# Prediction of Tooth Profile Deviation for WEDM Rigid Gears Based on ISSA-LSSVM
## 2 Methodology
### Performance Test of ISSA  
To assess the effectiveness and innovation of the ISSA, simulation experiments were conducted using four benchmark test functions listed in Table 1 [33]. Specifically, F1 and F2 represent unimodal functions, while F3 and F4 represent complex multimodal functions.  
The optimization performance of the ISSA algorithm was demonstrated by comparing it with PSO, GWO, and SSA.  
The following are the initial parameter settings for each optimization algorithm. The number of populations \(N\) and the number of iterations \(K\) remained consistent (\(N=30\), \(K=100\)). For PSO, the learning factors c1 and c2 were set to the same value (c1 = c2 = 1.5), with an elasticity coefficient of 0.8. The safety thresholds for the ISSA and SSA were defined as 0.8, and the discoverers and guards accounted for 20% each of the sparrow population. In the ISSA, the maximum coefficient of dynamic inertia weights is 0.9, while the minimum coefficient is 0.6.  
In order to minimize the influence of chance events and enhance the persuasiveness of the experiment, the average value and standard deviation were selected to assess the optimization ability and stability of each algorithm. The results of optimizing the benchmark functions using the ISSA, SSA, GWO, and PSO are presented in Table 2. The ISSA demonstrates superior global optimization ability for both unimodal and multimodal functions compared to the SSA, GWO, and PSO. Figure 3 illustrates the convergence curves of each algorithm, showing that the ISSA converges to the global optimal solution faster and performs well across various classes of test functions.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & **Dimension** & **Range** & \(\boldsymbol{F_{\min}}\) \\ \hline \(F_{1}(x)=\sum\limits_{i=1}^{d}\left(\sum\limits_{j=1}^{d}x_{j}\right)^{2}\) & 30 & [\(-\)30,30] & 0 \\ \(F_{2}(x)=\frac{d}{i=1}\left[100(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2}\right]\) & 30 & [\(-\)30,30] & 0 \\ \(F_{3}(x)=\sum\limits_{i=1}^{d}\left[x_{i}^{2}-10\cos(2\pi x_{i})+10\right]\) & 30 & [\(-\)600,600] & 0 \\ \(F_{4}(x)=\frac{1}{4000}\sum\limits_{i=1}^{d}x_{i}^{2}-\frac{d}{i=1}\cos \left(\frac{x_{i}}{\sqrt{i}}\right)+1\) & 30 & [\(-\)5.12,5.12] & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Selected 4 benchmark functions.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**F**} & \multirow{2}{*}{**Parameters**} & \multicolumn{4}{c}{**Optimization Algorithm**} \\ \cline{3-5}  & & **ISSA** & **SSA** & **GWO** & **PSO** \\ \hline \(F_{1}\) & Average value & \(6.86\times 10^{-291}\) & \(6.62\times 10^{-221}\) & 0.00218 & \(6.08\times 10^{3}\) \\  & Standard deviation & 0 & 0 & 0.00306 & \(2.28\times 10^{3}\) \\ \(F_{2}\) & Average value & \(3.17\times 10^{-6}\) & \(1.33\times 10^{-4}\) & 26.75 & \(2.78\times 10^{2}\) \\  & Standard deviation & \(5.83\times 10^{-6}\) & \(2.40\times 10^{-4}\) & 0.77846 & \(1.48\times 10^{2}\) \\ \(F_{3}\) & Average value & 0 & 0 & 6.99 & 27.20 \\  & Standard deviation & 0 & 0 & 5.08 & 7.43 \\ \(F_{4}\) & Average value & 0 & 0 & 0.00297 & \(4.07\times 10^{2}\) \\  & Standard deviation & 0 & 0 & 0.00685 & 32.34 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison and analysis of ISSA and other algorithms.