## 2 Materials and Methods
### The Proposition of RS-DFV
#### 2.2.3 RS-DFV Extraction  
\(\bullet\) DFV Extraction Tree  
Extracting the DFV of fault samples is another key step of the fault diagnosis method presented in this paper. The DFV is different from the traditional feature vector because of its unique structure. In a DFV, the importance of each feature item is not equal: some of them are valid feature items, and the others are invalid. More importantly, faults of different types had different valid feature items and invalid feature items, i.e., the same feature is of different importance for different faults. In addition, only the valid feature items of DFV need to be calculated from the sample data and the value of the invalid items are set subjectively, based on the requirement of the fault classification. All these differences considerably increased the difficulties of DFV extraction. Moreover, for a fault sample that has to be diagnosed, features that are valid items in the sample's DFV are unknown. This made the DFV extraction more difficult. Figure 4: (**a**) Build the feature selection tree; (**b**) the heuristic tree and the composition and structure of DFV. To overcome those problems in the DFV extraction mentioned above, a tree heuristic feature extraction method is put forward to extract the DFV in this paper, as illustrated in Figure 5.  
First, a DFV extraction tree (DET) that inherited the heuristic rules of the feature selection tree (FST) is put forward in this paper, as illustrated in Figure 5(3). The DET is constructed based on the FST: (1) The FST is traversed. When a non-leaf node is passed, continue; when a feature selection segment is passed, a feature calculation node for the DET is built; when a leaf node is passed, an end-node for the DET is built. (2) A connection relationship is established among the nodes for the DET, and consistency with the FST is ensured. (3) According to the value ranges of the child nodes that had the same parent node in the FST, heuristic rules that could guide the subsequent feature calculation for the DET are built. Then, for a new fault sample, the values of all valid feature items are calculated in turn according to the guidance of the (DET); hence, the valid feature items of the DFV are obtained. As illustrated in Figure 6, different faults have different valid features, but their valid features all could be accurately calculated through DET. Finally, the other feature items in the DFV are the invalid feature items of this fault sample, and all invalid features are subjectively assigned the same specific value. Therefore, the DFV of all fault samples could be acquired in this manner. * RS-DFV Extraction Tree  
Figure 5: DFV extraction based on the DFV extraction tree. Figure 6: DFV extraction. Although the tree heuristic feature extraction method overcomes the difficulties in DFV extraction, it could efficiently and accurately start the calculation of an unknown fault sample and successfully calculated the DFV of most of the fault samples. However, there is still another weakness in the DFV: The feature value obtained in the parent feature calculation node of the DFV extraction tree could not meet the demand of any one heuristic rule connected to this feature, and there is be no inspire information for the next step, as illustrated in Figure 7,b. This could lead to an abnormal termination in the DFV extraction and inaccurate calculation of the sample DFV. To overcome the problem mentioned above, a rough set is introduced to solve the boundary problem of heuristic rule in the DFV extraction tree, and a DFV extraction method based on the rough set and the DFV extraction tree is presented in this paper. As illustrated in Figure 7, a new DFV extraction tree combining the rough set and DET (RS-DET) is designed. The RS-DET increased an abnormal termination correction segment (ATCS) for each node that had more than one child nodes. When an abnormal DFV extraction termination occurred, the corresponding ATCS responded quickly and reasonably amended the value of the current heuristic feature to be instructive.