3.2.2. Analysis of CSvDE Scale Factor
As depicted in Equations (13) and (14), the maximum value of scale factor
should be determined preliminarily to construct the MsFFS of bearings for accurate fault diagnosis. For this,
Figure 8 presents the development trend of CSvDE average values for all training samples under 12 operation states with the increase in scale factor, in which the scale factor varies from 1 to 20. More specifically, the remaining three parameters of CSvDE can be set as
,
, and
[
35,
36]. It can be found from the figure that the values of CSvDE show a gradual decreasing tendency with the scale factor increasing, regardless of the working state. Moreover, when
, the curve of CSvDE tends to be stable and there is obvious overlap between the CSvDE values of different fault states, which indicates that the current value of
is appropriate for fault identification. In other words, the interval of the scale factor for the construction of the MsFFS can be determined as
.
In addition, to analyze the difference of CSvDE features of frequency components between 12 working states, the CSvDE values of all components under four different scale factors (
) are calculated, as shown in
Figure 9. It is obvious that the CSvDE values of the last few components are larger than those of the first several components when
, and a similar phenomenon occurs in the results of the other three factors. This is because the last few components with high frequency exhibit a stronger randomness and contain more information of fault states compared with the first several components. Moreover, it is worth noting that for the 12 states, the differences in CSvDE values of the last five components are more significant. However, there is obvious overlap between the CSvDE values of the other components. The relevant results indicate that the last five components show greater potential for identifying different operation states of bearings. Consequently, two groups of experiments can be conducted in this study, as described in
Table 2. In other words, from the perspective of feature construction, we try to explore the influence of MsFFS structure on the diagnosis accuracy. And most remarkably, much more attention should be paid to Experiment 1 to demonstrate the superior performance of the proposed method.
Experiment 1: Without the process of frequency component selection, all components obtained by SoVMD are used to construct the MsFFS.
Experiment 2: Based on the analysis results mentioned above, the last five components are considered to construct the MsFFS.
As depicted in Equation (21), the MsFFS can be effectively constructed for bearing fault diagnosis based on the extracted multiscale frequency components and CSvDE theory. Here, utilizing the results of the training set in Experiment 1 as an example, the following formula shows the constructed MsFFS to train the softmax classifier:
In addition, to validate the more superior performance of the proposed method, six other approaches are also employed for rolling bearing fault diagnosis, including the EMD-CSvDE, VMD-CSvDE, SoVMD-multiscale sample entropy (MSE) [
44], SoVMD-multiscale permutation entropy (MPE) [
45], support vector machine (SVM) [
46], and artificial neural network (ANN) [
47]. More specifically, the first four comparison methods are designed to analyze the contribution of each link in the process of fault diagnosis, i.e., SoVMD-based multiscale frequency component extraction and CSvDE-based MsFFS construction. As described in
Section 3.1, 10 trials with the same setup are implemented in two groups of experiments. Detailed descriptions about the parameter setup of seven methods in Experiment 1 are shown in
Table 3.
Considering the two different groups of experiments, the detailed diagnosis accuracies and FPRs of 10 trials for seven methods are presented in
Figure 10 and
Figure 11, respectively. Based on these results, the average accuracies and average false alarm rates of the seven methods in two experiments can be calculated and are listed in
Table 4. In addition, to compare the implementation efficiency, the average computation time of 10 trials for different methods is shown in
Table 5.
From the perspective of diagnosis accuracy and FPR of each trial, it can be seen from
Figure 10 and
Figure 11 that the accuracy and FPR of the proposed method are obviously superior to those of the other six approaches for both experiments. More comprehensively, as shown in
Table 4, we can observe that the average accuracy and average FPR of the proposed method in Experiment 1 are 96.70% and 0.30%, respectively, which are slightly superior to EMD-CSvDE, VMD-CSvDE, SoVMD-MSE, and SoVMD-MPE, and significantly superior to SVM and ANN. Specifically, compared with the other six approaches, the accuracy of the proposed method in Experiment 1 is improved by 4.51%, 2.08%, 4.68%, 6.23%, 55.04%, and 69.89%, respectively, and the FPR is reduced by 55.88%, 41.18%, 57.14%, 63.41%, 91.60%, and 92.4%, respectively. Similar results also appear in Experiment 2 depicted in
Table 4. Meanwhile, the standard deviations of the developed method for these two metrics are obviously smaller than those of the other approaches in any group of experiments, which confirms the stronger stability and robustness of the proposed method for bearing fault diagnosis. In addition, from the results presented in
Table 5, it can be found that the average computation time of the proposed method is slightly more than those of EMD-CSvDE, VMD-CSvDE, SoVMD-MSE, and SoVMD-MPE, while it is much more than those of SVM and ANN, regardless of Experiment 1 or Experiment 2. Compared with the other four combined methods, the process of adaptive parameter optimization and MsFFS construction with variable scale factors in the framework of the proposed method will take more time to improve diagnosis accuracy. In addition, without the consideration of the strategies of signal decomposition and MsFFS construction, the computational costs of SVM and ANN will be reduced compared to those of the other five diagnosis methods.
To give more details, the fault diagnosis results of the proposed method for the sixth trial in Experiment 1 and the corresponding multi-class confusion matrix are shown in
Figure 12 and
Figure 13, respectively. In
Figure 12, it can be seen clearly that a small number of predicted labels of testing samples deviate from the true labels, i.e., the phenomenon of misdiagnosis. More specifically, to intuitively reflect the accuracy rate and error rate, the multi-class confusion matrix can be further built based on the above-mentioned diagnosis results, as depicted in
Figure 13. It can be observed from this figure that the diagnosis accuracy of different operation states can reach 90% or even higher, especially for six states (Nora, RF_07, IRF_07, IRF_12, ORF_07, and ORF_15) with an accuracy of 100%. Moreover, an overall accuracy of 97% can be achieved by the proposed method for the sixth trial in Experiment 1, which indicates that the proposed method contributes to identify the different fault types and defect severities of the rolling bearing and also realizes satisfactory diagnosis accuracy as a whole.
Through the above experiment results, the relevant conclusions can be summarized as follows. (1) Among all the diagnosis approaches, the proposed method realizes the highest diagnosis accuracy and the lowest FPR whether in Experiment 1 or Experiment 2, which strongly confirms its superior performance on bearing fault diagnosis. (2) Compared with the last two approaches (SVM and ANN), the remaining methods (the proposed method, EMD-CSvDE, VMD-CSvDE, SoVMD-MSE, and SoVMD-MPE) can accomplish the task of fault diagnosis with higher accuracy and lower FPR. The main reason is that the multiscale frequency component extraction by different decomposition algorithms contributes to capture the inherent characteristics of the raw signal and further establish an effective feature space for accurate diagnosis. (3) Adopting the SoVMD algorithm, the diagnosis accuracy and FPR of the developed method can be significantly improved compared with EMD-CSvDE and VMD-CSvDE. This is because the optimal parameters of the decomposition process can be adaptively determined by the SoVMD method so that the multiscale frequency components can be effectively obtained without the influence of the mode mixing problem. (4) From the perspective of feature space construction, the diagnosis performance of the proposed method is more excellent and stable than the other two approaches are, including SoVMD-MSE and SoVMD-MPE. Because of the variable parameters of scale factor, the developed CSvDE method is helpful to construct the MsFFS more effectively and improve the diagnosis performance compared with the methods of MSE and MPE. (5) For the same method, the diagnosis results in Experiment 1 are slightly superior to those in Experiment 2, but it is worth noting that the calculation time in Experiment 1 is obviously more than that in Experiment 2. This is because a small amount of fault information can still be contained in the first five frequency components and may be useful for accurate diagnosis. Taking the fewer components into account, the computation costs of Experiment 2 can be decreased significantly.
Finally, to intuitively illustrate the superior performance on feature extraction of the proposed method, the extracted MsFFS can be reduced and visualized by the t-distributed stochastic neighbor embedding (t-SNE) algorithm. We use the results of the sixth trial in Experiment 1 as an example for analysis, and
Figure 14 shows the three-dimensional projections of the original samples and the MsFFS by t-SNE. It is obvious that the MsFFS obtained by the proposed method can effectively reveal essential information contained in the original samples and accomplish the fault state identification with high accuracy. The main reason is that the complex nonlinear relationships between the raw signal and the MsFFS can be constructed effectively based on the model architecture integrating the component mode with a variable scale factor. To sum up, the proposed method can achieve superior performance in capturing valuable features for accurate fault diagnosis.