A New Method of Fault Feature Extraction Based on Hierarchical Dispersion Entropy

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Introduction
As a key component of rotating machinery, the failure rate of rolling bearing accounts for 30%-40% of rotating machinery failure [1]. So it is very important to monitor and diagnose the fault of rolling bearing to ensure the safe and reliable operation of the rotating machinery. However, the rolling bearing often runs in different working conditions and worse environments, which makes the vibration signals have nonlinear and nonstationary characteristics. How to effectively extract fault features from the vibration signals is very crucial [2].
In recent years, many nonlinear dynamic analysis methods, such as approximate entropy [3], sample entropy [4], fuzzy entropy [5], and permutation entropy [6] are widely used in fault feature extraction due to their abilities to measure the complexity of time series. Zheng et al. [7] firstly realized the decomposition of vibration signals through local characteristicscale decomposition (LCD). en, the fault features of the modal components through fuzzy entropy are extracted to realize fault diagnosis of rolling bearing. Zhong et al. [8] extracted the fault features by sample entropy after empirical mode decomposition (EMD) of vibration signals to realize fault diagnosis of rolling bearing. Aiming at the parameter selection of multiscale permutation entropy, Wang et al. [9] proposed a method of optimizing parameters of multiscale permutation entropy to extract fault features. Li et al. [10] improved multiscale permutation entropy to extract fault features which were inputted into binary tree support vector machine to realize fault diagnosis of rolling bearing. Above entropies and their improvement are widely used in fault feature extraction, but the relationship between vibration amplitudes is not considered. So Rostaghi and Azami [11] proposed the dispersion entropy to characterize the complexity of time series which was proved better than the permutation entropy and sample entropy in the dataset fields of biology and finance. Because the dispersion entropy is not only efficient in the calculation but also it can describe the relationship between vibration amplitudes. Based on the dispersion entropy which can describe the relationship between vibration amplitudes, Mostafa [12] applied the dispersion entropy to the fault diagnosis of rotating machinery, which showed the dispersion entropy was better than the permutation entropy and the approximation entropy such as more stable in the detection of high SNR signals. However, the dispersion entropy could not extract multiscale fault features of signals. For this, Yi et al. [13] proposed the multiscale dispersion entropy which could realize fault feature extraction of vibration signals under multiscale. Luo et al. [14] proposed refined composite multi-scale dispersion entropy to realize fault diagnosis of rolling bearing by improving coarsening process. is method realized the feature extraction in the multiscale of vibration signals. However, literature [15,16] pointed out that multiscale entropy could only extract the feature information of vibration signals in the low-frequency band and ignore the feature information in the high-frequency band. But the fault information of vibration signals of rolling bearing not only exists in the low-frequency band but also in the high-frequency band. erefore, in order to achieve a comprehensive fault feature extraction of rolling bearing, a new fault feature extraction method named hierarchical dispersion entropy is proposed. e fault features of vibration signals of rolling bearing are extracted by the hierarchical dispersion entropy and inputted into the KNN classifier [17] to realize fault diagnosis.
e main contributions of this paper are as follows: (1) Compared with traditional methods, such as sample entropy, approximate entropy, and permutation entropy, dispersion entropy has the advantage of measuring the relationship between vibration signal amplitudes. In this paper, the hierarchical dispersion entropy is proposed by introducing hierarchical decompose based on dispersion entropy. Hierarchical dispersion entropy can extract feature information from the high-frequency band and low-frequency band. It overcomes the problem that multiscale dispersion entropy could not extract fault information from high-frequency band. (2) e hierarchical dispersion entropy could extract more fault features. So it is applied in the fault feature extraction of rolling bearing vibrations. e method of fault diagnosis of hierarchical dispersion entropy combined with KNN is realized.
is paper is organized as follows: Section 2 presents dispersion entropy and multiscale dispersion entropy. Section 3 is the theory of hierarchical dispersion entropy and its parameter selection and simulation analysis. Section 4 is the fault diagnosis of rolling bearing based on hierarchical dispersion entropy. Section 5 is experimental verification. Section 6 summarizes the conclusions.

Dispersion Entropy.
Dispersion entropy is a method which can characterize the irregularity of time series [11]. e greater the value of dispersion entropy, the higher the irregularity of time series. On the contrary, the smaller the value of dispersion entropy, the lower the irregularity of time series. e calculation steps of dispersion entropy for the time series x � x i , i � 1, 2, . . . , N are as follows: (1) e normal distribution function (1) is selected to map the time series x to y � y j , j � 1, 2, . . . , N and y j ∈ (0, 1).
where uandσ, respectively, represent mean and standard deviations of time series x. (2) y j is mapped into integer indices between [1, 2, . . . , c] by linear expression.
where round is the rounding function and cis the number of classes after mapping.
Forc m dispersion patterns, the probability of each dispersion pattern π v 0 v 1 ...v m− 1 is as follows: where number(

Multiscale Dispersion Entropy.
In 2002, Costa et al. [18] proposed a multiscale entropy method which characterized the complexity of time series under multiscale conditions. Azami et al. [19] introduced the multiscale analysis method and proposed the multiscale dispersion entropy. e calculation steps of multiscale dispersion entropy are as follows: (1) y τ (j)is obtained by coarsening original sequencex(i)as follows: where τis the scale factor, τ � 1, 2, . . . , n.
(2) According to the calculation process of dispersion entropy in Section 2.1, the multiscale dispersion entropy is obtained by analyzing the dispersion entropy of coarse-grained time seriesy τ (j).
us, it may be known that the key step of calculation of multiscale dispersion entropy is the coarse-grained of original time series. e coarsening process is to sum time series according to different scales and then average them to form a new time series. According to the analysis of literature [15,16], we know that multiscale dispersion entropy only obtains the low-frequency information of time series, but ignores the high-frequency information. In fact, the fault information contained in vibration signals not only exists in the high-frequency band but also in the low-frequency band.

Hierarchical Dispersion Entropy.
Hierarchical dispersion entropy is proposed by introducing hierarchical analysis. e first step is to hierarchically decompose vibration signals. e second step is to get the dispersion entropy of different hierarchical nodes. e detailed calculation process is as follows: Step 1. Vibration signals are decomposed hierarchically as follows: (1) For the obtained vibration signals, the operatorQ j is defined as follows: where j � 1 or 0, the form of the matrix Q j is determined by the length of vibration signals. erefore, the following operators are defined as follows: Q 0 and Q 1 represent the low-frequency and highfrequency parts of the first layer of vibration signal decomposition. According to the operator Q 0 and Q 1 , the reconstructed vibration signal is as follows: where j � 0, 1, 2, . . . , 2 n− 1 .
{ }is reconstructed. e integerecan be expressed byc j as follows: (3) According to the vector [c 1 , c 2 , . . . , c n ], the nodes of each layer decomposition of vibration signalu i are defined as follows, and the hierarchical structure is shown in Figure 1.
Step 2. e calculation function of hierarchical dispersion entropy of different nodes is as follows: where data are the original vibration signals, kis the number of decomposition order, mis the embedded dimension, cis the number of mapping class, and d is the time delay. It can be seen from the definition of operators that Q 0 and Q 1 represent the low frequency and the high frequency, which Shock and Vibration 3 are equivalent to the low pass and the high pass in wavelet analysis. e dispersion entropies of u (1,0) , u (2,0) , and u (3,0) on the left of Figure 1 are equivalent to the dispersion entropies of different scales in multiscale dispersion entropy. In essence, the dispersion entropy of u k,0 is the dispersion entropy of different scales in multiscale, which represents the low-frequency part of time series. e high-frequency part is on the right of Figure 1, which is ignored by the multiscale entropy. It can be seen that hierarchical dispersion entropy can not only extract the lowfrequency information of vibration signals but also extract the high-frequency information. erefore, the ability of hierarchical dispersion entropy is better than multiscale dispersion entropy for extracting fault features of rolling bearing from the theoretical analysis.

Parameter Selection.
In the process of calculating hierarchical dispersion entropy, the related parameters need to be set in advance. ese parameters are decomposition order k, embedding dimension m, mapping classes c, and time delay d. When the k value is small, the frequency band of vibration signals would be larger and the detailed fault feature information from low frequency to high frequency could not be obtained. If the k value is large, the calculation efficiency of the algorithm would be reduced and the points involved in the calculation would be reduced. In general, the value of decomposition k � 3 is selected [20]. At the same time, according to the influence of the parameters of the dispersion entropy in literature [12], the embedded dimension m � 2, mapping classes c � 5, and time delay d � 1 are selected.

Simulation Analysis.
e irregularity of Gaussian white noise in different frequency bands is basically unchanged [21]. erefore, the feasibility of hierarchical dispersion entropy is verified by analyzing the hierarchical dispersion entropy of Gaussian white noise signal and compared with multiscale dispersion entropy of Gaussian white noise signal. e Gaussian white noise signal with a length of N � 2048 is generated by simulation as shown in Figure 2. e sample frequency of the Gaussian white noise is 12 kHZ. e spectrum diagram can be obtained by Fourier transforming as shown in Figure 3. e parameter settings k � 3 and τ � 8 are used to extract the hierarchical dispersion entropy and multiscale dispersion entropy. At each scale, two kinds of entropy values are calculated for 100 groups of white noise generated randomly. e error bars are shown in Figure 4. From Figure 4, we can see that the hierarchical dispersion entropy at different hierarchical nodes is basically unchanged. It is consistent with the fact that the complexity of Gaussian white noise in different frequency bands is basically unchanged. However, the multiscale dispersion entropy decreases with the scale  increase which is not consistent with the fact that the complexity of Gaussian white noise in different frequency bands is basically unchanged. At the same time, the error bar of multiscale dispersion entropy is small when the scale is small. With the increase of scale, the error bar of multiscale dispersion entropy is larger. But the error bar of hierarchical dispersion entropy is basically unchanged. It means that the stability of multiscale dispersion entropy is poor and the stability of hierarchical dispersion entropy is better. To sum up, the hierarchical dispersion entropy could represent the complexity of signals and its performance is better than multiscale dispersion entropy.

Fault Diagnosis of Rolling Bearing Based on
Hierarchical Dispersion Entropy e complexity of vibration signal is different in a different position and damage degree of rolling bearing. erefore, different fault features of vibration signals of rolling bearing can be extracted by the proposed hierarchical dispersion entropy. e fault diagnosis of rolling bearing is realized by inputting hierarchical dispersion entropies of vibration signals into the KNN classifier. e implementation steps of fault diagnosis are shown in Figure 5.

Experimental Verification
e rolling bearing experimental data of the Case Western Reserve University of the United States are selected [21]. e test system is shown in Figure 6 and the bearing is 6205-2RSJEMSKF deep groove ball bearing. Different fault settings of bearing inner ring, outer ring, and rolling element are made by electrodischarge machining (EDM) technology. e information of fault data is shown in Table 1.  Figure 7.

Fault Diagnosis of Different
We can see from Figure 7 that, it is difficult to directly identify different fault types of rolling bearing only through time-domain waveform. Firstly, the multiscale dispersion entropy of 120 samples (30 samples for each state) are extracted, and the mean value of the multiscale dispersion entropy is shown in Figure 8, where the scale τ � 8 is set and other parameters remain unchanged.
It can be seen from Figure 8 that multiscale dispersion entropy of vibration signals under normal conditions of rolling bearing is larger than that under fault conditions. is is due to the periodic impact of vibration signals in fault states is more regular than the randomness of vibration signals in the normal state. e multiscale dispersion   entropies of the balling and the inner ring are larger than that of the outer ring and are close to each other when the scale is small. Because the outer ring is fixed, but the balling and the inner ring run together with the bearing. is leads to balling and inner ring vulnerable to more noise interference in the path transmission and more irregularities. e extracted multiscale dispersion entropy is inputted into the KNN classifier for fault diagnosis. We set the number of nearest neighbors K � 2, where classification label 1 represents NR, classification label 2 represents IRF, classification label 3 represents ORF, and classification label 4 represents BF. e diagnosis results are shown in Figure 9.
From Figure 9, we can see that one sample of ball fault is considered as the inner ring fault. e overall diagnosis accuracy is 99.17%. e dispersion entropy is extracted from the same sample and is inputted into the KNN classifier for fault diagnosis as shown in Figure 10. Except for the scaleτ, the other parameters are the same as the multiscale dispersion entropy. e diagnosis accuracy is 75% and there are serious errors for the classification of inner ring and balling. Due to the rotation of inner ring and balling together, the fault impact signals interfere with each other. Dispersion entropy and multiscale dispersion entropy are not easy to extract the distinguishing features and are easy to cause misjudgment. e fault features of rolling bearing are extracted by the proposed hierarchical dispersion entropy. e decomposition order is set k � 3 and the other parameters are the same as the multiscale dispersion entropy and dispersion entropy. e hierarchical dispersion entropies of samples are shown in Figure 11.
Firstly, from Figure 11, we know that hierarchical dispersion entropy in normal state decreases with the increase of nodes in general. is indicates that the feature information mainly exists in the low frequency of vibration signals. Secondly, hierarchical dispersion entropy of vibration signals in node � 1 (low frequency part) and node � 8 (high-frequency part) is large in fault state. is indicates that the fault feature exists not only in the low frequency but also in the high frequency.
irdly, the hierarchical dispersion entropy of the same node in different states is basically unchanged.
is means that the stability of     Figure 12. We can see from Figure 12 that the accuracy of diagnosis result is 100% which is higher than 99.17% of multiscale dispersion entropy and 75% of dispersion entropy. e diagnostic accuracy of the three methods is shown in Table 2. e effective feature extraction of inner ring and balling is realized. erefore, hierarchical dispersion entropy is better than multiscale dispersion entropy and dispersion entropy for fault feature extraction.     more similar which make them more difficult to distinguish. At the same time, the two methods are less able to distinguish the signals with stronger similarity. us, it is not feasible to diagnose different damage faults directly by dispersion entropy and multiscale dispersion entropy. Similarly, the accuracy of fault diagnosis with different damage degrees by using hierarchical dispersion entropy is 98.33%, and the results are shown in Figure 16. ere is an error classification between the inner ring and the balling. e diagnosis accuracy of the three methods is shown in Table 3. Compared with the dispersion entropy and multiscale dispersion  entropy, the accuracy is greatly improved. is shows that hierarchical dispersion entropy can realize the diagnosis of different damage fault degrees and has a stronger ability to extract fault features. At the same time, the fault diagnosis accuracy of 100% can be achieved for the outer ring 2, the balling 1, and the Balling 3 with strong noise interference.

Fault Diagnosis of Different
is further shows that hierarchical dispersion entropy has better fault feature extraction ability under strong noise.
To summarize, the fault diagnosis ability of dispersion entropy for different positions and damage degrees is poor. So it is not feasible to directly extract the fault features of vibration signals. e multiscale dispersion entropy has a high diagnostic accuracy of fault diagnosis for different positions, but its discrimination ability of different damage degrees is not satisfied. Hierarchical dispersion entropy not only achieves good results for fault diagnosis of different positions but also achieves satisfactory results for fault diagnosis of different damage degrees. Because it can extract the fault feature information of vibration signals in high-and low-frequency bands simultaneously. So hierarchical dispersion entropy has obvious advantages for fault features extraction of complex signals and is better than dispersion entropy and multiscale dispersion entropy. Finally, for the same dataset, the accuracies of the proposed method and other methods are shown in Table 4. It can be seen that the proposed method in this paper has a high-fault diagnosis rate and a good diagnostic accuracy in fault diagnosis of multiclassification.

Conclusions
In this paper, a rolling bearing fault diagnosis method based on hierarchical dispersion entropy and KNN is proposed. e main conclusions are as follows: (1) Based on the dispersion entropy, this paper proposed the hierarchical dispersion entropy by combining with hierarchical decomposition. Data Availability e datasets analysed during the current study are available in the https://csegroups.case.edu/bearingdatacenter/pages/ download-data-file link to datasets.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.