# A New BDS-2 Satellite Clock Bias Prediction Algorithm with an Improved Exponential Smoothing Method
## 4 Results and Discussion
### Short-Term Forecast Performance Analysis
Figure 8 shows the boxplot changes of 6 h prediction accuracy (RMS) and stability (Range) using different models, where Figure 8a is the boxplot changes of RMS of prediction errors, and Figure 8b is the boxplot changes of Range of prediction errors. Figure 9 shows the changes of the average accuracy (RMS), stability (Range), and improvement percentage for 6 h prediction using different models, where Figure 9a is the changes of the average of RMS and Range, Figure 9b is the improvement of the ES2 + GM (ES3 + GM) model compared to the ES2 (ES3) model The 6 h prediction statistics for the different satellites with different models are listed in Table 2, and the statistics of the average forecast precision (RMS), stability (Range), and improvement of the different models for 6 h prediction are shown in Table 3.  
From Figures 6-9 and Tables 2 and 3, we can see that in terms of a single model, the average forecast accuracy of the ES3 model is roughly the same as that of the ES2 model, while the average forecast accuracy of the ES3 model is slightly better than that of the ES2 model. The average forecast accuracy of the GM model is worse than that of the ES2 model. As for the combined model, the GM model is used to learn the fitting errors of the ES model and forecast the values of the fitting errors. The comparative analysis between the combined models (ES2 + GM and ES3 + GM) and the single ES models (ES2 and ES3) show that the prediction results of the combined ES model can boost the forecast performance of the ES model. Especially, the performance enhancement of the ES3 model is remarkable and their prediction accuracy has been further heightened. Compared with the single ES (ES2 and ES3) model, the average forecast accuracy of the ES2 + GM (ES3 + GM) model has been increased by 13.30% (25.30%) from 3.68 ns (3.59 ns) to 3.19 ns (2.68 ns), and the average forecast stability of the ES2 + GM (ES3 + GM) model has been increased by 14.10% (32.60%) from 1.42 ns (2.98 ns) to 1.22 ns (2.01 ns). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Strategies**} & \multirow{2}{*}{**Criteria**} & \multicolumn{2}{c}{**GEO**} & \multicolumn{2}{c}{**IGSO**} & \multicolumn{2}{c}{**MEO**} \\ \cline{3-8}  & & **C 01** & **C 05** & **C 10** & **C 16** & **C 11** & **C 12** \\ \hline \multirow{3}{*}{GM} & RMS & 36.90 & 43.91 & 16.38 & 3.77 & 12.93 & 3.51 \\  & Range & 73.10 & 68.98 & 20.45 & 7.05 & 19.64 & 4.09 \\  & RMS & 8.79 & 7.95 & 9.08 & 1.89 & 3.39 & 2.24 \\  & Range & 21.38 & 11.00 & 9.09 & 5.22 & 3.96 & 3.27 \\  & RMS & 7.10 & 3.72 & 3.78 & 1.40 & 1.95 & 1.13 \\  & Range & 17.08 & 9.86 & 7.36 & 3.22 & 3.82 & 2.94 \\  & RMS & 6.14 & 3.16 & 3.91 & 1.27 & 1.67 & 0.81 \\  & Range & 5.73 & 6.86 & 4.61 & 2.79 & 3.00 & 2.48 \\  & RMS & 9.21 & 8.11 & 9.01 & 2.96 & 8.41 & 2.93 \\ ES3 & Range & 20.06 & 29.00 & 19.58 & 6.58 & 14.69 & 6.85 \\  & RMS & 8.30 & 7.36 & 5.67 & 1.04 & 4.47 & 1.18 \\  & Range & 15.75 & 27.69 & 15.96 & 4.82 & 14.03 & 4.88 \\  & RMS & 5.41 & 3.75 & 4.61 & 0.67 & 2.21 & 1.05 \\ ES3 + GM + SW 1 & Range & 6.09 & 13.49 & 13.12 & 3.35 & 8.20 & 4.53 \\ \hline \hline \end{tabular}  
* 1: ES + SW implies the two exponential smoothing model with the sliding window, ES2 + GM + SW implies the combined model of the ES2 + GM with the sliding window, ES3 + SW implies the three exponential smoothing model with the sliding window, and ES3 + GM + SW implies the combined model of the ES3 + GM using the sliding window.  
\end{table} 
Table 4: Statistics of the different models for 24 prediction (ns).