## 2 Materials and Methods
### The Proposition of RS-DFV
#### 2.2.2 The Tree Heuristic Feature Selection  
\(\bullet\) Feature evaluation based on information granulation and neighboring clustering  
Different feature selection methods required different feature evaluation criteria, but the excellent traditional feature evaluation methods could not provide the best heuristic knowledge for the feature selection method proposed in this paper. So, a feature evaluation method based on information granulation and neighboring clustering is put forward for tree heuristic feature selection (THFS). In this feature evaluation method, information granulation and neighboring clustering were used to  
Figure 2: (**a**) The structure of DFV; (**b**) the valid and invalid features of DFV. delete the features that were obviously ineffective in fault distinguishing at first; then, the remaining features were evaluated through Equations (8)-(11). The specific steps of this feature evaluation method are introduced below. Step 1: The value range of one feature is divided into many small granules evenly in accordance with the same criterion, as illustrated in Figure 3(1)-2. The granules that included some feature values of samples are black granules, and the others are white granules. Step 2: The adjoining granules of the same type are merged to a larger one through the neighboring clustering method, as illustrated in Figure 3(2)-3. The black granules that included very few samples and the white granules with a tiny length are outliers (Figure 3(4)). Step 3: The black outlier is first changed to be white and amalgamated with white granules adjacent to it. Then, the white outlier is changed to be black and amalgamated with black granules adjacent to it, as illustrated in Figure 3(4)-5. Step 4: If there is only one black granule or two different black granules both with some samples of the same type, this feature is considered to be ineffective in fault distinguishing and deleted. Step 5: The remaining features are evaluated through Equations (8)-(11):  
\[L_{i} ={x_{i}}^{+}-{x_{i}}^{-} \tag{8}\] \[L_{i(i+1)} ={x_{i+1}}^{-}-{x_{i}}^{+}\] (9) \[V_{i(i+1)} =\frac{L_{i(i+1)}}{\frac{1}{2}(L_{i}+L_{i+1})}\] (10) \[P(x) =\sum_{i}^{N-1}V_{i(i+1)} \tag{11}\]  
where \(L_{i}\) is the length of the No. \(i\) black granule, \({x_{i}}^{+}\) is its upper bound, and \({x_{i}}^{-}\) is its lower bound. \(L_{i}\) (\(i\)+1) is the length of the white granule between the No. \(i\) and No. (\(i\)+1) black granules. \(V_{i}\) (\(i\)+1) is the separability between the No. \(i\) and No. (\(i\)+1) black granules, and \(N\) is the number of the black granules. \(P(x)\) is the score of this feature, and a large \(P(x)\) displayed its high separability among the black granules. * Feature selection based on the tree heuristic search strategy  
Figure 3: Information granulation and neighboring clustering. The structure establishment of the feature vector is a very important basis for DFV. To adapt to the uniqueness of DFV, this study proposes a feature selection method based on THFS to establish the structure of DFV. The workflow of THFS is illustrated in Figure 4.  
The most efficient feature (\(x_{kl}\)) for the sample space is first chosen through the feature evaluation method based on information granulation and neighboring clustering, which is introduced above, and the black granules are obtained. Each black granule includes some fault types, and the samples of these types make a fault subspace. Then, the sample space is made the root node, the subspaces are made the leaf nodes, and the \(x_{kl}\) value range of each leaf node is marked. Thus, the local structure connected to the root of the heuristic tree is established, as illustrated in Figure 4a. If a leaf node includes faults of more than one type, its subtree is built using the same method. When all leaf nodes contain faults of only one type, the heuristic tree has completed its growth and gained complete tree, as illustrated in Figure 4b. The optimization feature subset is obtained through traversing the heuristic tree.