## 3 Proposed Methodology  
### Design of Hierarchical Multi-Label Classification Strategy  
The occurrence of simultaneous failure means that multiple different single failure modes break out at the same time, and the category label necessarily contains multiple various elements representing the corresponding single failure modes. Simultaneous fault identification not only has the characteristics of multi-label but also has the characteristics of multi-class. The most frequently used methods to solve multi-class classification problems are summarized below:1. Calculate the occurrence probability of all categories, and the category corresponding to the maximum value is chosen as the prediction category;
2. Based on binary classification, the classification problem with multiple classes can be skillfully converted into several sub-problems with two classes to be solved, and then the multiple binary classification results are effectively combined. Considering the lack of ample simultaneous fault samples, this paper proposes a novel hierarchical multi-label classification strategy as shown in Figure 1. The hierarchical multi-label classification strategy solves the classification problem with multi-class by combining a series of sub-classifiers. As shown in Figure 1, at the first level, each classifier only focuses on the identification of one fault category. For diagnostic problems with m fault types, there are m classifiers presented as [C\({}_{1}\), C\({}_{2}\),..., C\({}_{m}\)] in which C\({}_{i}\) is utilized to recognize the \(i\)th fault type. At the second level, each classifier at the previous level is segmented into several sub-classifiers which can be trained by single fault samples. Such as, the \(i\)th classifier Ci in the first level is further expressed as dichotomization [C\({}_{i1}\), C\({}_{i2}\),..., C\({}_{im}\)] (\(m\
eq i\)) in which \(m-1\) element represents a set of sub-classifiers focusing on two classes. ### Design of Hierarchical Multi-Label Classification Strategy  
SBELM has the characteristic of probability output. By calculating the predicted probability distribution of an unknown sample x, the fault diagnosis algorithm based on SBELM can provide a conditional probability of all failures for sample x, that is, to achieve the probability output. For simultaneous fault diagnosis, any two single failure modes may have some correlation between them. Since the traditional SBELM can only be used to solve the classification problem of two classes. To address the multi-class classification problem, this paper combines the proposed hierarchical multi-label classification strategy (HMLC) with SBELM to construct the model based on HMLC-SBELM. The architecture of the model based on HMLC-SBELM is shown in Figure 2.  
The model is constructed of m combinational SBELM (C-SBELM) which can be expressed as [C-SBELM\({}_{1}\), C-SBELM\({}_{2}\),..., C-SBELM\({}_{m}\)]. That is, the simultaneous fault diagnosis model based on HMLC-SBELM outputs a probabilistic vector [\(p_{1}\), \(p_{2}\),..., \(p_{m}\)] in which each element pi is the output of C-SBELMi th to measure the occurrence possibility of the \(i\)th fault mode. According to the strategy of HMLC, C-SBELMi is composed of m-1 independent two-class sub-classifiers [SBELM\({}_{i1}\), SBELM\({}_{i2}\),..., SBELM\({}_{im}\)], and each classifier SBELM\({}_{ij}\) is trained by training samples of the \(i\)th fault and training samples of jth fault. The output of SBELM\({}_{ij}\) is expressed as \(p_{\bar{ij}}\) which is used to judge the probability that an unknown sample belongs to the \(i\)th or the \(j\)th fault mode. Figure 1: The structure of hierarchical multi-label classification strategy. Due to the complementarity between SBELM\({}_{ij}\) and SBELM\({}_{ji}\), that is, \(p_{ij}=1-p_{ji}\), the diagnostic model based on HMLC-SBELM contains \(m(m-1)/2\) two-class sub-classifiers. For an unknown sample \(x\), SBELM\({}_{ij}\) can predict the possibility of associating the category ti for sample \(x\).