*3.5. Simulation 5*

In order to prove the stability of RDE for synthetic signal, we carried out stability testing by using the cosine signal cos(200πt) under 10 dB. The data length of each sample is 2000, we calculated the 5 entropies of 100 samples. The five entropies of cosine signal under 10 dB are shown in Figure 7. As shown in Figure 7, the entropy values of the same category are at the same level and have very little difference.

**Figure 7.** The five entropies of cosine signal under 10 dB.

To more intuitively compare the stability of the five entropies, the complexity feature boxplots of five entropies for cosine signal under 10 dB are shown in Figure 8. As shown in Figure 8, PE, W-PE, RPE, and DE have obvious fluctuations, however, RDE has the smallest fluctuation range compared to the other four entropies. The mean and standard deviation of the five entropies for the cosine signal under 10 dB are shown in Table 7. As shown in Table 7, RDE has the smallest standard deviation compared with the other four entropies. The experimental results show that RDE has better stability than the other four entropies under noisy conditions.

**Figure 8.** The complexity feature boxplots of five entropies for cosine signal under 10 dB.

**Table 7.** The mean and standard deviation of five entropies for the cosine signal under 10 dB.


### **4. Application for Real Sensor Signals**

### *4.1. Simulation 1*

In order to compare the ability of five entropies to distinguish real sensor signals, we carried out complexity testing by using three kinds of ship signals, termed as ship 1, ship 2, and ship 3. Each sample was 5000 points with a sampling frequency of 44.1 kHz. The five entropy distributions of three kinds of ship are shown in Figure 9, and each kind of ship includes 100 samples. As shown in Figure 9, compared with the distributions of PE, W-PE, and RPE, the distributions of DE and RDE were easier to distinguish the three kinds of ship signals.

The complexity feature boxplots of five entropies for three kinds of ship are shown in Figure 10, and the mean and standard deviation of five entropies for three kinds of ship are shown in Table 8. As shown in Figure 10 and Table 8, compared with the other four entropies, RDE had the smallest fluctuation range and standard deviation for each ship signal. The experimental results show that RDE has better stability for ship signals.

**Table 8.** The mean and standard deviation of five entropies for three kinds of ship.


**Figure 9.** The five entropy distributions for three kinds of ship.

To further prove the distinguishing ability of RDE, we used a support vector machine (SVM) to distinguish the three kinds of ship signals; the classification results by five entropies for three kinds of ship are shown in Table 9. As seen in Table 9, PE and W-PE have a recognition rate of less than 95%; DE and RPE have a recognition rate of more than 95%; RDE has the highest recognition rate of up to 99%. The experimental results show that RDE has better distinguishing ability for ship signals.

**Figure 10.** The complexity feature boxplots of five entropies for three kinds of ship.

**Table 9.** The classification results by five entropies for three kinds of ship.


*4.2. Simulation 2*

Like simulation 1 in Section 4.1, we carried out complexity testing by using three kinds of rolling bearings signals, termed as fault 1, fault 2, and fault 3, which come from the Case Western Reverse Laboratory [25]. Each sample is 2000 points with a sampling frequency of 12 kHz. The mean and standard deviation of the five entropies for three kinds of fault are shown in Table 10, and each kind of fault includes 50 samples. As shown in Table 10, for PE, W-PPE, RPE, and DE, the mean values of fault 1 and fault 2 are very close, which makes it difficult to distinguish the two faults; for RDE, there are obvious differences in the mean values of the three faults, and it has the smallest standard deviation compared to the other four entropies. The experimental results show that RDE has better stability for rolling bearing signals.


**Table 10.** The mean and standard deviation of five entropies for three kinds of fault.

To further prove the distinguishing ability of RDE for rolling bearing signals, we used an SVM to distinguish three kinds of rolling bearing signals; the classification results by five entropies for three kinds of rolling bearing signals are in Table 11. As seen in Table 11, PE and W-PE have a recognition rate of less than 80%; RPE has a recognition rate of more than 80%; DE and RDE have a recognition rate of less than 95%; RDE has the highest recognition rate of up to 100%. The experimental results show that RDE has better distinguishing ability for rolling bearing signals.

**Table 11.** The classification results by five entropies for three kinds of rolling bearing signals.


### **5. Conclusions**

This paper proposed a new complexity measure for analyzing time series and termed RDE. A large number of simulation experiments was carried out to verify the effectiveness of this complexity measure. Its main contributions are as follows:


Overall, as an effective complexity metric, RDE could be used to analyze more real sensor signals in different fields.

**Author Contributions:** Development of theoretical approach, Y.L. and X.G.; numerical analysis, Y.L., X.G. and L.W.; writing—original draft preparation, Y.L.; writing—review and editing, Y.L. and X.G.

**Funding:** The authors gratefully acknowledge the support of the National Natural Science Foundation of China (No. 11574250).

**Conflicts of Interest:** The authors declare no conflicts of interest.

### **References**


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