# A Performance Prediction Method Utilizing Time-Dependent Subsystem Transfers between Family Systems
## 6 Examples of Verification
### FAST Performance Prediction
According to Formula (17), the RMSE value of the time-varying inter-family transfer prediction method is 35.57 (accounting for 1.598% of the total amount in the hydraulic actuator cluster system). \[RMSE=\sqrt{\frac{\sum\limits_{i=1}^{n}{(p_{i}-\alpha_{i})^{2}}}{n}} \tag{17}\]  
where \(p\) is the predicted value and \(\alpha\) is the actual value. Figure 8: The number of subsystems in the fifth state interval and their relative error. Figure 9: The number of subsystems in the sixth state interval and their relative error. According to Formula (18), the R\({}^{2}\) value is 0.9906. \[R^{2}=\frac{\sum\limits_{i=1}^{n}{(\hat{y}_{i}-\overline{y})^{2}}}{\sum\limits_{ i=1}^{n}{(y_{i}-\overline{y})^{2}}} \tag{18}\]  
where \(\hat{y}\) is the predictive value, \(y\) is the actual value, and \(\overline{y}\) is the average value. According to Formula (19), the MAE value is 22.1 (accounting for 0.9932% of the total amount in the hydraulic actuator cluster system). \[\delta_{r}=\frac{\left|ST_{r}^{t_{i+1}}-\hat{ST}_{r}^{t_{i+1}}\right|}{ST_{r}^ {t_{i+1}}}\times 100\% \tag{16}\]0  
where \(\hat{y}\) is the predictive value, \(y\) is the actual value, and \(n\) is the number of samples. According to Formula (20), the MAPE value is 13.29% (the model's average prediction error accounts for 13.29% of the true values). \[\delta_{r}=\frac{\left|ST_{r}^{t_{i+1}}-\hat{ST}_{r}^{t_{i+1}}\right|}{ST_{r}^ {t_{i+1}}}\times 100\% \tag{16}\]1  
where \(\hat{y}\) is the predictive value, \(y\) is the actual value, and \(n\) is the number of samples. The summary of the aforementioned accuracy indicators can be found in \[\delta_{r}=\frac{\left|ST_{r}^{t_{i+1}}-\hat{ST}_{r}^{t_{i+1}}\right|}{ST_{r}^ {t_{i+1}}}\times 100\% \tag{16}\]2  
This method is observed to meet the requirements for predicting inter-family transfer performance, indicating its credibility in life prediction through time-varying inter-family transfer. Due to the extensive observation range of the FAST telescope, which enables a multitude of scientific observation tasks, diverse targets necessitate distinct observation strategies for determining their reflective surface shape. Due to spatial constraints, the observation strategy requirements in this case study have been simplified. The remaining lifespan of the hydraulic actuator group system has been transformed into a correlation curve between the number of subsystems meeting accuracy criteria and transfer times. By integrating this curve with Equation (14) and preliminary observations, valuable insights regarding the projected residual life can be obtained. Taking the 240 th moment in the data as an example, the transition matrix from 239 to 240 is as follows:  
\[T_{t_{229}\to t_{240}}=\left[ \begin{array}{cccccc}0.93&0.05&0.01&0&0&0.01\\ 0.02&0.91&0.06&0.01&0&0\\ 0.01&0.09&0.86&0.01&0.01&0.02\\ 0&0.07&0.24&0.24&0.07&0.38\\ 0&0&0.03&0.08&0.21&0.68\\ 0&0&0&0.02&0.02&0.96\\ \end{array} \right]\]  
The relationship between the number of actuators meeting the accuracy requirements for prediction at 240 time intervals and the number of transfer times can be derived using Formula (13) based on the 240 time series system state data, as illustrated in Figure 10. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Accuracy Metrics** & RMSE & R\({}^{2}\) & MAE & MAPE \\ \hline
**Value** & 35.57 & 0.9906 & 22.1 & 13.29\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Accuracy metrics. The prediction curve is subject to temporal variations, particularly in the presence of significant changes in the control state or environment. Consequently, the state matrix undergoes substantial modifications, necessitating synchronous updates of both the prediction curve and life prediction values.