# A Performance Prediction Method Utilizing Time-Dependent Subsystem Transfers between Family Systems
## 6 Examples of Verification
### FAST Performance Prediction
According to the prediction process shown in Figure 3, based on expert experience, design requirements, and on-site usage needs, the absolute error values of the optimal state and failure state can be determined to be \([0,250]\); that is, when the extension error is 0 \(\upmu\)m, the actuator subsystem is in the optimal state. When the extension error exceeds 250 \(\upmu\)m, it is determined that the actuator subsystem has failed. Considering computational complexity, the interval for dividing the family system is selected as 50 \(\upmu\)m and the intervals from the optimal state to the failure state are \([0,\,50)\,,[50,\,100)\,,\,[100,\,150)\,,\,[150,\,200)\,,\,[200,\,250)\,,\) and \((250,\,+\infty)\,,\) respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**DEVICE ID**} & **CONTROL** & **CONTROL** & **CONTROL** & **CONTROL** \\  & **THEORY** & **ACTUAL** & **TEMPERATURE** & **PRESSURE** \\ \hline
1 & 1,070,000 & 1,070,000 & 26.17 & 2777 \\
2 & 961,943 & 962,634 & 25.48 & 2293 \\
3 & 1,000,000 & 1,000,000 & 25.98 & 2383 \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\
2223 & 445,522 & 445,620 & 26.05 & 3700 \\
2224 & 389,577 & 389,675 & 26.2 & 5125 \\
2225 & 395,675 & 395,731 & 25.14 & 4397 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Partial data of actuator monitoring data at a certain time. The actuator monitoring data obtained during a specific observation period using the FAST telescope were analyzed, resulting in a total of 259 datasets. At the initial moment \(t_{0}\), multiple random moments were selected from the monitoring data for prediction and comparison to validate the efficacy of the approach. The randomly selected initial moments include the 5 th, 29 th, 41 st, 65 th, 88 th, 126 th, 143 rd, 162 nd, 211 th, and 240 th, totaling 10 moments. The comparison between the prediction and actual values at each moment is shown in Table 2, where \(\hat{ST}_{1}^{t_{i+1}}\), \(\hat{ST}_{2}^{t_{i+1}}\), \(\hat{ST}_{3}^{t_{i+1}}\), \(\hat{ST}_{4}^{t_{i+1}}\), \(\hat{ST}_{5}^{t_{i+1}}\), and \(\hat{ST}_{6}^{t_{i+1}}\) are the predicted values for each state interval at the next time of the 10 initial moments, and \(ST_{1}^{t_{i+1}}\), \(ST_{2}^{t_{i+1}}\), \(ST_{3}^{t_{i+1}}\), \(ST_{4}^{t_{i+1}}\), and \(ST_{6}^{t_{i+1}}\) are the actual values for each state interval at the next time of the 10 initial moments. To evaluate the prediction accuracy of the model, the relative error \(\delta\) is used for judgment:  
\[\delta_{r}=\frac{\left|ST_{r}^{t_{i+1}}-\hat{ST}_{r}^{t_{i+1}}\right|}{ST_{r}^ {t_{i+1}}}\times 100\% \tag{16}\]  
Here, \(ST_{r}^{t_{i+1}}\) is the actual number of subsystems within the \(r\)-th state interval and \(\hat{ST}_{r}^{t_{i+1}}\) is the predicted number of subsystems within the \(r\)-th state interval. Figure 3: Flow chart of time-varying inter-family transfer life prediction. Figure 5: The number of subsystems in the second state interval and their relative error. Figure 6: The number of subsystems in the third state interval and their relative error. Figure 7: The number of subsystems in the fourth state interval and their relative error. The average relative error of the prediction detected using this method during the random sampling period is 13.3%, as concluded from comparing the rounded predicted values of inter-family transfer in Table 2 with the actual values. Among the predicted values for 60 state intervals at 10 random times, the relative error of 40 predicted values is within 10%. The root mean square error (RMSE), fitting accuracy (R\({}^{2}\)), mean absolute error (MAE), and mean absolute percentage error (MAPE) of the conventional evaluation prediction model are calculated. The aforementioned accuracy metrics are applicable to regression models, enabling the assessment of the model's conformity to the observed data and the quantification of disparities between actual observations and predicted values.