## 3 An Instance-Based Learning (IBL) Approach
### RUL Estimation
#### 3.4.3 Estimate Ensemble  
Once synthetic data are generated, we can try to find the best match between the test instance and the learning instance by the similarity measures of all real and synthetic data. As the synthetic learning sequences are generated in the discrete way (with a set of scaling factors), the estimation result may be biased from the actual RUL if it is only based on one synthetic sequence. Hence, we propose an ensemble method based on multiple synthetic learning sequences to achieve the best out-come of assessing RUL. Figure 6: Similarity evaluation. (**a**) Similarity matching between the learning instance Bearing1-1 and the test instance Bearing1-3; (**b**) Similarity measure function. As illustrated in Figure 7, multiple synthetic feature sequences are generated from the original feature sequence of the learning instance (e.g., Bearing1-1) with different scaling factor \(\alpha\). From each synthetic sequence, an RUL estimate, \(RUL(\alpha)\), for the test instance (e.g., Bearing1-3) can be obtained with the corresponding minimum similarity measure \(d_{min}(\alpha)\). A scaling factor \(\alpha^{*}\) for the best similarity matching sequence can be identified from all learning sequences including the original and synthetic ones. After that, the final RUL estimation can be aggregated from the best matching sequence and the multiple synthetic sequences, whose scaling factors are evenly distributed on both sides of \(\alpha^{*}\), for example, two sequences with \(\alpha>\alpha^{*}\) and two sequences with \(\alpha<\alpha^{*}\), as shown in Figure 7. The formula for computing the ensemble RUL estimate is defined as Equation (16). In the averaging formula of the ensemble method over \(M\) number of the selected sequences, each RUL estimate is weighted by its corresponding similarity measure, \(d_{min}\). \[RUL_{e}=\frac{\sum_{m=1}^{M}(RUL(\alpha_{m})/d_{min}(\alpha_{m}))}{\sum_{m=1}^ {M}(1/d_{min}(\alpha_{m}))} \tag{16}\]  
In this example as illustrated in Figure 7, nine synthetic sequences are generated from the original feature sequence of Bearing1-1, with the scaling factor \(\alpha\) of 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90, and 0.95, respectively. Table 1 presents the results of a similarity evaluation between the nine synthetic sequences and the selected segment in the range of [300, 1800] (time cycles) of Bearing1-3. The best similarity matching sequence can be identified as \(\alpha=0.75\). Based on the seven selected synthetic feature sequences with \(\alpha=0.60\), 0.65, 0.70, 0.75, 0.80, 0.85, and 0.90, the ensemble RUL estimate, \(RUL_{e}\), can be computed as 642 (time cycles) using Equation (18). The actual RUL at the last time cycle 1800 (i.e., the length of future sequence) of Bearing1-3 is \(RUL_{a}=573\) (time cycles). Thus, our proposed IBL approach can achieve a good performance with the relative error \(E_{r}=-12.07\%\), which is calculated using the formula Equation (17). \[E_{r}=(RUL_{a}-RUL_{e})/RUL_{a} \tag{17}\]  
In summary, the procedure for the RUL estimation method with data augmentation is described as follows:  
<Procedure-II: Data Augmentation and RUL Estimation> \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \(\alpha\) & 0.55 & 0.60 & 0.65 & 0.70 & 0.75 & 0.80 & 0.85 & 0.90 & 0.95 \\ \(d_{min}(\alpha)\) & 0.2384 & 0.1857 & 0.1434 & 0.1190 & 0.1137 & 0.1225 & 0.1400 & 0.1621 & 0.1814 \\ \(RUL(\alpha)\) & 423 & 473 & 523 & 578 & 635 & 693 & 756 & 823 & 887 \\ \hline \end{tabular}
\end{table} 
Table 1: Similarity evaluation for RUL estimation between Beaing1-3 and Bearing1-2. Figure 7: The ensemble method for RUL estimation of Bearing1-3 using Bearing1-1. 1. Data Augmentation (1.a) Specify the pair of comparison feature segments within the test instance and learning instance, \(P_{k}\) and \(L_{k}\) (i.e., \(\alpha\) = 1.0). (1.b) Generate multiple synthetic sequences \(L_{k}\) (i.e., \(\alpha\) > 1.0 or \(\alpha\) < 1.0) from the original one using the segment scaling method described in Section 3.4.1. 2.