## 3 Proposed Method
### Example  
The proposed method is explained using a specific example for better understanding, and Step 1 in Equation (4) is applied to divide the stages. First, the minor growth stage (\(s=1\)) starts with \(k_{0}=1\). Figure 3 shows the regression results for this stage, where the  
Figure 2: Key concept of dividing the stages: (**a**) minor growth stage; (**b**) moderate growth stage.  
black dots represent degradation data obtained every 100 cycles, and the blue lines depict the linear regression results using these data. Figure 3a shows the first regression result using four data points collected up to 300 cycles (\(k=4\)), while Figure 3b shows the final regression result in this stage, using 17 data points collected up to 1600 cycles (\(k=17\)). Fourteen regression results, obtained every 100 cycles from 300 to 1600 cycles, are displayed in Figure 3c. In the proposed method, the criterion for dividing the stages is based on the slope of the regression model, as illustrated in Figure 3d (in the figure, A and B indicate the slopes in Figures 3a and 3b, respectively). As shown, the slope increases consecutively four times from 1200 to 1600 cycles, indicating that 1600 cycles (\(k=17\)) mark the end of the minor growth stage.  
From the results in Figure 2, the moderate growth stage (\(s=2\)) begins at \(k_{0}=17\), and the first regression is performed at 1900 cycles (\(k=20\)) after three additional data points are collected (see \(k=k_{0}+3\) in Equation (4)). The first regression result for the moderate growth stage is shown in Figure 4a. In the figure, the gray dots represent data used to divide the minor growth stage, while only the black dots are used to establish the criteria for the newly started moderate growth stage. The regression result using the four data points is shown as the magenta line in the figure. Similarly, Figure 4b shows the final regression result for this stage (\(k=27\)), using 11 data points collected every 100 cycles from 1600 to 2600 cycles, while Figure 4c presents eight regression results obtained every  
Figure 3: Dividing results for the minor growth stage: (**a**) linear model at \(k=4\); (**b**) linear model at \(k=17\); (**c**) linear models from \(k=4\) to \(k=17\); (**d**) slopes of the linear models.  
100 cycles from 1900 to 2600 cycles. Figure 4d depicts the slope of the regression models during the moderate growth stage, which increases consecutively four times from 2200 to 2600 cycles. With the moderate growth stage ending at 2600 cycles, Step 2 proceeds to predict the RUL of this example.  
Step 1 ended at 2600 cycles (\(k=27\)), and Step 2 in Equation (5) starts with \(k_{0}=27\), assuming \(y_{thres}=0.05\). The first prediction in the accelerated growth stage starts at 2600 cycles, and the result is shown in Figure 5a. In the figure, the prediction result, denoted by the red line, is obtained using the most recent three data points, represented by the black dots. The predicted EOL, which is when the red line reaches the green horizontal line (\(y_{thres}\)), is 3260 cycles. Therefore, the RUL becomes 660 cycles (\(=3260-2600\)), as indicated by the red diamond marker at 2600 cycles in Figure 5d. For reference, the gray curves in Figure 5a-c represent the true degradation behavior, and the black diagonal line in Figure 5d represents the true RUL. The last prediction in this stage is performed at 3000 cycles (\(k=31\)), also using the most recent three data points, as shown in Figure 5b. The RUL prediction is indicated by the red diamond marker at 3000 cycles in Figure 5d. The red lines in Figure 5c show five prediction results performed every 100 cycles from 2600 to 3000 cycles. The final result obtained from the proposed method is the RUL prediction for the entire accelerated growth stage, as shown in Figure 5d.  
Figure 4: Dividing results for the moderate growth stage: (**a**) linear model at \(k=20\); (**b**) linear model at \(k=27\); (**c**) linear models from \(k=20\) to \(k=27\); (**d**) slopes of the linear models.