## 4 Experiment and Discuss
### Comparative Analysis and Experimental Results
#### 4.3.3 Performance Evaluation of the Diagnostic Framework
For the diagnostic model with a one-to-all strategy, inseparable regions (indecision region) are easy to appear between different failure categories, and the existence of inseparable regions is prone to cause fault misclassification in simultaneous fault diagnosis and further reduce the diagnostic accuracy. By longitudinally comparing the average classification accuracy of these diagnostic models, models based on SVM are lower than models based on ELM and KELM. The main reason is that the ELM-based model searches for the optimal solution for classification in a large feature space, while the SVM-based model tries to search for the solution in a linear plane, which enables the ELM-based model to obtain better classification solutions than SVM. The accuracy of the KELM-based diagnostic model with a one-to-all strategy is 93.26% which is higher than the proposed model with a one-to-all strategy in the single fault diagnosis. It can be concluded that KELM inherits the characteristics of high generalization from ELM. Moreover, KELM has its own advantages, that is, hidden-layer nonlinear feature maps are represented as a kernel function. (4) Admissibility of simultaneous fault diagnosis results  
As listed in Table 5, the diagnostic accuracy of the contrastive models does not exceed 90%, meaning that 10% of the simultaneous failure samples are misclassified. Due to the characteristics of the probability output of the proposed model based on HMLC-SBELM, before the final diagnosis is obtained based on the optimal discriminative threshold, the model will output a probability output vector in which each element of the vector represents the probability of a certain single failure mode. For simultaneous fault diagnosis, a partial matching between the diagnostic results with actual multiple failures is also valuable. If the occurred single failure mode is still contained in the diagnostic result vector of a misclassified simultaneous-failure sample, it indicates that the result is also acceptable to some extent. To evaluate the admissibility of simultaneous fault diagnosis model, focusing on 500 simultaneous failure samples in the testing set, analyze the output vector of 56 misclassified samples. A misclassified sample with gear hard point failure (C2) and gear crack failure (C3) is taken as an example, the probatilistic output vector is [0.013, 0.825, 0.629, 0.004, 0.001, 0.033, 0.108]. By using the discriminative threshold 0.71, the result vector is [0, 0, 0, 0, 0, 0, 1] and the sample is recognized as single fault C2. Although the decision result is wrong, it is not difficult to find out that the probability output value of C3 is 0.629, which is still very close to the decision threshold. On the other hand, the decision result still correctly identifies one of the actually emerged single-failure modes. The analysis illustrates that the proposed model based on HMLC-SBELM is highly acceptable and can provide superior technical support and contribution of the practical simultaneous fault diagnosis. Analysis of efficiency and stability  
In addition to the high diagnostic accuracy, the execution time is another key criterion for measuring the model's performance. Therefore, the training time and execution time of the testing set are compared between these five models to measure the model's efficiency. Thet results are shown in Figures 10 and 11. As shown in Figure 10, the training time of the KELM-based model is the shortest, while the training time of the HMLC-SBELM-based model takes the longest training time. The main reason is that the HMLC-SBELM-based model approximates the output weight in an iterative way, and the iterative process is relatively time-consuming. In contrast, SVM, ELM, and KELM all solve the output weights directly by analytical computation, so the training time is relatively short. Although the HMLC-SBELM-based model is time-consuming in the training stage, the training time remains around 0.5 s, which is completely acceptable in practice.