## 5 Experimental Verification and Analysis
### Case Study I: Fault Diagnosis of Rolling Bearing
#### 5.1.2 Analysis Results of RMSDE  
The proposed RMSDE is used to analyze the experimental data of the above five different fault types and compare them with several types of multi-scale traditional entropy-based feature extraction methods. For any entropy method, when the embedding dimension \(m\) is too small, the vector formed contains fewer matching patterns, which makes it impossible to detect the dynamic characteristics of the time series effectively. When \(m\) is too large, the computing time will increase, and the computing efficiency will decrease. For RMSDE, when the sign numbers \(c\) and \(\varepsilon\) are set too small, it cannot accurately reflect the changes in the subtle dynamic characteristics of the data, so it cannot accurately extract the fault characteristics. When \(c\) and \(\varepsilon\) are set too large, RMSDE will be easily affected by noise. In order to achieve optimal results, the embedding dimension \(m\) of RMSDE is set to 5, and the two sign numbers \(c\) and \(\varepsilon\) are taken as 7 and 20, respectively. It should be noted that the influence mechanism of the parameter setting of the entropy method on the results is complicated because there are interactions among various parameters. Based on the principle of obtaining optimal and optimal processing results, the parameters of the traditional entropy methods are mainly chosen according to the references [8; 10; 11]. The embedding dimension m of both MSE and MFE is taken as 2, the similarity tolerance \(r\) is taken as 0.15, the fuzzy power n of MFE is taken as 2, and the embedding dimension m of MPE is taken as 6. Each entropy method is used to process each sample data, and the entropy values of the first 20 scales calculated are used as the initial eigenvalues, and then the average value of the initial eigenvalues of the same fault type under each scale is calculated, and the results obtained are shown in Figure 12.  
Figure 11: The vibration signals of each rolling bearing: (**a**) C1, (**b**) C2, (**c**) C3, (**d**) C4, and (**e**) C5. Figure 12: _Cont._It can be seen from Figure 12 that the processing results of bearing vibration signals in different health states by several methods have certain differences. Analyzing the ranking of the overall entropy value, the overall ranking of the estimated results of different entropy methods is not the same. However, in addition to the phenomenon of high aliasing in MPE, the result curves of other methods show certain separability. It is also proved that in the field of bearing fault diagnosis, signals of different fault types have different dynamic complexity, and fault feature extraction based on the entropy calculation method is effective. For example, from the vibration signal example shown in Figure 11, it can be seen that the periodic pulse characteristic of the signal of C4 is the most significant, indicating that the dynamic complexity is the lowest. Therefore, the entropy value calculated by several entropy methods for this fault-type signal is kept at a low value. It is worth noting that C4 and C5 in Figure 12a and C2 and C3 in Figure 12b show strong separability, which demonstrates that if the fault location is the same, different processing methods will make the estimation method extract different features from the signal. Comparing the fault entropy characteristic curves of several methods in Figure 12, it is found that the RMSDE method proposed in this paper is more stable than the traditional entropy method to extract the health state characteristics with the smallest error and fluctuation. Only in scale factor \(\tau\) at 10 to 12 the entropy values of C2 and C3 intersect. In the traditional entropy calculation method, the MPE method is almost completely ineffective; only when the scale factor \(\tau\) is equal to 1, the extracted eigenvalues have a certain degree of local division. Both MSE and MFE have certain separation effects, but the effect is relatively general.