## 2 Methods
### Arima-LSTM Method for RUL Prediction  
When compared with the traditional ANN method, the LSTM algorithm can fully capture nonlinear trends and correlations within data. However, there are some disadvantages associated with the single LSTM model, including a slow convergence speed, a large influence of outliers, and local minima [35]. The ARIMA model is suitable for extracting the linear trend in a data series [36; 37]. The research [38; 39; 40] shows that the combination of the ARIMA and LSTM models can improve the accuracy of trend prediction. Therefore, the present study used a hybrid ARIMA-LSTM model to predict the RUL of an HSV. (1) ARIMA  
The ARIMA model is the concept of predicting a future time series value according to the historical time series data. This method is valid when the current value within a time series has a linear relationship with a past value of the time series and random interference. The model can be expressed as ARIMA (\(p\), \(d\), \(q\)), which is a combination of auto-regression AR(\(p\)), moving average MA(\(q\)), and the degree of differencing \(d\):  
\[\left(1-\sum_{i=1}^{p}\varphi_{i}L^{i}\right)(1-L)^{\text{d}}\chi_{t}=\left(1 +\sum_{i=1}^{q}\theta_{i}L^{i}\right)\epsilon_{t} \tag{9}\]  
where \(L\) is the lag operator, \(\varphi_{i}\) and \(\theta_{i}\) represent the parameters of the AR and MA parts, respectively, and \(\epsilon_{t}\) represents the error terms. The ARIMA process consists of four steps:  
1. Sequence stabilization processing: A non-stationary data series requires differential processing for stabilization. The difference coefficient \(d\) is used to construct a stable data series. 2. Model order determination: The ARIMA requires the determination of three parameters. Parameter \(d\) is determined in step (a), whereas parameters \(p\) and \(q\) are determined through ACF and PACF diagrams. AIC is used to select a set of optimal model parameters. The expression of the AIC is shown in Equation (10). \[AIC=2k-2\ln(L) \tag{10}\]  
Model test: The validity of the model can be verified by assessing whether the model residual is a white noise sequence. Prediction: A future value of the time series can be predicted according to the historical time series. LSTM usually includes three gates: input gate; forgetting gate; output gate. Figure 2 is a structure of the LSTM model [25]. The LSTM model constructs a relationship from the input sequence to an output sequence using Equations (11)-(15). Within these equations, \(\sigma\) is the activation function, and \(\tanh\) is the tangent hyperbolic function. Within these two functions, \(h,c\), and \(t\) represent hidden states, memory cell state, and time steps, respectively. As such, \(h_{t-1}\) is the previous hidden state, \(C_{t}\) is the current memory cell state, and \(\odot\) indicates element-wise multiplication. \[i_{t}=\sigma(w_{i}[h_{t-1},X_{t}]+b_{i}) \tag{11}\]  
\[o_{t}=\sigma(w_{o}[h_{t-1},X_{t}]+b_{o}) \tag{12}\]  
\[f_{t}=\sigma\Big{(}w_{f}[h_{t-1},X_{t}]+b_{f}\Big{)} \tag{13}\]  
\[c_{t}=f_{t}\odot c_{t-1}+i_{t}\odot\tanh(w_{c}[h_{t-1},X_{t}]+b_{c}) \tag{14}\]  
\[h_{t}=o_{t}\odot\tanh(c_{t}) \tag{15}\]  
The HSV health index curve contains both linear and nonlinear elements. Therefore, the use of a single model is inadequate for prediction. The present study constructed a hybrid model. Firstly, raw data were filtered using the ARIMA model with the nonlinear law included in the model residuals. The residuals were then input into the LSTM model to extract the nonlinear law. Finally, the predictions based on the ARIMA-LSTM model were superimposed to obtain the final prediction results. Figure 3 shows the workflow of the hybrid models. Figure 2: Dataflow of an LSTM cell.