# Remaining Useful Life Estimation of Bearings Using Data-Driven Ridge Regression
## 7 Performance Evaluation
### RUL Evaluation  
Figure 6 presents the health indicator and RUL of true measurements, adaptive sliding window method, and Bayesian methods using the exponential and the linear degradation models of Bearing B_6 as a function of experimental time. In Figure 6b, we also show the \(\alpha\) bound of the true RUL where \(\alpha\) is set to 20%. Figure 6a depicts that the health indicator of Bearing B_6 gradually increases as a function of experimental time. Both the adaptive sliding window method and the Bayesian method using the linear degradation model follow the true health indicator. On the contrary, the Bayesian method using the exponential degradation model fails to estimate the health indicator. The main reason is the fundamental model limits of the exponential degradation model with respect to the slowly degraded health indicator of Bearing B_6. By comparing Figure 6a,b, we observe that predicting the health indicator is not enough to estimate the RUL value due to the complex interaction between machine components with high uncertainty during whole experimental times. In fact, some experimental cases show sudden performance degradation when it gets closer to failure, as shown in Figure 2. Hence, it is infeasible to provide an accurate estimation of RUL at the beginning of the experimental operations. However, the adaptive sliding window method still provides the reasonable estimation of RUL within the \(\alpha\)-bound of the true RUL for Bearing B_6 as it is closer to failure, as shown in Figure 6b. Both Bayesian methods using different models generally provide the conservative RUL prediction during the experimental time. Although the Bayesian method using the exponential degradation model does not provide the  
Figure 5: Health indicator of training cases Bearing A_1, Bearing A_2, Bearing B_1, Bearing B_2, Bearing C_1, Bearing C_2 as a function of experimental time. Figure 6: Health indicators and RUL of true measurements, adaptive sliding window method, and Bayesian methods using the exponential and the linear degradation models of Bearing B_6. reasonable RUL estimation for BearingB_6 due to the fundamental model limit, the performance of different methods considerably depends on the degradation trends of the real health indicator of each case, as we show below. We use the error ratio metric to evaluate the RUL prediction of different methods. Let us denote \(y_{i}\) and \(\hat{y_{i}}\) as the actual RUL to be predicted and the estimated RUL of \(i\)th dataset, respectively. The error ratio of of \(i\)th dataset is defined by  
\[R_{i}=\frac{y_{i}-\hat{y_{i}}}{y_{i}}\,. \tag{8}\]  
The effects of RUL underestimation and overestimation are not the same in practice. While early predictions of RUL, \(R_{i}>0\), are considered as the reasonable estimation with the deduction to early removal, the RUL estimation that exceeded actual RUL, \(R_{i}<0\), incurs more severe consequences. Based on this observation, the IEEE PHM Data Challenge defines a scoring function  
\[\text{Score}=\frac{1}{11}\sum_{i=1}^{11}A_{i} \tag{9}\]  
where  
\[A_{i}= \begin{cases}\exp(-\text{ln}(0.5)(P_{i}/5))&P_{i}\leq 0\\ \exp(+\text{ln}(0.5)(P_{i}/20))&P_{i}>0\end{cases}  \tag{10}\]  
and \(P_{i}\) denotes the percent error equal to Equation (8) multiplied by 100 for \(i\)th testing set [45]. The accuracy of the RUL prediction becomes more critical as the machine approaches to failure over time. To analyze the RUL prediction performance over time, we evaluate the error ratio of different solutions in each interval of 10% based on the time sequences to the whole experimental time. In this evaluation, we show the RUL prediction between 60% and 100% of the experimental time of each case since the first 40% of data are not a critical factor for the practical purpose of the maintenance. For instance, the error ratio between 90% and 100% of the experimental time means the average error ratio of the last 10% of the experimental data.