# BearingCog: A Bearing Fault Diagnosis Method under Variable Operational Conditions
## 4 Experimental Results and Discussion
### Classifier Model Optimization  
To discuss the optimization of the classifier, three kernel functions, including linear kernel, polynomial kernel, and radial basis function (RBF) kernel, are imported into the SVM model for comparison. The classification results are listed in Table 7. As seen from Table 7, the accuracy of the RBF kernel model is a little higher than the other two models. However, it is noted that the RBF kernel costs more training time. The reason is that the radial basis kernel function has the advantage of having fewer parameters than the polynomial kernel function. Therefore, it can simplify the complexity of the model selection. Combined with the above analysis, the radial basis kernel function shows excellent local performance when classifying samples with close distances. Therefore, the radial basis is selected as the kernel function for the SVM multi-class classifier. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Bearing State**} & **Fault Free** & **Little Fault** & **Medium Fault** & **High Fault** \\ \cline{2-5}  & **Accuracy** & **Accuracy** & **Accuracy** & **Accuracy** \\ \hline Linear kernel & 95.6\% & 92.7\% & 94.3\% & 93.7\% \\ Polynomial kernel & 95.7\% & 93.3\% & 94.6\% & 94.1\% \\ Radial basis function kernel & 96.8\% & 94.5\% & 96.7\% & 97.6\% \\ \hline \hline \end{tabular}
\end{table} 
Table 7: SVM performance results based on different kernel functions. Figure 8: The projection effect of test samples under different data feature samples: (**a**) based on dimensionless time domain characteristics, (**b**) based on EMD decomposition energy, (**c**) based on VMD decomposition energy, and (**d**) based on multi-information fusion. The penalty factor C amongst this is used to represent the penalty factor for the deviation, which is caused with the misclassification of samples. By adjusting the penalty factor C value, the empirical risk and confidence range can be balanced, which ensures the generalization of the classifier. To be specific, when the parameter C is large, it means that all constraints must be strictly met. It would guarantee the fitting degree of samples but decrease the generalization ability of the classifier. Conversely, when the parameter C is small, the complexity of the learning process is simplified, but the empirical risk would be increased. Moreover, the parameter \(\sigma\) is used to control the gaussian distribution width of the samples. When \(\sigma\) is far less than the minimum distance between the training samples, all the training samples become support vectors. In other words, the classifier is over-fitting, which decreases the generalization ability of the model. Conversely, when \(\sigma\) is much greater than the maximum distance between the training samples, the classifier sorts all the training samples into one group. Therefore, when using the radial basis as a kernel function, the penalty factor C value and parameter \(\sigma\) should be carefully selected to improve both learning ability and generalization ability of the classifiers. Inspired by this, the grid search algorithm is imported to optimize the parameter of the radial basis kernel function. The main steps are listed as follows: Firstly, an initial range and the moving step for both the parameters C and \(\sigma\) are given. Secondly, the SVM model is trained by moving the step of the given parameter. Then, the test result of the training classifier model is verified. As an alternative approach, the quadratic optimization strategy is imported to improve the efficiency of the searching. In detail, a large step value is first preset to obtain the rough range of optimized parameters during grid search. Then, a small step is adopted to refine the search. To ensure the efficiency and precision, it is noted that the step value is set in exponential form with a base of natural constants.