## 4 Simulations and Discussion
### The Prediction Accuracy of the BA-ELM  
The high fitting accuracy in the training samples cannot illustrate the performance of the BA-ELM model comprehensively. In practical operations, the BA-ELM is to be utilized to predict the systematic error of the star centroid at random positions. Therefore, the prediction accuracy is a critical parameter to evaluate the BA-ELM model. Firstly, the prediction accuracy is defined in this section.  
Let \(\hat{x}_{g}\) be the actual star centroid position obtained by the CoM method, and let \(\delta_{BA-ELM}(\hat{x}_{g})\) be the predicted systematic error of the BA-ELM model with input \(\hat{x}_{g}\); we have the compensated star centroid \(\hat{x}_{g}^{c}\) as  
\[\hat{x}_{g}^{c}=\hat{x}_{g}-\delta_{BA-ELM}(\hat{x}_{g}). \tag{43}\]  
Then, the prediction error \(\xi\) of the BA-ELM model can be expressed by the following equation:  
\[\xi=\hat{x}_{g}^{c}-x_{0}. \tag{44}\]  
where \(x_{0}\) is the ideal star centroid position.  
After obtaining a trained BA-ELM model with optimal parameters, its prediction accuracy was calculated. In the simulations, 600 positions on the image plane were selected randomly, and then we made a star spot centroid traverse these positions sequentially. The coordinates in the \(x\) direction of these selected positions ranged from 1 to 512.  
Figure 10 shows the respective results of the prediction accuracy in the 600 random positions under \(\sigma_{PSF}\) = 0.3 and \(\sigma_{PSF}\) = 0.671. The red lines were formed by connecting all ideal coordinates (\(x_{0}\), \(\sigma_{\hat{x}_{g}}\)) successively, while the blue lines were formed by connecting the compensated coordinates (\(\hat{x}_{g}^{c}\), \(\delta_{BA-ELM}(\hat{x}_{g})\)). From Figure 10a2,b2, which are enlarged pictures relative to Figure 10a1,b1, we can see that the blue lines nearly the overlap red lines; this means that each compensated star position \(\hat{x}_{g}^{c}\) is extremely close to its corresponding ideal position \(x_{0}\) in all test positions. The prediction error is illustrated in Figure 11.  
Figure 9: The fitting error of the BA-ELM compensation method. (**a**) Gaussian width \(\sigma_{PSF}\) = 0.3. (**b**) Gaussian width \(\sigma_{PSF}\) = 0.671.  
As shown in Figure 11, the maximum prediction errors of the BA-ELM were \(2.5\times 10^{-7}\) pixels and \(3.0\times 10^{-7}\) pixels when the \(\sigma_{PSF}\) values were 0.3 and 0.671, respectively. The test results indicate that the proposed compensation method based on the BA-ELM model can achieve excellent prediction accuracy under different Gaussian widths.  
Figure 11: The prediction error of the BA-ELM compensation method. (**a**) Gaussian width \(\sigma_{PSF}\) = 0.3. (**b**) Gaussian width \(\sigma_{PSF}\) = 0.671.  
Figure 10: Experiment results of random positions. (**a1,a2**) Gaussian width \(\sigma_{PSF}\) = 0.3. (**b1,b2**) Gaussian width \(\sigma_{PSF}\) = 0.671.  
In Figure 12, the 10 brightest stars were selected, and their ideal position in the \(x\) direction (IX), position in the \(x\) direction before compensation (XBC), position in the \(x\) direction after compensation (XAC), and errors in the \(x\) direction before compensation (EXBC) and after compensation (EXAC) are listed in Table 2.  
In Figure 13, the 10 brightest stars were selected, and their ideal position in the \(x\) direction (IX), position in the \(x\) direction before compensation (XBC), position in the \(x\) direction after compensation (XAC), and errors in the \(x\) direction before compensation (EXBC) and after compensation (EXAC) are listed in Table 3.  
Through comparing the EXBC and EXAC values in Tables 2 and 3, it was noticed that the proposed BA-ELM compensation method can drastically reduce the systematic error and achieve high accuracy in estimating star centroid locations.