}}{\sigma^{4}} \tag{2}\]  
where \(x\) indicates signal, \(\mu\) indicates the mean value of \(x,\sigma\) is the standard deviation of \(x\).  
However, it is not completely accurate to use the kurtosis criterion to fix the number of modes and choose the mode with the largest value for power spectrum analysis. This is because the fault frequency does not necessarily occur in the mode with the largest kurtosis under the strong noise interference but could occur in the mode with the other largest values.  
Aiming at the shortage of kurtosis, VMD with kurtosis mean is proposed, which uses kurtosis mean to measure the mean value of fault information under different \(k\). By calculating the kurtosis mean of the modes with larger kurtosis values at different \(k\), the maximum kurtosis mean values at different \(k\) are compared to obtain the number of modes. When the mode number is \(k<5\), \(\left[\frac{K}{2}\right]\) modes with larger values are chosen to calculate the kurtosis mean value. When the mode number \(k\geq 5\), 3 modes with larger values are chosen to calculate the kurtosis mean value. When \(K\) is less than 5, the problem of less mode number should be considered, so the number of selected IMF should be the number rounding half of \(k\). This selection method can not only retain the fault information greatly under strong noise interference but also avoid the interference of abnormal modes on mode number selection. When \(k\geq 5\), three modes are selected for analysis. Because for weak fault signals, the possibility of fault information appearing in IMF with the second or third larger value should be considered, but under normal conditions, fault information will not appear in modes with lower values. Therefore, the kurtosis mean value of the three larger kurtosis values of IMF can represent the proportion of fault information in the signal.  
The expression of the kurtosis mean is shown in Equation (3).  
\[kurm_{mean}=\frac{1}{N}\sum_{i=1}^{N}kur_{i} \tag{3}\]  
where \(N\) represents the number of IMFs.  
The specific implementation steps of the KMVMD are given as follows.  
Step 1: Determine the number of modes.  
Step 2: Decompose fault signals using VMD in order to obtain IMFs in different \(k\). \(I_{k}=\{IMF_{i}\}(k=2,\ldots,k_{max},i=1,\ 2,\ldots,k)\);  
Step 3: Calculate the kurtosis of each IMF, the set of kurtosis of IMF is obtained. \(kur_{k}=\{kur(IMF_{i})\}(k=2,\ldots,k_{max},i=1,2,\ldots,k)\);Step 4: In \(kur_{k}\), \(j\) larger kurtosis is selected, so the set of larger kurtosis is obtained. \(kur_{k,j}\big{(}k=2,3,4,j=1,\cdots,\big{[}\frac{k}{2}\big{]};k=5,6,\cdots,k_{ max},j=1,2,3\big{)}\). When \(k<5\), \(j=\big{[}\frac{k}{2}\big{]}\), When \(k\geq 5\), \(j=3\); Step 5: Calculate the mean of larger kurtosis. \(kur\_mean_{k}=\Big{\{}\frac{1}{jmax}\sum_{j=1}^{jmax}kur_{K\_j}\Big{\}}\)\(\Big{(}k=2,3,4,j=1,\cdots,\big{[}\frac{k}{2}\big{]};k=5,6,\cdots,k_{ max},j=1,2,3\big{)}\); Step 6: Finding the maximum value \(max\_kur\_mean_{k}\) in \(\{kur\_mean_{k}\}\); Step 7: The mode number corresponding to \(max\_kur\_mean_{K}\) is the number of modes for VMD, and the modes with larger kurtosis values are chosen to obtain the reconstructed signal.