# Remaining Useful Life Estimation of Bearings Using Data-Driven Ridge Regression
## 7 Performance Evaluation
### RUL Evaluation
Figure 7 presents the normalized histogram of error ratios in each interval of 10% from 60% to 100% of the experimental time using different approaches for all cases. The adaptive sliding window method, Bayesian methods using the exponential and the linear degradation models, and adaptive Bayesian method provide considerably different error ratios of the RUL estimation as a function of time. One interesting observation is that the overall error ratio using Bayesian methods is gradually shifted from the positive error ratio to 0 as a function of time. It means that these methods provide a better underestimation of RUL prediction as the machine approaches to failure. While the histogram of the adaptive Bayesian method is similar to the one of the exponential degradation model for the error ratio around 0, the large error ratio greater than 0.6 is reduced thanks to the linear degradation model. The RUL prediction error ratio of the adaptive Bayesian method is improved compared to the ones using either exponential degradation model or linear degradation model. While the error ratio using Bayesian methods is spread around 0.5, the one using the adaptive sliding window method is concentrated around 0. We observe that most absolute error ratios between 90% and 100% of the experimental time are less than 0.1 or larger than 1 for all cases. The main reason for the high negative or positive ratio is the small value of the true RUL of Equation (8) as the machine approaches to the end. The score of the adaptive sliding window method is 0.6028, greater than that of the adaptive Bayesian method (0.4605). Note that the score values are 0.2554 and 0.3827 for Bayesian methods using the exponential and the linear degradation models, respectively. The score of the recently developed multiscale CNN method [10] is 0.3624, while the adaptive sliding window method gives 0.3117 for five test sets of the operating condition A. Note that [PERSON] et al. [10] only provided the score value of the operating condition A. However, our proposed scheme significantly reduces the computation complexity while guaranteeing the robustness against the possible uncertainty. Note that DL models are vulnerable to input uncertainties or adversarial attacks [14; 15]. Mean absolute error (MAE) and mean absolute percentage error (MAPE) are used to evaluate prediction performance with the following formula:  
\[\text{MAE} =\frac{1}{N}\sum_{i=1}^{N}\left|y_{i}-\hat{y}_{i}\right|\text{,} \tag{11}\] \[\text{MAPE} =\frac{1}{N}\sum_{i=1}^{N}\left|\frac{y-\hat{y}_{i}}{y_{i}} \right|\text{,} \tag{12}\]  
where \(N\) is the number of samples. Note that we do not show the root-mean-squared error since it has a similar trend with the MAE results. Figure 8 shows MAE and MAPE between true RUL value and predicted RUL obtained by adaptive sliding window method, Bayesian methods using the exponential and the linear degradation models, and adaptive Bayesian method for all cases. We observe that various approaches have considerably different MAEs of the RUL estimation dependent on cases. Since we do not reuse any specific model parameters of the training sets to evaluate the testing sets, the performance of the training sets does not necessarily better than the ones of testing sets. The adaptive sliding window method provides the smallest MAE for 12 cases out of 17 cases. It outperforms all cases with operating condition C and B except the case of Bearing B_7. The RUL MAE using the adaptive sliding window method is only worse than other approaches for Bearing A_3, Bearing A_4,  
Figure 7: Normalized error ratio histogram of the RUL prediction of adaptive sliding window method, Bayesian methods using the exponential and the linear degradation models, and adaptive Bayesian method as a function of the experimental time. We split the error ratio of different solutions in each interval of 10% based on the time sequences with respect to the entire experimental time to analyze the performance of the RUL prediction over time. Bearing A_5, Bearing A_7, Bearing B_7.