# Fault Feature Extraction of Parallel-Axis Gearbox Based on IDBO-VMD and t-SNE
## 5 Simulation Testing  
For testing the effectiveness of the proposed algorithm improvements, grey wolf optimization (GWO), DBO, the sparrow search algorithm (SSA), and IDBO were selected for comparative optimization on test functions. The test functions are listed in Table 1, where \(F_{1}\)-\(F_{3}\) are single-peak functions, and \(F_{4}\)-\(F_{6}\) are multipeak functions, all with a dimensionality of 30. To ensure fair testing, each algorithm utilized a population size of 30, a maximum iteration count of 500, and was independently tested on the six test functions 10 times to obtain average fitness convergence curves, evaluating their convergence speed. The experimental results are presented in Table 1, using criteria such as the best value, average value, and standard deviation for evaluation.  
From Figure 3, it can be observed that under the same parameter settings, IDBO exhibits a faster convergence speed compared to DBO. In Figure 3c, the convergence speed of IDBO is slightly lower than DBO in the early stages, but it shows improvement in solution accuracy, and its average optimization capability is more stable. A comparison with GWO and SSA also reveals that DBO has good convergence speed and optimization capability. When comparing these four algorithms simultaneously, it is evident that IDBO's convergence speed is significantly higher than the other three. Moreover, it requires the least number of iterations. As the iteration count increases, the convergence curves of DBO, the SSA, and GWO gradually stabilize, and optimization accuracy starts to decrease. This indicates that the three improvement strategies applied to IDBO result in a noticeable enhancement in convergence speed and an improvement in global search capability, showcasing the advantage in local optimization capability.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Test Functions** & **Dimension** & **Range** & **Optimal Value** \\ \hline \(F_{1}(x)=\sum\limits_{i=1}^{n}x_{i}^{2}\) & 30 & \([-100,100]\) & 0 \\ \(F_{2}(x)=\sum\limits_{i=1}^{n}\left(\frac{i}{\sqrt{-1}}\,x_{i}\right)^{2}\) & 30 & \([-100,100]\) & 0 \\ \(F_{3}(x)=\sum\limits_{i=1}^{n}\left[100\left(x_{i+1}-x_{i}^{2}\right)^{2}+(x_ {i}-1)^{2}\right]\) & 30 & \([-30,30]\) & 0 \\ \(F_{4}(x)=\sum\limits_{i=1}^{n}-x_{i}\sin\left(\sqrt{|x_{i}|}\right)\) & 30 & \([-500,500]\) & 0 \\ \(F_{5}(x)=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}x_{i}^{2}} \right)-\exp\left(\frac{1}{n}\sum\limits_{i=1}^{n}\cos(2\pi x_{i})\right)+20+\epsilon\) & 30 & \([-32,32]\) & 0 \\ \(F_{6}(x)=\frac{\pi}{n}\left\{10\sin(\pi y_{1})+\sum\limits_{i=1}^{n}\left(y_{1 }-1\right)^{2}\left[1+10\sin^{2}(xy_{1+i})\right]^{2}\right\}+\sum\limits_{i=2} ^{n}u(x_{i},10,100,4)\) & & \\ \(y_{i}=1+\frac{x_{i}+1}{4}\) & 30 & \([-50,50]\) & 0 \\ \(u(x_{i},a,k,m)=\left\{\begin{array}{ll}k(x_{i}-a)^{m},x_{i}>a\\ 0,-a<x_{i}<a\\ k(-x_{i}-a)^{m},x_{i}<-a\end{array}\right.\) & & \\ \hline \end{tabular}
\end{table}
Table 1: Test functions.  
Table 2 reveals that IDBO did not find the optimal solution only in the case of testing the \(F_{6}\) function. Moreover, its performance in terms of average and standard deviation is similar to DBO. However, as shown in Figure 3, IDBO demonstrates the fastest convergence speed, swiftly locating the optimal values and exploring them in-depth, showcasing higher optimization accuracy. For the \(F_{1}\)-\(F_{5}\) tests, IDBO outperforms other algorithms, showing significant improvements in average optimization capability, precision, and standard deviation. In summary, in the convergence curves of most test functions, the IDBO algorithm exhibits excellent performance. It maintains an absolute advantage in convergence speed  
while achieving high convergence accuracy, reflecting a reasonable balance between global search capability and local development capability.