# A New BDS-2 Satellite Clock Bias Prediction Algorithm with an Improved Exponential Smoothing Method
## 4 Results and Discussion
### Medium-Term Forecast Performance Analysis
The changes in the bar chart for 24 h forecast accuracy (RMS) and stability (Range) using different models are shown in Figure 11, where Figure 11a is the changes of the forecast accuracy (RMS) for the different satellites, and Figure 11b is the changes of the forecast stability (Range) for the different satellites. Figure 12 shows the boxplot changes of 24 h prediction accuracy and stability using different models, where Figure 12a is the boxplot changes of RMS of prediction errors, and Figure 12b is the boxplot changes of Range of prediction errors. Figure 13 reveals the changes of the average accuracy (RMS), stability (Range), and improvement percentage for 24 h prediction, where Figure 13a is the changes of the average of RMS and Range, Figure 13b is the improvement of the ES2 + SW, ES2 + GM + SW, ES3 + SW, and ES3 + GM + SW model compared to the ES2, ES2 + SW, ES3, and ES3 + SW model, respectively. The 24 h prediction statistics for the different satellites with different models are exhibited in Table 4, and the Statistics of the average forecast precision (RMS), stability (Range), and improvement of the different models for 24 h prediction are shown in Table 5.  
From Figures 10-13 and Tables 4 and 5, it can be seen that the medium-term forecast performance of the ES2 model is slightly better than that of the ES3 model with respect to single-model forecasting. These mean that the ES3 model is more suitable for short term forecast, while the ES2 model is more suitable for medium-term forecast by comparison with the abovementioned parts. Compared with the prediction performances of the ES2 model and the ES3 model, the GM model has a slightly worse performance in medium-term prediction. The forecast performance of both the ES2 model and ES3 model has been boosted to a great degree after using the SW processing. Compared with the single ES2 (ES3) model, the average forecast accuracy of the combined model of the ES2 (ES3) model and the SW processing has been increased by 42.80% (31.00%) from 5.56 ns (6.77 ns) to 3.18 ns (4.67 ns), and the average forecast stability has been increased by 17.90% (14.10%) from 8.99 ns (16.13 ns) to 7.38 ns (13.86 ns). The aforementioned comparative analysis shows that the use of the SW processing strategy can significantly boost the forecast performance of the ES model, especially for the ES2 model, both in terms of forecast accuracy and forecast stability. The forecast accuracy and stability of these fusion prediction algorithms are further improved by using the GM model to learn the fitting errors of the ES model in the modeling process followed based upon the SW processing. Compared with the single ES2 (ES3) model that does not use the GM model to learn the fitting error, the average forecast accuracy of these fusion algorithms is increased by 11.00% (36.80%) from 3.18 ns (4.67 ns) to 2.83 ns (2.95 ns), and the average forecast stability is increased by 43.20% (41.30%) from 7.38 ns (13.86 ns) to 4.19 ns (8.13 ns). Overall, the GM model is used to learn the fitting error of the ES model, and then the fusion prediction algorithm with the SW processing can significantly improve the performance of the ES model. In particular, the performance of the medium-term prediction of the ES3 model has been improved greatly. In addition, both single algorithm and fusion algorithm have better prediction performance for recently launched satellites than for early launched satellites.