## 4 Simulation and Results
### Results Analysis  
This paper classifies the sample data by the Lssvmlab v 1-8 toolbox in Matlab R2014a environment, of which the Gaussian RBF is used as the kernel function and the \"One vs. One\" classification algorithm is used to build the LSSVM multi-classifiers model.  
This paper establishes the initial model \(H_{0}\) first; the regularization parameter \(\theta\) is set to 10 and the kernel parameter \(\sigma^{2}\) is 0.2. Furthermore, the Bayesian inference is used to optimize the parameters of the model and to obtain the optimal parameters (\(\theta_{\text{MP}}\), \(\sigma^{2}_{\text{MP}}\)) = (175.214, 6.17); then, the optimal model is obtained by re-training the training set with the optimal parameters \(\theta_{\text{MP}}\) and \(\sigma^{2}_{\text{MP}}\). Finally, the testing set is input into the optimal classifier model and obtains the posteriori probability and classification.  
The posteriori  
Figure 5 is the multi-classification diagram based on Bayesian theory, Figure 5a is the multi-classification diagram of the training set, and Figure 5b is the multi-classification diagram of the testing set. \(x^{1}\), \(x^{2}\) are the feature vectors whose dimension is reduced by the RBF kernel function.  
Compared the LSSVM algorithm in Bayesian theory with the LSSVM algorithm and the standard SVM algorithm, the results are as shown, in Table 5.  
Among the LSSVM algorithm and the standard SVM algorithm, the RBF is the kernel function, the classification algorithm is the \"One vs. One\", the difference is that the optimization method of the regularization parameter \(\theta\), and the kernel parameter \(\sigma^{2}\) is the 10 times cross-validation method. O is the total number of test samples, and Percent is the proportion of the well-judged test samples and the total test samples. The results show that the LSSVM in Bayesian theory has higher generalization ability and good modeling effect.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**The Fault Types** & **The Short Circuit** & **The Open Circuit** & **The Abnormal Aging** & **The Normal** & **The Actual Fault Type** & **The Fault Type of Diagnose** \\ \hline
1\# & 0.4249 & 0.1282 & 0.1614 & 0.2855 & 1 & 1 \\
2\# & 0.3926 & 0.1028 & 0.1780 & 0.3266 & 1 & 1 \\
3\# & 0.3751 & 0.1315 & 0.1931 & 0.3003 & 1 & 1 \\
4\# & 0.3553 & 0.2988 & 0.2049 & 0.1410 & 1 & 1 \\
5\# & 0.4285 & 0.1271 & 0.1631 & 0.2813 & 1 & 1 \\
6\# & 0.4093 & 0.1497 & 0.1593 & 0.2817 & 1 & 1 \\
7\# & 0.4248 & 0.1251 & 0.1812 & 0.2689 & 1 & 1 \\
8\# & 0.3837 & 0.1518 & 0.1625 & 0.3020 & 1 & 1 \\
9\# & 0.4405 & 0.1216 & 0.1478 & 0.2901 & 1 & 1 \\
10\# & 0.4340 & 0.1202 & 0.1516 & 0.2942 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The final posteriori probabilities of 10 test sample groups in short-circuits state.  
Figure 5: The multi-classification diagram. (**a**) the multi-classification diagram of the training set; (**b**)the multi-classification diagram of the testing set.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**The Fault Diagnose Models**} \\ \hline
**The LSSVM Algorithm in Bayesian Theory** & **The LSSVM Algorithm** & **The Standard SVM Algorithm** \\ \hline O & Percent & O & Percent & O & Percent \\
40 & 97.5\% & 40 & 92.5\% & 40 & 90.0\% \\ \hline \hline \end{tabular}
\end{table}
Table 5: The comparison of three models’ performance. LSSVM: Least Squares Support Vector Machine; SVM: Support Vector Machine.