## 4 Case Study for Comparison
### Results  
First, 100 data sets with a noise level of \(1\times 10^{-3}\) m are used, and the prediction results obtained by applying the proposed method to these data are shown in Figure 8. Figure 8a presents the distribution of the first prediction cycle, denoted as \(k_{0}\) in Step 2 of Equation (5), from 100 data sets, which is caused by the random noise in the data. In the figure, the most frequent first prediction cycle is 2600 cycles, and the majority of the data (82 out of 100) fall within the range of 2000 to 2700 cycles. The red lines in Figure 8b show the RUL predictions from the 100 data sets, while the black diagonal line represents the true RUL. It can be seen that as the cycle increases, the prediction results become closer to the true RUL. The result shown in Figure 5, which was used to explain the proposed method in the previous section, is 1 of these 100 predictions, where the prediction begins at 2600 cycles. The same data sets are used to predict RUL using the ANN with different numbers of hidden nodes (\(n_{H}\)). The prediction is initiated at the same cycle used in Step 2 of the proposed method for each data set (see Figure 8a), and the RUL prediction results for \(n_{H}=1,\ n_{H}=2\) and \(n_{H}=3\) are presented in Figure 9a, Figure 9b, and Figure 9c, respectively. While RUL is consistently over-predicted for all data sets when \(n_{H}=1\), the trend of RUL predictions becomes more erratic and fluctuates significantly when \(n_{H}=3\), indicating increased uncertainty in the prediction. In contrast, the results using \(n_{H}=2\)  
Figure 8: Prediction results using the proposed method with 100 data sets at a noise level of 1 mm: (**a**) distribution of the first cycle for prediction; (**b**) RUL prediction. Figure 7: Illustration of the network model used in this study. exhibit reduced fluctuation compared to using \(n_{H}=3\) while avoiding the overprediction observed with \(n_{H}=1\). This balance between stability and accuracy suggests that \(n_{H}=2\) provides the most reliable RUL predictions, and it is now fixed in the ANN model. As shown in Figure 9a-c, the prediction results from the ANN are erratic, primarily due to variations in the initial settings for weights and biases in each run. To mitigate this uncertainty, the ANN is run 30 times for each of the 100 data sets, and the medians of these 30 results for each data set are presented in Figure 9d. This result is considerably more stable than the single-run outcome shown in Figure 9b. It should be noted that both Figure 9b,d represent results using \(n_{H}=2\). From this point forward, all results from the ANN will be presented as the median of 30 repeats using \(n_{H}=2\). Figure 10 represents 1 of the 100 predictions shown in Figure 9d, which can be compared with the results from the proposed method in Figure 5. These figures demonstrate that the degradation and RUL predictions from the proposed method are consistently closer to the true values throughout all prediction cycles. To compare 100 prediction results from the proposed method and the ANN, the relative errors at each cycle are calculated using the following equation:  
\[e=\frac{\left|t_{\text{EOL}}^{\text{T}}-t_{\text{EOL}}^{\text{P}}\right|}{t_{ \text{EOL}}^{\text{T}}}\times 100 \tag{8}\]  
where T and P indicate true and predicted values, respectively. Since the magnitude of errors using RUL becomes sensitive as the remaining cycles decrease, the EOL is used for error calculation instead. The true EOL is 3045 cycles, and the prediction errors from the two methods are shown in Figure 11. In the figure, the solid red line represents the median error across available prediction results at each cycle, while the dashed and dotted lines  
Figure 9: RUL prediction results using the ANN with 100 data sets at a noise level of 1 mm: (**a**) \(n_{H}=1\); (**b**) \(n_{H}=2\); (**c**) \(n_{H}=3\); (**d**) median of 30 repeats with \(n_{H}=2\). represent the 5 th and 95 th percentiles of the error, respectively.