## 4 Experiment and Discuss
### Comparative Analysis and Experimental Results
#### 4.3.3 Performance Evaluation of the Diagnostic Framework  
(1) Training of HMLC-SBELM-based model  
During the training of the diagnostic model based on HMLC-SBELM, due to the probability output characteristic of SBELM, the diagnostic model outputs a probability vector represented as \(p=[p_{1},\,\ldots,p_{m}]\) in which m is the number of a single fault. The prediction result is the failure mode corresponding to the largest element in the probabilistic output vector \(=[p_{1},\,\ldots,\,p_{7}]\). As indicated in Table 3, the training accuracy of the diagnostic model based on HMLC-SBELM which is trained by using is 99.5%. Moreover, the values of each element \(p_{i}\) of the probability output vector \(p\) are significantly different and relatively scattered. It indicates that the trained diagnostic model has high-performance reliability, and very few samples are misdiagnosed. Through further analysis, it is found that in the probability output vector of the misclassified sample, although the probability value corresponding to the actual category is not the maximum value of the seven elements, it is also ranked relatively high and relatively close to the probability value corresponding to the predicted category. (2) Determine the optimal decision threshold for the diagnostic model  
To achieve the identification of various different forms of failure modes, using the validation set an optimization algorithm is used to optimize an appropriate discriminative threshold \(\varepsilon^{\star}\) which is in \((0,1)\). The PSO is utilized to optimize the discriminative threshold by performing 50 trials to obtain the average value of the objective function. After performing multiple iterations, the optimization of the decision threshold using PSO yields, small objective functions, and the corresponding \(F_{me}\) value is 0.923 which indicates that the model performance is obviously high. In addition, the standard deviation  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**No. ** & **Actual Category** & \(p_{1}\) & \(p_{2}\) & \(p_{3}\) & \(p_{4}\) & \(p_{5}\) & \(p_{6}\) & \(p_{7}\) & **Predicted Result** \\ \hline
1 & 1 & 0.7218 & 0.0256 & 0.0672 & 0.1005 & 0.1026 & 0.0052 & 0.2735 & 1 \\ \hline
2 & 1 & 0.7237 & 0.2096 & 0.0034 & 0.1128 & 0.0007 & 0.1265 & 0.1542 & 1 \\ \hline
3 & 1 & 0.7623 & 0.0028 & 0.1205 & 0.0005 & 0.1037 & 0.0039 & 0.0126 & 1 \\ \hline
4 & 1 & 0.8906 & 0.2118 & 0.0087 & 0.0132 & 0.0005 & 0.0458 & 0.0001 & 1 \\ \hline
5 & 1 & 0.7314 & 0.0075 & 0.3008 & 0.0009 & 0.0386 & 0.0012 & 0.0006 & 1 \\ \hline
6 & 1 & 0.7209 & 0.1022 & 0.1102 & 0.0026 & 0.0065 & 0.0725 & 0.0214 & 1 \\ \hline \multicolumn{8}{c}{...} \\ \hline
2447 & 7 & 0.1115 & 0.0009 & 0.0077 & 0.1026 & 0.0059 & 0.0437 & 0.7601 & 7 \\ \hline
2448 & 7 & 0.0021 & 0.0097 & 0.0382 & 0.0815 & 0.1027 & 0.0829 & 0.7268 & 7 \\ \hline
2449 & 7 & 0.2091 & 0.0038 & 0.0006 &of 50 trials is 1.79 \(\times\) 10\({}^{-3}\) which reaches a lower level, indicating that PSO shows a stable performance in the decision threshold optimization problem. To contrast with the existing machine learning algorithms, the discriminative thresholds of models based on some typical and representative methods are, respectively optimized using PSO. The optimal decision thresholds and corresponding Fme are shown in Table 4. As shown in Table 4, the optimal decision threshold of the diagnostic model based on HMLC-SBELM is 0.71. Moreover, the classification accuracy index Fme of the HMLC-SBELM-based model is 0.923, which is superior to other contrastive models with an increase of about 3% to 10%.