## 4 Results and Discussion
### RUL Prediction Based on ARIMA-LSTM  
The current study used the performance health index \(g\) of a water hydraulic HSV to predict the RUL of the valve. The data processing procedure that followed is described below. 1. Obtaining data: The initial degradation was regarded as the prediction starting cycle. The end prediction cycle was regarded to be the failure state of the HSV. 2. Data normalization processing: The performance degradation index \(g\) data were normalized so that the input value of the ARIMA-LSTM ranged between [0, 1]. This was done to allow faster and more stable convergence of the ARIMA-LSTM model. The normalization formula used was: \[X_{norm}=\frac{X-X_{min}}{X_{max}-X_{min}}\] (21) where \(X\) is the original data, \(X_{norm}\) is the normalized data, and \(X_{max}\) and \(X_{min}\) are the maximum and minimum value in the original data, respectively. 3. Data inverse normalization: Equation (22) was used when evaluating the model after training to restore (inversely normalize) the normalized data. This was done to evaluate the error of the predicted value of the model. \[X=X_{norm}(X_{max}-X_{min})+X_{min}\] (22)  
Figure 10: Confusion matrix of prediction and true states. 4. Estimation of prediction accuracy: The absolute error and relative error of RUL prediction of the HSV is used to quantitatively evaluate the prediction accuracy of the ARIMA-LSTM model: \[RUL(Kp)=|EOP-Kp|\] (23) \[RUL(ae)=|EOP-EOL|\] (24) \[RUL(re)=\frac{|(EOP-Kp)-(EOL-Kp)|}{EOL-Kp}\times 100\%\] (25) where \(RUL(ae)\) represents the absolute error of RUL, \(RUL(re)\) is the relative error of RUL, \(EOL\) is the operation cycle at end of life, \(EOP\) is the predicted operation cycle at end of life, and \(Kp\) is the operation cycle when the prediction begins. Within the use of the ARIMA model for prediction, the unit root test method is used to assess the stationarity and to determine parameter \(d\) for the ARIMA model. Figure 11 shows the sequence diagram of the original data after first-order difference processing. The results showed that the sequence data stabilized, except for individual noise points. Table 2 shows the unit root test results. The statistical value of \(-118.8474\) was less than the critical statistical value of three confidence levels (1%, 5%, 10%), and the \(p\) value was less than 0.05. Therefore, the difference coefficient of the ARIMA model (d) was 1. The AIC was used to determine \(p\) and \(q\) for the ARIMA model. The value of the AIC is inversely proportional to the fitting performance. However, the performance of the model is proportional to the model complexity, and the increase in the likelihood may also result in model over-fitting. The values of parameters were limited to [1; 10] to limit the scale of the model and to avoid over-fitting, and the most appropriate value was found through circulation. Table 3 shows the AIC values corresponding to different \(p\) and \(q\). As shown in Table 3, a minimum AIC of \(-2.3043\) was obtained under \(p\) and \(q\) of 8 and 9, respectively. Therefore, the determined model was ARIMA (8, 1, 9). \begin{table}
\begin{tabular}{c c} \hline
**Items** & **Value** \\ \hline Test statistic & \(-118.8474\) \\ \(p\) value & \(0.0001\) \\
1\% level & \(-3.430571\) \\
5\% level & \(-2.861522\) \\
10\% level & \(-2.566801\) \\ \hline \end{tabular}
\end{table} 
Table 2: Results of unit root test. Figure 11: Sequence diagram after first-order difference processing. The model code was written in the Matlab program language. The parameters of the LSTM model were determined and set according to experience and comparison of multiple residual experiments. The hidden layer comprised three layers, containing 30, 300, and 30 neurons, respectively, whereas the number of iteration epochs was 800. Dropout technology was used in model training to further prevent the occurrence of overfitting. The loss rate of the experimental model was set to 0.2, i.e., 20% of the nodes were discarded in each weight update. The learning rate was set to 0.005.