# Research on Distributed Autonomous Timekeeping Algorithm for Low-Earth-Orbit Constellation
## 4 Experiment and Results Analysis
### Prediction Error Evaluation  
The prediction error between the autonomous time reference and the actual system time is a critical evaluation metric, and different datasets exhibit varying behaviors across different algorithms. The most commonly used statistical prediction models include the linear model, quadratic polynomial model, and gray prediction model [27; 28; 29], which are suitable for clock error modeling and prediction in cases of small drift terms, large drift terms, and noticeable exponential trends, respectively.  
Figure 11 presents the 20-day prediction error curve. The error of an individual atomic clock increases over time, and the 20-day prediction error of a poorly performing satellite clock can reach up to 600 ns. Such an error is intolerable if the system is synchronized to this satellite. In contrast, the 20-day prediction error of the time reference established by the D-KPW (Control) algorithm is only 3.40 ns, allowing the system to synchronize with it and achieve high precision.  
Figure 12 illustrates the root mean square error (RMSE) of the 20-day predictions for the time scale (\(TA\)) and all the satellites involved in the calculation. The RMSE of \(TA\) is 1.55 ns, outperforming that of all the individual satellites included in this analysis.  
Figure 11: Twenty-day forecast residual.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & **1000 s** & **10,000 s** & **100,000 s** & **1,000,000 s** \\ \hline Equal weight & \(1.07\times 10^{-13}\) & \(2.39\times 10^{-14}\) & \(1.20\times 10^{-14}\) & \(5.43\times 10^{-14}\) \\ ALGOS & \(7.50\times 10^{-14}\) & \(1.15\times 10^{-14}\) & \(6.02\times 10^{-15}\) & \(2.01\times 10^{-14}\) \\ AT1 & \(2.09\times 10^{-14}\) & \(1.58\times 10^{-14}\) & \(1.62\times 10^{-14}\) & \(8.15\times 10^{-14}\) \\ Kalman & \(1.72\times 10^{-13}\) & \(4.94\times 10^{-14}\) & \(2.09\times 10^{-14}\) & \(1.21\times 10^{-14}\) \\ D-KPW & \(4.49\times 10^{-14}\) & \(1.06\times 10^{-14}\) & \(8.27\times 10^{-15}\) & \(4.19\times 10^{-15}\) \\ D-KPW & \(7.40\times 10^{-15}\) & \(7.86\times 10^{-15}\) & \(7.83\times 10^{-15}\) & \(2.30\times 10^{-15}\) \\ (Control) & \(7.40\times 10^{-15}\) & \(7.86\times 10^{-15}\) & \(7.83\times 10^{-15}\) & \(2.30\times 10^{-15}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Allan deviation under several sampling times.  
To further verify the prediction performance of the time reference established by the D-KPW (Control) algorithm, a comparison is made with the time references established by the equal weight average algorithm, AT1 algorithm, ALGOS algorithm, Kalman filter algorithm, and D-KPW algorithm.  
Figures 13 and 14 display the prediction residual curves and the RMSE of the six algorithms for 3 days, 5 days, 10 days, and 20 days, respectively. Table 4 shows that the RMSE of the prediction error for the time reference established by the D-KPW (Control) algorithm reaches \(7.82\times 10^{-10}\) s, \(8.01\times 10^{-10}\) s, \(1.03\times 10^{-9}\) s, and \(1.55\times 10^{-9}\) s for 3, 5, 10, and 20 days, respectively, outperforming the other algorithms in comparison. The RMSE of the 20-day prediction residuals generated by the D-KPW (Control) algorithm is 1.55 ns, with a prediction accuracy that is 78%, 65%, 65%, 66%, and 67% higher than those of equal-weight averaging, ALGOS, AT1, the Kalman filter, and D-KPW, respectively. In the system's independent operation mode, the D-KPW (Control) algorithm achieves the minimum absolute deviation in the time reference, maintaining a high level of synchronization. This indicates that the time reference established by the D-KPW (Control) algorithm offers better predictability.