## 3 Proposed Online Fault Diagnosis Model
### DSK Construction by Selecting Discriminant Fault Signatures of Detected Fault Modes  
Discriminant features of candidate data classes are important to effectively distribute class samples and improve classification accuracy. However, the classifier knowledge varies depending on the number of emerging fault classes. In a high-dimensional feature space, features differ in their ability to effectively discriminate a given fault condition. Irrelevant or redundant features degrade the performance of predictive models (e.g., supervised and unsupervised classifiers). To address this issue, we use a discriminant fault signature selection process to build a DSK of detected fault modes.  
Feature selection involves the evaluation of feature subsets based on processes that are generally either wrapper or filter based [31]. However, diagnostic performance can be improved by combining the advantages of both the wrapper-and filter-based approaches [15; 32; 33]. In this study, we introduce a hybrid feature selection algorithm that uses an effective combination of filter and wrapper methods to select the most discriminant signature of emerging faults. In the selection process, extracted features of detected fault classes (\(N_{detected\_class}\times N_{sample}\times N_{feature}\)) are divided into two halves, where one half is used for the filter selection process and the other half is used for the wrapper selection process (\(N_{feature}\) is the size of the original feature vector). In the filter selection process, (\(N_{detected\_class}\times N_{sample}\)/2 \(\times N_{feature}\)) are randomly divided into a \(k\)-fold cross validation \(k_{cr}\), where \(k=2\). Optimal feature sets are selected using sequential forward selection (SFS) with feature subset evaluation. The filter selection process selects 10 suboptimal feature subsets from \(N_{iteration}\times k_{cr}\), where \(N_{iteration}=5\).  
The 10 suboptimal subsets are combined to create a feature occurrence histogram. Subsequently, in the wrapper selection process, the algorithm selects feature combinations from different levels of the feature occurrence histogram and estimates the average classification accuracy of the \(k\)-NN classifier. In the classification process, 50% of the (\(N_{detected\_class}\times N_{sample}\)/2 \(\times N_{feature}\)) are used as a training set and  
Figure 6: Calculation of \(N_{OFreqR,i}\), \(N_{IFreqR,i}\), and \(N_{RFreqR,i}\). (**a**) Defect frequencies of BCI; (**b**) defect frequencies of BCO; (**c**) defect frequencies of BCR.  
the rest are used as a resting set. The wrapper selection process picks the best feature combination depending on the maximum average classification accuracy. Figure 7 illustrates the overall flow of the DSK construction by selecting discriminant fault signatures of detected fault modes.  
Usually, the SFS algorithm evaluates different subsets and selects an optimal subset based on an objective value. In this study, we adopt the maximum class separability (MCS) feature subset evaluation algorithm proposed by [PERSON] et al., which is illustrated in Figure 8 [PERSON] et al. (2018). According to this algorithm, the final objective value is calculated as Equation (5).  
\[objective\_value=\frac{between\_class\_separability}{within\_class\_compactness} \tag{5}\]  
Figure 8: Feature subset evaluation algorithm: (**a**) Within-class compactness value, and (**b**) between-class distance value.  
Figure 7: The flow of DSK construction by selecting discriminant fault signatures of detected fault modes.