## 4 Case Study for Comparison
### Results
It is clearly shown that the error from the proposed method in Figure 11a is much smaller and more precise than that from the ANN in Figure 11b. The percentiles of the prediction errors in Figure 11 are listed in Table 1 for cycles ranging from 2600 to 3000. While the proposed method provides results for all percentiles starting at 2600 cycles, the ANN fails to provide predictions at the 2600 and 2700 cycles, indicating that the ANN was unable to predict RUL for many data sets at those points. Additionally, the proposed method shows comparable accuracy at an earlier time step, as indicated by the comparison of percentiles: the results for the \(k\)-th cycle from the proposed method correspond to those of the \((k+1)\)-th cycle from the ANN. Moreover, it is important to highlight that the computational cost for 100 data sets with the proposed method is significantly lower, taking only 52 s compared to 2815 s (approximately 47 min) for the ANN, achieving a 54-fold reduction in computational cost. From the prediction results in Figure 11 and Table 1, using the data with a noise level of 1 mm in Figure 6a, the proposed method provides more accurate and consistent predictions than those of the ANN while achieving a 54-fold reduction in computational cost. The ANN, by contrast, struggles to provide results for some cycles and percentiles, as indicated by the missing values in Table 1. To assess whether similar results are obtained under a different noise level, a sensitivity analysis is conducted using data with a noise level of 2 mm in Figure 6b. First, the proposed method is applied, and Figure 12 shows the distribution of the first cycle for RUL prediction (Step 2 in Equation (5)). The figure  
Figure 11: Prediction errors using 100 data sets at a noise level of 1 mm: (**a**) proposed method; (**b**) ANN. Figure 10: Prediction results using the ANN with data at a noise level of 1 mm: (**a**) degradation prediction; (**b**) RUL prediction. displays a total count of 97 instead of 100, indicating that the proposed method was unable to predict the RUL for 3 out of 100 data sets. This occurs because the large noise in the data causes a delay in the prediction starting point (the algorithm stopped at Step 1 in Equation (4) for these three data sets). The prediction delay is evident when comparing Figure 12 with Figure 8a, as the distribution of the first cycle shifts to the right as the noise level increases. The majority of the data (81 out of 97) fall within the range of 2300 to 3000 cycles in Figure 12, which represents a 300-cycle delay compared to the range of 2000 to 2700 cycles for the data with a noise level of 1 mm in Figure 8a. This shift is a natural phenomenon as the noise level increases, and the proposed method remains robust and effective even at a noise level of 2 mm, compared to the ANN. Figure 13 and Table 2 demonstrate the validity of the proposed method by comparing it with the ANN. Figure 13a,b show the RUL prediction results from the proposed method and the ANN, respectively. Although the prediction uncertainty in Figure 13a appears larger than in Figure 13b, this is misleading, as the ANN fails to provide prediction results at the early stage. This is further supported by comparing the prediction errors in Figure 13c,d. The prediction errors using the proposed method in Figure 13c are significantly lower than those using the ANN in Figure 13d. Additionally, it is important to note that the medians of prediction errors from the proposed method remain consistent across two different noise levels, as shown by the solid red (2 mm noise level) and blue (1 mm noise level) lines in Figure 13c. In contrast, the median prediction error from the ANN using a noise level of 2 mm is significantly higher than that using a noise level of 1 mm, as shown in Figure 13d. These results indicate that the proposed method outperforms the ANN in terms of both accuracy and robustness across different noise levels.