## 3 An Instance-Based Learning (IBL) Approach
### RUL Estimation
#### 3.4.1 Data Augmentation  
In machine learning applications, a data augmentation strategy is often used to generate additional learning data by modifying the historical samples [18, 19, 20]. By increasing the learning datasets, classification models can be trained to be more robust and generalized without over-fitting. For example, in image analysis, learning samples may be acquired under a limited set of conditions. However, classification may need to be applied to images under conditions beyond that of the learning samples, such as different orientation, location, scale, brightness, etc. These situations can be accounted for by training the classification model with additional image samples, which may be generated by synthetically modifying existing samples through translation, rotation, scaling, and illumination. Augmenting the learning datasets can result in a trained classification model to be more robust and generalized with-out over-fitting. Taking a similar approach, we propose a segment scaling method in order to generate synthetic feature sequences by modifying the learning instance in the IBL approach for bearing RUL estimation. We take the learning instance Bearing1-1 as an example. Figure 5b shows the 1 st PC feature sequence over time. Based on the magnitude trend of the feature curve, three typical stages within the lifespan of the learning instance can be observed: (i) normal stage where no obvious degradation occurs, the magnitude remains nearly constant; (ii) degradation stage where a gradually changing trend is clear; (iii) close-to-failure stage where the bearing is close to failure as signaled by steep or abrupt changes in the feature curve. The close-to-failure stage may last a short period, while the degradation stage can last quite long. Figure 5a shows the 1 st PC feature sequence of the test instance. If we can determine the equivalent location of the test instance relative to the lifespan of the learning instance, we can estimate the RUL of the test instance. As illustrated in Figure 5a, the proposed segment scaling method starts by identifying a defect-degradation segment in the first PC feature sequence of the test instance Bearing1-3. The specified segment for the test instance begins on entering the degradation stage until the current time cycle. Correspondingly, specify a degradation segment of length \(T_{l}\) within the feature sequence of the learning instance. The learning degradation segment should present similar trending as the test segment. For the learning instance, the specified segment also begins on entering the degradation stage until ending the entire degradation stage. Understandably, the learning degradation segment should encompass the test segment in magnitude so that it is possible to determine the relative location of the test case degradation. Hence, the following condition Equation (7) should be satisfied for the specified test and learning segments. \[\left\{ \begin{array}{l}x_{s}^{(p)}\leq x_{s}^{(l)},\ for\ increasing\ trend\\ x_{s}^{(p)}\geq x_{s}^{(l)},\ for\ decreasing\ trend\end{array} \right. \tag{7}\]  
where \(x_{s}^{(p)}\) and \(x_{s}^{(l)}\) are prognostic feature values of the test and learning instances at the ending times of their respective segments, as illustrated in Figure 5a,b. The learning degradation segment is then stretched (\(\alpha>1\)) or shrunk (\(\alpha<1\)) in length with different scaling factor \(\alpha\), as illustrated Figure 5b-d. By scaling, multiple segments representing different prognostic feature sequences from the given learning instance can be generated.