## 2 Methodology
### Feature Selection  
Among the extracted features, it is necessary to find the most informative ones for more efficient and accurate diagnostic performance. For this purpose, a score metric is introduced, which accounts for the two following aspects: First, the features should present a consistent trend as the fault severity rises, which is an important factor as the health indicator. Second, Spearman correlation (named _corr_ here) is employed to evaluate this, which is defined as  
\[corr=\frac{cov(rg_{X},rg_{Y})}{\sigma_{rg_{X}}\sigma_{rg_{Y}}} \tag{5}\]  
where \(rg_{X}\) and \(rg_{Y}\) denote the rank variables of feature \(X\) and fault severity \(Y\). The \(corr\) is calculated for each component. Next, the features should have good separability of the individual faults and their severities. This can be evaluated by the Fisher discriminant ratio (FDR) defined as follows:  
\[FDR=V_{W}{}^{-1}V_{B} \tag{6}\]  
\[V_{B}=\sum_{k=1}^{K}(m_{k}-m)(m_{k}-m)^{T} \tag{7}\]  
\[V_{W}=\sum_{k=1}^{K}\sum_{n\in C_{k}}(x_{n}-m_{k})(x_{n}-m_{k})^{T} \tag{8}\]  
where \(V_{B}\) and \(V_{W}\) represent the variances of between-class and within-class, respectively, in which each class denotes the individual fault with different severities. By combining the two metrics, a hybrid score is defined for each component. For example, assume that we have three components A, B, and C, and we have acquired signals for each component under the fault conditions with the number of severities being \(L_{A}\), \(L_{B}\), and \(L_{C}\), respectively. For example, if \(L_{A}=4\), it means that the fault is applied to the component A with 4 severities: normal (no fault), small, medium, and large degree of fault. Then the total number of conditions is \(L=L_{A}\times L_{B}\times L_{C}\). For each fault condition, a large number of features is obtained by decomposing the signal through WPD and extracting the features. For an individual feature, the correlation of component A can be defined as  
\[corr_{A}=\left|\sum_{j=1}^{L_{C}}\sum_{i=1}^{L_{B}}corr_{A|i,j}\right| \tag{9}\]  
where \(corr_{A|i,j}\) represents the correlation of the feature with respect to the fault severity A when B and C are at the severity level \(i\) and \(j\), respectively. Since this is performed for \(i=1\) to \(L_{B}\) and \(j=1\) to \(L_{C}\) to result in \(L_{B}\times L_{C}\) number of \(corr_{A}\), these are summed to obtain a single representative value. Then the result is replaced by absolute value since the \(corr_{A}\) can be negative. The same process is applied to obtain \(FDR_{A}\) to evaluate the separability of fault severity of component A, which is defined as  
\[FDR_{A}=\sum_{j=1}^{L_{C}}\sum_{i=1}^{L_{B}}FDR_{A|i,j} \tag{10}\]  
Once the two metrics \(corr_{A}\) and \(FDR_{A}\) are calculated, they are combined into a single score by introducing weight parameter \(\alpha\). Before proceeding, however, the metric values are normalized since they differ in scale. Min-max algorithm is applied to the whole features so that they vary within the interval [0, 1], by assigning 0 and 1 to the minimum and maximum values. Then, the hybrid score for component A is given by  
\[score_{A}=\alpha\times normalize(corr_{A})+(1-\alpha)\times normalize(FDR_{A}) \tag{11}\]where the symbol _normalize_ indicates the metrics are normalized by min-max algorithm. In this way, the score is calculated for all the components A, B, and C for each feature. The reason why we need to consider the two metrics together is illustrated in Figure 3, which shows three cases where, in Figure 3a, _corr_ is low but _FDR_ is high, in Figure 3b, _corr_ is high but _FDR_ is low, and in Figure 3c, both are high. While Figure 3a shows high separability, it is less useful since it is not consistent with respect to the fault progression. Likewise, Figure 3b is also less useful due to the large variance, even though the _corr_ is high. For this reason, the feature should have higher values in both metrics.