# A Novel Rolling Bearing Fault Diagnosis Method Based on MFO-Optimized VMD and DE-OSELM
## 4 VMD Optimization Based on the Improved MFO  
### Optimal Process  
Modal aliasing may likely occur if the VMD decomposition process is accompanied by improperly chosen parameters [40]. To prevent the influence of artificial parameters, the improved MFO algorithm is implemented in the parameter selection process of VMD to achieve the adaptive selection of the critical parameters of VMD.  
This paper employs envelope entropy to describe the coefficient characteristics of signals. A smaller envelope entropy indicates more regular fault pulses [27]. Thus, the minimum average envelope entropy (MAEE) is utilized as the fitness function, which is defined as:  
\[p_{i}(n)=\frac{a_{i}(n)}{\sum_{n=1}^{N}a_{i}(n)} \tag{19}\]  
\[H_{en}(i)=-{\sum_{n=1}^{N}p_{i}(n)\log_{2}(p_{i}(n))} \tag{20}\]  
\[MAEE=\min_{(K_{A})}\biggl{\{}\frac{1}{K}{\sum_{i=1}^{K}H_{en}(i)}\biggr{\}} \tag{21}\]  
where; \(a_{i}(n)\) is the envelope signal of the \(i\)--\(th\) modal component, \(p_{i}(n)\) is the normalized form of \(a_{i}(n)\), and \(H_{en}(i)\) is the envelope entropy value of the \(i-th\) modal component.  
After the original signal has been subjected to VMD decomposition, it gives rise to several modal components. The efficacy of the decomposition can be evaluated through determination of the average envelope entropy of each modal component. A decrease in the average envelope entropy value indicates a reduction in noise signals in the decomposed modal components, yielding more consistent fault pulses.  
The specific process of the MFO-optimized VMD proposed in this paper is as follows:  
1. The moth population and flame matrix are initialized, and MAEE was used as fitness function.
2. VMD decomposition was performed on the position of each moth to obtain the fitness value of each moth. The moth population was sorted according to the fitness value, and the former flameno flames were selected to construct the matrix.
3. Select the first m flame and any flame for \(p\) times cross recombination, and, if the fitness value of the new flame is better than the original flame, the original flame is replaced.
4. Equation (14) was used to update the moth population position, and VMD decomposition was performed on the signal in the new position of each moth to calculate the fitness value of each moth and update the moth population and flame population.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Test Function** & **Optimization Method** & **Optimal Value** & **Average Value** & **Standard Deviation** \\ \hline \(f_{1}\) & MFO & \(1.805\times 10^{3}\) & \(2.267\times 10^{4}\) & \(1.206\times 10^{4}\) \\  & PSO & \(1.733\times 10^{3}\) & \(6.425\times 10^{3}\) & \(4.266\times 10^{3}\) \\  & GWO & \(4.375\times 10^{-30}\) & \(1.048\times 10^{-22}\) & \(3.188\times 10^{-22}\) \\  & Improved MFO & 0 & 0 \\ \(f_{2}\) & MFO & \(1.786\times 10^{-5}\) & 37.001 & 24.516 \\  & PSO & 9.450 & 17.299 & 7.075 \\  & GWO & \(5.2215\times 10^{-53}\) & \(3.995\times 10^{-52}\) & \(3.899\times 10^{-52}\) \\  & Improved MFO & 0 & 0 \\ \(f_{3}\) & MFO & \(4.436\times 10^{-6}\) & 366.667 & 441.262 \\  & PSO & 97.068 & 297.065 & 92.301 \\  & GWO & \(9.625\times 10^{-93}\) & \(9.744\times 10^{-18}\) & \(3.885\times 10^{-87}\) \\  & Improved MFO & 0 & 0 & 0 \\ \(f_{4}\) & MFO & 0.9313 & 17.3889 & 5.0531 \\  & PSO & 4.001 & 5.156 & 0.563 \\  & GWO & \(7.105\times 10^{-15}\) & \(1.042\times 10^{-14}\) & \(2.787\times 10^{-15}\) \\  & Improved MFO & 0 & 0 & 0 \\ \hline \end{tabular}
\end{table}
Table 1: Test results of the four algorithms.  
(5) The algorithm determines whether the pre-determined number of iterations has been reached. In case it has not, the algorithm repeats the process in steps 2 to 4. When the predetermined number of iterations has been reached, the iteration stops, and the optimal configuration of parameters is presented as output.