# A Hierarchical Decision Fusion Diagnosis Method for Rolling Bearings
## 2 Method
### Decision Fusion Based on the Voting Method  
The idea of decision fusion comes from ensemble learning. Ensemble learning is not a single machine learning algorithm, but a method completing the task by building and combining multiple machine learning algorithms, which can be applied for classification problem integration, regression problem integration, feature selection integration, abnormal detection integration and so on [31; 32]. For feature selection, [PERSON] et al. [33] applied the data set with selected features to five base learners and aggregated the outputs obtained by the base classifier by a weighted voting method to predict the final output. A deep neural network algorithm, support vector machine (SVM) and decision tree c4.5 base classifiers were adopted by [PERSON] et al. [31], and the decision strategy based on the maximum number of votes was used to identify and classify botnets. Each ensemble learner produces a result, so it is necessary to integrate the results of each base learner to obtain the final decision result. A simple and effective way is to obtain the final decision result of multiple machine learning algorithms by the voting method. The voting method mainly includes the majority voting method [34] and the weighted voting method [33]. Inspired by the ensemble learning method, this paper makes use of the majority voting method to fuse the diagnosis results of multiple BPNN classifiers trained by a single feature set, which are extracted from all layers based on wavelet packet decomposition. Then, they are sorted from most to least. Finally, based on the principle that the minority obeys the majority, the diagnosis result with the largest number of votes is regarded as the decision fusion diagnosis result. The decision fusion algorithm based on the voting method is shown in Figure 3.  
Figure 3: Decision fusion algorithm based on the voting method. In Figure 3, \(m\) characteristics \(F_{1}\), \(F_{2}\),..., \(F_{m}\) are extracted from the vibration signals and are normalized to \(X_{1}\), \(X_{2}\),..., \(X_{m}\) as the input of \(m\) BPNN models \(Net_{1}\), \(Net_{2}\),..., \(Net_{m}\), and then \(h\) kinds of fault modes of bearings are recognized, The output results \(D_{1}\), \(D_{2}\),..., \(D_{m}\) are Synthesized into a diagnostic matrix \(D_{h\times m}\). Then, each row of the matrix \(D_{h\times m}\) is summed in order to get the scoring matrix \(Q_{h\times 1}\) = {\(Q_{1}\); \(Q_{2}\);... ; \(Q_{h}\)}. Each \(Q_{i}\) (\(i\) = 1, 2,..., \(h\)) represents the score of the fault mode \(i\), respectively. Finally, the highest score \(Q_{r}\) is the output, which is the maximum of the output \(Q_{1}\), \(Q_{2}\),..., \(Q_{h}\). Setting fault discrimination matrix \(C\):  
\[\{C_{1}\); \(C_{2}\);... ; \(C_{h}\)} = I_{h\times h} \tag{11}\]  
where \(C_{i}\) (\(i\) = 1, 2,..., \(h\)) represents the fault mode and \(I_{h\times h}\) is an \(h\)-dimensional identity matrix. \(h\) is the total number of bearing state modes, which determines the number of output layer nodes. The corresponding fault mode of the fault discrimination matrix \(C\) is found according to the angle mark \(r\) of the highest score, which is the decision fusion diagnosis result of the current state. It can be seen that decision fusion is the fusion of multiple bearing fault diagnosis results based on a single time-domain feature. It is not necessary to screen the features, and the diagnostic results under each feature contribute to the final decision results. The decision fusion method takes into account the diagnostic advantages of each single time-domain feature for rolling bearing signals, and the diagnostic results based on the reasoning of each single time-domain feature will be comprehensively analyzed. To some extent, it can avoid the misdiagnosis and missed diagnosis of rolling bearing faults caused by the sudden change of environmental or human factors, which is helpful in reflecting the healthy condition of rolling bearings more comprehensively and accurately.