# Fault Diagnosis of Rolling Bearings Based on Acoustic Signals in Strong Noise Environments
## 2 Methodologies
### Sboa  
The Secretary Bird Optimization Algorithm (SBOA) [26] is a new meta-heuristic optimization algorithm with the advantages of robustness, high efficiency, and fast convergence. It was inspired by the hunting behavior of the secretary bird. The algorithm steps are as follows:  
Firstly, initialize the secretary bird's position using the following formula:  
\[X_{i,j}=lb_{j}+r*\left(ub_{j}-lb_{j}\right) \tag{1}\]where \(X_{i,j}\) represents the value of the \(i\)th secretary bird in the \(j\)th dimension, \(lb_{j}\) and \(ub_{j}\) are the lower and upper bounds of the \(j\)th dimension, respectively, and \(r\) is a random number within the range of [0, 1].  
The SBOA represents the secretary bird group by the following formula:  
\[X=\begin{bmatrix}x_{1,1}&x_{1,2}&\cdots&x_{1,j}&\cdots&x_{1,Dim}\\ x_{2,1}&x_{2,2}&\cdots&x_{2,j}&\cdots&x_{2,Dim}\\ \vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\ x_{i,1}&x_{i,2}&\cdots&x_{i,j}&\cdots&x_{i,Dim}\\ \vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\ x_{N,1}&x_{N,2}&\cdots&x_{N,j}&\cdots&x_{N,Dim}\\ \end{bmatrix}_{N\times Dim} \tag{2}\]  
where \(N\) is the number of secretary birds and \(Dim\) is the dimension of the problem variable, which is equal to 4 in this paper. The SBOA is used to optimize the FMD parameters: the number of modes \(n\), the filter length \(L\), the number of frequency band cuts \(K\), and the cycle  
Figure 1: Illustration of the FMD algorithm.  
period \(m\). Each secretary bird represents a parameter combination, and the objective function value is given by the following formula:  
\[F=\begin{bmatrix}F_{1}\\ \vdots\\ F_{1}\\ \vdots\\ F_{N}\end{bmatrix}_{N\times 1}=\begin{bmatrix}F(X_{1})\\ \vdots\\ F(X_{i})\\ \vdots\\ F(X_{N})\end{bmatrix}_{N\times 1} \tag{3}\]  
where \(F\) is the objective function value vector.  
In this paper, the envelope entropy value is used as the objective function. By comparing the objective function values, the secretary bird with the smallest envelope entropy value is selected as the best candidate solution. The position and objective function value of the secretary bird are updated in each iteration, so the best candidate solution needs to be re-determined in each iteration.  
The test function described by Formulas (4)-(7) is employed to compare the performance of the Snake Optimization Algorithm (SO) [27], Whale Optimization Algorithm (WOA) [28], and Secretary Bird Optimization Algorithm (SBOA). Figure 2 illustrates the convergence curve of the average efficiency value for the selected function obtained from the SBOA. This algorithm demonstrates a rapid convergence speed, strong stability, and high robustness.  
\[F_{1}(x)=\sum\limits_{i=1}^{n}x_{i}^{2} \tag{4}\]  
\[F_{2}(x)=\sum\limits_{i=1}^{n}\left|x_{i}\right|+\lim\limits_{x\to \infty_{i}=1}^{n}\left|x_{i}\right| \tag{5}\]  
\[F_{3}(x)=\sum\limits_{i=1}^{n}\left.ix_{i}^{2}+\text{random}\right.(0,1) \tag{6}\]  
\[F_{4}(x)=-20\exp\left(-\left(\frac{1}{5}\right)\sqrt{\frac{1}{n\sum_{i=1}^{n-1 }x_{i}^{2}}}\right) \tag{7}\]  
Figure 2: Fitness values convergence curves for selected functions under different optimization methods. (**a**) Convergence curve of the F1 function; (**b**) convergence curve of the F2 function; (**c**) convergence curve of the F3 function; (**d**) convergence curve of the F4 function.