Refined time-shift multiscale normalised dispersion entropy and its application to fault diagnosis of rolling bearing

Physica A xxx (xxxx) xxx

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Refined time-shift multiscale normalised dispersion entropy and its application to fault diagnosis of rolling bearing ∗

Jinde Zheng a,b , , Haiyang Pan a , Qingyun Liu a , Keqin Ding c a

School of Mechanical Engineering, Anhui University of Technology, Maanshan, Anhui, 243032, China School of Mechanical and Manufacturing Engineering, UNSW Sydney, NSW 2052, Australia c China Special Equipment Inspection and Research Institute, Beijing, 100029, China b

article

info

Article history: Received 21 May 2018 Received in revised form 14 April 2019 Available online xxxx Keywords: Multiscale entropy Multiscale permutation entropy Dispersion entropy Refined time-shift multiscale dispersion entropy Rolling bearing Fault diagnosis

a b s t r a c t Multiscale entropy (MSE) and multiscale permutation entropy (MPE) are two effective nonlinear dynamic complexity measurement methods of time series that have been applied to many areas for complexity feature extraction in recent years. To overcome the inherent defects of sample entropy and permutation entropy, together with the coarse graining process used in MSE and MPE, based on the recently proposed dispersion entropy (DisEn), the refined time-shift multiscale normalised dispersion entropy (RTSMNDE) is proposed here for the complexity measurement of time series. In the RTSMNDE method, first, to expand DisEn to the multiscale framework, the time-shift multiscale method is used to replace the traditional coarse graining multiscale method. The refining approach is adopted to alleviate the fluctuation of DisEns in larger scale factors, and a normalisation operation is implemented on all entropies to restrain the influence of the parameters on RTSMNDE values. Furthermore, the RTSMNDE is compared with the multiscale dispersion entropy (MDE) by analysing synthetic simulation signals to verify its effectiveness. Based on that, an intelligent fault diagnosis method for rolling bearings is proposed by combining the RTSMNDE for fault feature extraction with the particle swarm optimisation support vector machine for feature classification. Finally, the proposed method is applied to rolling bearing experimental data analysis, and the analysis results show that the proposed method can effectively diagnose the locations and degrees of rolling bearing failures and obtain a higher recognition rate than those of the MPE and MDE-based methods. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Rolling bearings are widely used in rotary machinery equipment and play an increasingly important role in many industrial production departments. It is of great importance to study the condition-monitoring technologies and faultdiagnosis methods of rolling bearings to ensure the safe operation of the whole rotating machinery equipment and its key parts [1]. The kernel step of the rolling bearing fault diagnostic is to extract fault feature information from vibration signals. As most vibration signals from mechanical systems are nonlinear and non-stationary, it is necessary to select the appropriate signal processing method to deal with these types of signals. With the development of nonlinear dynamics, many nonlinear dynamic analysis methods, especially the entropy-based irregularity or complexity measurement tools, have been employed to reflect the fault information and extract fault ∗ Corresponding author at: School of Mechanical Engineering, Anhui University of Technology, Maanshan, Anhui, 243032, China. E-mail address: [email protected] (J. Zheng). https://doi.org/10.1016/j.physa.2019.123641 0378-4371/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: J. Zheng, H. Pan, Q. Liu et al., Refined time-shift multiscale normalised dispersion entropy and its application to fault diagnosis of rolling bearing, Physica A (2019) 123641, https://doi.org/10.1016/j.physa.2019.123641.

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features from vibration signals of rolling bearings and other machinery equipment [1]. The most often used entropy-based irregularity or complexity measurement methods mainly include the approximate entropy, sample entropy (SampEn), fuzzy entropy, permutation entropy (PE), multiscale entropy (MSE), and multiscale permutation entropy (MPE) [2–7], and they have been applied to mechanical fault detection and diagnosis by related scholars and fruitful research results have been achieved. For example, the approximate entropy was introduced to nonlinear fault feature information extraction of mechanical failure by Yan et al. [1]. PE was applied to monitor the state of a mechanical system by comparing it with Lempel–Ziv complexity and the results validate the effectiveness of PE in fault information representation [8]. In the literature [9], MSE-based features were constructed and applied to reflect the fault information of rolling bearings by Zhang et al. by combination with the adaptive network-based fuzzy inference system. Further, MPE was applied to the fault feature representation of vibration signals of rolling bearings by Li et al. [10]. In addition, in the literature [11,12] the improved MSE method called multiscale fuzzy entropy was used for fault feature extraction of machinery systems. However, these entropy-based irregularity or complexity measure methods still have some inherent defects. For example, the self-matching of the template is considered in approximate entropy, which causes that the estimated entropy value is smaller than the expected one. The computation of sample entropy is time-consuming and the similarity measurement of the template changes suddenly with the use of the Heaviside step function. The amplitude information of the time series is ignored in PE computation. The coarse graining process used in MSE and MPE in nature is linear mean filtering, which will cause the loss of important information of the analysed signals [13]. Recently, dispersion entropy (DisEn) was proposed by Rostaghi and Azami [14,15] as a new nonlinear dynamic indicator for irregularity measurement to overcome the shortcomings of SampEn and PE. They also expanded the DisEn to the multiscale framework in [16], obtaining the multiscale dispersion entropy (MDE). Compared with SampEn and PE, the amplitude of the time series is considered by DisEn, and its computation cost is much lower than that of SampEn and PE. In particular, DisEn has a stronger anti-noise capability than that of PE and SampEn, because the slight change in amplitude does not change the corresponding class label defined in DisEn. However, the original coarse graining multi-scale approach used in MDE generally results in the obtained coarse grained time series becoming shorter with the increase in scale factor, and thus, the MDE values in different scale factors will exhibit large fluctuation [13]. To overcome the above shortages of MDE, the refined time-shift multiscale normalised dispersion entropy (RTSMNDE) is proposed here for irregularity and complexity measure of time series. In the RTSMNDE method, the coarse grainingbased multiscale method used in MSE, MPE, and MDE is replaced by the new time-shift multiscale method that constructs the new time-shift multiscale time series based on the amplitude information of the original time series. This can effectively preserve the important constructing information of the original data without loss of information [17]. Further, the refining method is used to avoid the fluctuation of DisEn values at larger scale factors, and lastly, the normalisation operation is implemented to restrain the influence of different parameter selections on the RTSMNDE. Owing to the interference of the background or environmental noises, the vibration acceleration signals of the rolling bearing are generally random signals obeying a normal distribution [18]. The irregularity and dynamic complexity will change once the rolling bearing works with failure and the characteristic information related with fault of the rolling bearing vibration signal is often distributed over different scales because of the complexity of the mechanical system. When we obtain the RTSMNDE values related with fault complexity, an intelligent fault diagnosis method for rolling bearings is proposed based on the RTSMNDE and particle swarm optimisation support vector machine (PSOSVM) [19,20], which is often used for automatic identification of failure feature modes. Finally, the proposed fault diagnosis method for rolling bearings is applied to experimental data analysis by comparing it with the MPE and MDE-based fault diagnosis ones. The analysis results show that the proposed method can effectively distinguish the fault severity and categories of rolling bearings and obtain a higher identifying rate and better performance of rolling bearing fault diagnosis than those of the MPE and MDE-based fault diagnosis methods. The rest of this paper is organised as follows. The algorithms of dispersion entropy and multiscale dispersion entropy are reviewed in Section 2. The refined time-shift multiscale normalised dispersion entropy is presented and the comparison analysis of the RTSMNDE with MDE is investigated in Section 3. The RTSMNDE and PSOSVM-based fault diagnosis method for rolling bearings is proposed in Section 4, along with the application to experimental data and a comparison analysis. Finally, conclusions are drawn in the final section. 2. Review of dispersion entropy and multiscale dispersion entropy 2.1. Dispersion entropy DisEn is a nonlinear dynamic analysis method for the irregularity measure of time series. The relationship of neighbouring amplitude values of the original data ignored in PE will be taken into account by DisEn. In addition, the operation of the embedded vector rank and distance calculation between two different delay vectors in SampEn is abandoned by DisEn. In particular, DisEn has a stronger anti-noise ability than SampEn and PE because the small change in amplitude does not alter the corresponding class label in DisEn. For a given univariate time series X = {x1 , x2 , . . . , xN } with length N, the steps of DisEn can be described as follows [14]: Please cite this article as: J. Zheng, H. Pan, Q. Liu et al., Refined time-shift multiscale normalised dispersion entropy and its application to fault diagnosis of rolling bearing, Physica A (2019) 123641, https://doi.org/10.1016/j.physa.2019.123641.

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(1) The original time series X is mapped into Y = {y1 , y2 , . . . , yN } by using a normal cumulative distribution function (NCDF) shown as yj =

1

∫

xj

e

√

σ 2π

−(t −µ)2 2σ 2

dt

(1)

−∞

where yi ∈ (0, 1), and µ and σ represent the mean and standard deviation of time series X , respectively. (2) For each member of the mapped signal, all elements of Y (yi , j = 1, 2, . . . , N) are mapped to c classes with integer indices from 1 to c by using the linear transform zjc = R(c · yj + 0.5)

(2)

zjc

where denotes the jth member of the classified time series, c means the number of classes, and R(·) represents the rounding function. Although step (2) is linear, the whole mapping procedure is nonlinear for the use of NCDF in step (1). m,c (3) For given embedding dimension m and time delay d, the time series zi can be obtained according to { c the } m,c m,c c c zi = zi , zi+d , . . . , zi+(m−1)d , i = 1, 2, . . . , N − (m − 1)d. Each time series zi can be mapped to a dispersion pattern πv0 v1 ...vm−1 , where zic = v0 , zic+d = v1 , . . . , zic+(m−1)d = vm−1 . The number of possible dispersion patterns that can be assigned m,c to each time series zi is equal to c m as the signal has m members, and each member can be one of the integers from 1 to c. (4) For each c m potential dispersion pattern πv0 v1 ...vm−1 , the relative frequency can be obtained by m,c

Number i ⏐i ≤ N − (m − 1)d, zi

{ ⏐

p(πv0 v1 ...vm−1 ) =

has type πv0 v1 ...vm−1

N − (m − 1)d

} (3)

where the ‘Number’ in the above formula means counting the number of i that meets the conditions in the curly braces m,c and p(πv0 v1 ...vm−1 ) shows the number of dispersion patterns of πv0 v1 ...vm−1 that is assigned to zi , divided by the total number of embedded signals with embedding dimension m. (5) Based on the definition of Shannon entropy, the DisEn of X is computed by m

DisEn(X , m, c , d) = −

c ∑

p(πv0 v1 ...vm−1 ) ln(p(πv0 v1 ...vm−1 ))

(4)

π =1

It can be found from the algorithm of DisEn that when all distribution patterns p(πv0 v1 ...vm−1 ) have equal probabilities, DisEn acquires the largest entropy value ln(c m ), and a typical example is the Gaussian white noise. In contrast, when the probability of the distribution pattern p(πv0 v1 ...vm−1 ) is unitary, i.e., only one value is not zero, DisEn obtains the smallest value, which indicates that the time series is completely predictable and a typical example is the periodic signal with low frequency. 2.2. MDE algorithm Based on DisEn, the MDE algorithm can be described as follows [15]. (1) For a normalised univariate data W = {w1 , w2 , . . . , wL } with length L, the original signal W is firstly divided into ⌊L/τ ⌋ non-overlapping segments with length τ . Then, the average of each segment is calculated to derive the coarse graining time series. This process, named coarse graining, can be shown as follows: (τ )

yj

=

1

τ

jτ ∑

wa , 1 ≤ j ≤ ⌊L/τ ⌋

(5)

a=(j−1)τ +1

where τ is called the scale factor and y(1) ( i.e., τ = 1) is the original data. When τ > 1, the original data is divided into τ coarse graining time series y(τ ) with length ⌊N /τ ⌋ (where ⌊N /τ ⌋ represents the largest integral smaller than N /τ ). (2) DisEn is calculated for each coarse graining data under the same parameters as MDE(W , τ , m, c , d) = DE(y(τ ) , m, c , d)

(6)

DisEns over different time scales are drawn as a function of the scale factor, and this process is called MDE analysis. MDE overcomes the defects of DisEn, which only measures the irregularity of the time series on a single scale. However, the coarse graining-based multiscale process used in MDE heavily depends on the length of the time series, and the entropy fluctuation over multiple scales will increase with the increase in the scale factor. In addition, the amplitude values of the coarse graining time series are obtained by calculating the mean of all data points in each non-overlapping segment, which inevitably leads to the loss of potentially useful information. Please cite this article as: J. Zheng, H. Pan, Q. Liu et al., Refined time-shift multiscale normalised dispersion entropy and its application to fault diagnosis of rolling bearing, Physica A (2019) 123641, https://doi.org/10.1016/j.physa.2019.123641.

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Fig. 1. Calculation process of RTSMNDE.

3. Refined time-shift multiscale normalised dispersion entropy 3.1. RTSMNDE method In this subsection, RTSMNDE is proposed aiming at the above problems of MDE, and the detailed steps of RTSMNDE can be described as follows: (1) For a normalised time series X = {x(i), i = 1, 2, · · · , N } with length N, suppose β and k are positive integers and β = 1, 2, . . . , k. The kth scale time series is defined by β

Xk = xβ , xβ+k , xβ+2k , . . . , xβ+nβ k ,

(

)

(7)

where nβ = ⌊(N − β) /k⌋ (nβ represents the largest integer no more than (N − β) /k). It can be seen from Eq. (7) that β and k stand for the initial time point and time interval of the new time series, respectively, i.e., for a given time interval k, the kth new time series are constructed by using the k time shift from the β th time point. For convenience, k is still called scale factor. β (2) For a given scale factor k, k new time series Xk can be obtained with different starting points β (β = 1, 2, . . . , k), β β and thus, the pk (πν0 ν1 ...νm−1 ) of each Xk can be obtained according to steps (1)–(4) of DisEn’s computation. Then, we define ∑k β p(πv0 v1 ...vm−1 ) = β=1 pk (πv0 v1 ...vm−1 ). (3) RTSMDE of X = {x(i), i = 1, 2, · · · , N } can be defined as m

RTSMDE(X , m, c , d, k) = −

c ∑

p(πv0 v1 ...vm−1 ) · ln(p(πv0 v1 ...vm−1 ))

(8)

π =1

where k = 1, 2, . . . , τm , and τm is the largest scale factor. Because the maximum of DisEn is ln(c m ), we define the normalised RTSMDE, i.e., RTSMNDE by RTSMDE/ln(c m ), to restrain the influence of the different parameters m and c on DisEn of all scales. Compared with MDE, RTSMNDE (or RTSMDE) combines all the information of the new time series in the same scales and, most of all, its amplitudes of time-shift multiscale sequences are parts of the original time series, which can avoid the phenomenon of ‘‘neutralisation’’ caused by the coarse graining process. Simultaneously, we use the refining method to average the probability of each time-shift time series in the same scale factor, which can avoid the undefined entropy of MSE and MDE on a certain scale. The calculation process of RTSMNDE can be described as shown in Fig. 1. Please cite this article as: J. Zheng, H. Pan, Q. Liu et al., Refined time-shift multiscale normalised dispersion entropy and its application to fault diagnosis of rolling bearing, Physica A (2019) 123641, https://doi.org/10.1016/j.physa.2019.123641.

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Fig. 2. Waveforms of Gaussian white noise and 1/f noise (a) Gaussian white noise and (b) 1/f noise.

3.2. Parameter selection and comparison analysis The calculation of RTSMDE is related with the following parameters: embedding dimension m, class number c, and time delay d. The Ref. [14] recommended that, in general, m obtains a value of 2 or 3, d = 1, and c obtains an integer between 3 and 9. We will investigate the influence of these parameters in the following part. Because the superiority of MDE over MSE and MPE in terms of time cost and robustness has already been verified in the literature [16], in this study we only compare RTSMNDE with the MDE method. First, we use Gaussian random white noise and 1/f noise as examples to verify the performance of the proposed method. Without loss of generality, the MDE and RTSMNDE of 20 groups of Gaussian random white noises and 1/f noises (shown in Fig. 2) are estimated, and their mean and standard deviation curves are computed for comparison when m = 2 or 3, c = 3, 6, or 9, and d = 1. Fig. 3(a–b) shows the MDE and normalised MDE of the 1/f noise for different m and c, while Fig. 3(c–d) shows that of the white noise for different m and c. RTSMDEs of the 1/f noise and white noise for different m and c without normalisation are depicted in Fig. 3(e–f), while RTSMNDEs of the 1/f noise and white noise are presented in Fig. 3(g–h). First, it can be seen from Fig. 3(a, c, e, and f) that the MDE and RTSMDE of the white noise and 1/f noise will increase with the increase in parameters c or m, and thus, the estimated DisEn values are greatly influenced by c and m. However, as shown in Fig. 3(b, d, g, and h), the normalised MDE and RTSMNDE are less affected by the parameters c and m. In addition, much more information will be considered by DisEn for a smaller m and c, and thus, the RTSMNDEs of the white and 1/f noises are very close and it is difficult to distinguish them from each other. However, a larger c will cause the RTSMNDEs of the white and 1/f noises to be smaller than the desired value. According to Ref. [14], generally, we set m = 3 and c = 6. Second, the MDE of the Gaussian white noise decreases monotonically with the increase in scale factor, while that of the 1/f noise is maintained constant. In the RTSMNDE analysis, with the increase in scale factor, the mean curves of RTSMNDE of the white noises are very stable and close to 1, while that of the 1/f noise monotonically increases. Although there is a very little difference in MDE and RTSMDE (or their normalisation) for the white and 1/f noises, they both indicate that the changing trend of the white noise and 1/f noise has an obvious difference. The 1/f noise is more complex than the white noise in construction, which is consistent with our understanding of the white and 1/f noises. However, the trend of the normalised MDE curve of the white noise decreases (from the normalised maximum 1) with the increase in scale factor, while the RTSMNDE of the white noise is stable at the maximum value 1 in all scale factors. The normalised MDE of the 1/f noise is nearly stable at a constant value in all scale factors, while RTSMNDE increases (to the maximum value 1), and these conclusions can also be clearly seen from Fig. 5(a–b). The white noise is less complex than the 1/f noise and MDE’s result is consistent with that of MSE. However, the result of RTSMNDE may be more easily understandable from the perspective of the complexity relationship with the changing trend of entropy values. That is, for a given signal, if its RTSMNDE is stable at a constant value without large fluctuation with the increase in scale factor, it may have the same constructed mode information and have no new information increase in all scales, such as white noise. If its RTSMNDE is increasing with the increase in scale factor, it means that the signal may have more new complexity information in the larger scales, such as 1/f noise. Next, the comparison results of Fig. 3 also demonstrate that with the increase in scale factor, the standard deviations of the normalised MDE of 20 groups of random signals are larger than that of RTSMDE, which means that RTSMNDE is more stable and has stronger robustness than MDE. Finally, the normalised MDE and RTSMNDE of the white and 1/f noises under the selected parameters (m = 3, c = 6, and d = 1) are shown in Fig. 4 for a clearer comparison, from which it can be seen that the white and 1/f noises are distinguished clearly by both the normalised MDE and RTSMNDE, and they have different change trends with the increase in scale factor. Please cite this article as: J. Zheng, H. Pan, Q. Liu et al., Refined time-shift multiscale normalised dispersion entropy and its application to fault diagnosis of rolling bearing, Physica A (2019) 123641, https://doi.org/10.1016/j.physa.2019.123641.

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Fig. 3. MDE, RTSMDE, and RTSMNDE comparisons of white and 1/f noises under different m and c. (a) MDE of 1/f noise for different m and c without normalisation; (b) normalised MDE of 1/f noise for different m and c; (c) MDE of white noise for different m and c without normalisation; (d) normalised MDE of white noise for different m and c; (e) RTSMDE of 1/f noise for different m and c without normalisation; (f) RTSMDE of white noise for different m and c without normalisation; (g) RTSMNDE of 1/f noise for different m and c; and (h) RTSMNDE of white noise for different m and c.

Please cite this article as: J. Zheng, H. Pan, Q. Liu et al., Refined time-shift multiscale normalised dispersion entropy and its application to fault diagnosis of rolling bearing, Physica A (2019) 123641, https://doi.org/10.1016/j.physa.2019.123641.

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Fig. 3. (continued).

Please cite this article as: J. Zheng, H. Pan, Q. Liu et al., Refined time-shift multiscale normalised dispersion entropy and its application to fault diagnosis of rolling bearing, Physica A (2019) 123641, https://doi.org/10.1016/j.physa.2019.123641.

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Fig. 4. Normalised MDE and RTSMNDE of white and 1/f noises under the selected parameters (m = 3, c = 6, and d = 1). (a) Normalised MDE of white and 1/f noises and (b) RTSMNDE of white and 1/f noises. Table 1 Class labels and notations of rolling bearing experiment data. Fault type

Fault size (mm)

Class label

Fault type

Fault size

Class label

Normal (Norm) Inner ring slight fault (IR1) Inner ring moderate fault (IR2) Inner ring severe fault (IR3) Outer ring slight fault (OR1)

0 0.1778 0.3556 0.5334 0.1778

1 2 3 4 5

Outer ring moderate fault (OR2) Outer ring severe fault (OR3) Ball element slight fault (B1) Ball element moderate fault (B2) Ball element severe fault (B3)

0.3556 0.5334 0.1778 0.3556 0.5334

6 7 8 9 10

4. RTSMNDE-based fault diagnosis method of rolling bearings 4.1. Proposed method Based on the RTSMNDE and PSOSVM, the detailed steps of the proposed fault diagnosis method for rolling bearing are described as follows: (1) Assume that we have L states of a rolling bearing composed of different fault classes and degrees, and the sample numbers of each of the states are M1 , M2 , . . . , ML . All samples of each state are randomly divided into Ml /2 training samples and Ml /2 testing samples (l = 1, 2, . . . , L). (2) The RTSMNDEs of all training and test samples are calculated with the largest scale factor τm = 20, embedding dimension m = 3, class label c = 6, and time delay d = 1. Then, the first several features that contain the most important fault information are selected as sensitive fault features, and thus, L sensitive fault feature sets are obtained. (3) The sensitive fault feature sets of the training samples of L states are input to the PSOSVM-based L-class fault classifier for training. (4) The sensitive fault features of the testing samples of L states are input to the trained PSOSVM-based L-class fault classifier above for testing and diagnostics. The outputs of the multi-classifier are used to differentiate the working states of the rolling bearing. 4.2. Experimental data analysis In this subsection, experimental data of the Bearing Data Center of Case Western Reserve University are used to verify the effectiveness of the proposed fault diagnosis method for rolling bearings [21]. In the test, the used bearings are 62052RS deep groove ball bearings, and the electrical discharge machining technology is used to set the single point failure to the rolling bearings. The fault locations include the local rolling ball element (Ball) fault, outer ring (OR) fault, and inner ring (IR) fault with different diameters of 0.1778 mm, 0.3556 mm, and 0.5334 mm, with the same depth of 0.2794 mm. The vibration acceleration signals are collected under the following conditions: rotating speed 1730 r/min, load 2.2 kW, and sampling frequency 12 kHz. A detailed description of the experimental data can be found in [22]. Considering the normal rolling bearing (Norm), ten working states of the rolling bearing are under our consideration, and the types of vibration acceleration signals of each state are shown in Fig. 5. The class labels of the experiment data of the rolling bearings are presented in Table 1. The above ten states of the experiment data include different fault categories and severities of the rolling bearing, and each state contains 30 samples with length of 2048 points; thus, a total of 300 samples are obtained. Firstly, the RTSMNDEs Please cite this article as: J. Zheng, H. Pan, Q. Liu et al., Refined time-shift multiscale normalised dispersion entropy and its application to fault diagnosis of rolling bearing, Physica A (2019) 123641, https://doi.org/10.1016/j.physa.2019.123641.

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Fig. 5. Time domain waveform of vibration signals of rolling bearing.

Fig. 6. RTSMNDE of vibration signals of a rolling bearing under different states.

of all 300 samples are calculated with m = 3, c = 6, and d = 1. The corresponding mean and standard deviation curves of each state are shown in Fig. 6, from which it is difficult to distinguish the ten states of the rolling bearings directly by observing the RTSMNDE curves. Secondly, all 30 samples of each class are randomly divided into 15 training samples and 15 testing samples, and thus, a total of 150 training samples and 150 testing samples are obtained. When the rolling bearing has a local failure, the amplitude of the high frequency carrier of vibration signals is modulated by the fault characteristic frequency. Given that the information related with the fault location and fault severity generally locates in the high frequency, the extracted RTSMNDE values in the first five scale factors are seen as sensitive fault features. Then, the sensitive fault features of 150 training samples are input to the PSOSVM-based multi-fault classifier for training and testing. In the PSOSVM parameter selection, the local and global search ability parameters of particle swarm c1 and c2 are set to 1.5 and 1.7, the largest number of initial evolutions is set to 200, and the parameters c and g in the SVM are respectively ranging from 0.1 to 100 and from 0.01 to 1000. Lastly, the sensitive fault features of the testing samples are input to the trained PSOSVM multi-fault classifier for testing and identifying, and the corresponding outputs are given in Fig. 7. It can be seen from Fig. 7 that the proposed method can identify the ten states of the rolling bearing with different fault locations and severities (the labels are listed in Table 1) and the corresponding fault identification rate is 100%. Please cite this article as: J. Zheng, H. Pan, Q. Liu et al., Refined time-shift multiscale normalised dispersion entropy and its application to fault diagnosis of rolling bearing, Physica A (2019) 123641, https://doi.org/10.1016/j.physa.2019.123641.

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Fig. 7. Outputs of the proposed method of all testing samples. Table 2 Identification comparison for different numbers of fault features.

a

Selected sensitive features

Misclassified samples

Identification rate

First 2, 3 entropies First 5,6,7 entropies First 4, 8, . . . , 25 entropies

9(2)→3a 0 9(1)→3

98.67% 100% 99.33%

Means that two samples in class 9 are misclassified to class 3.

Next, the influence of the number of sensitive fault features on the recognition rate is further studied. The vibration signal of a rolling bearing often indicates that the natural frequency component is modulated by the components with a fault characteristic frequency and its multiplications. Therefore, the important fault feature information generally concentrates at the high frequency, and thus, we select the RTSMNDE values in the first several scales to construct the sensitive fault feature. In fact, from Fig. 6 it can also be seen that the ten states of the rolling bearings have more clear distinguishing ability in the relative smaller scale factors than that in the larger ones. To select an appropriate number of sensitive fault features, we take the first 2, 3, . . . , 25 scale factors of the RTSMNDE values as the sensitive fault features, respectively, and the same number of training and testing samples are used to train and test the PSOSVM multi-fault classifier using the same parameter selections as in the above process. The corresponding outputs and identifying rates of the testing samples are presented in Table 2, from which it can be found that when the features consist of the first 2 or 3 RTSMNDEs, two samples of the ninth class are wrongly classified to the third class, namely, two samples of the rolling bearing with moderate ball element failure are misclassified into the ones with moderate inner ring fault, and the fault recognition rate is 98.67%. When the features are composed of 5, 6, and 7 RTSMNDE values, all testing samples are rightly classified, and the corresponding fault identification rate is 100%. When the feature consists of the first 4 or 8, 9, . . . , 25 RTSMNDE values, one testing sample of a rolling bearing with a moderate rolling ball element fault is misclassified into the class of moderate inner ring fault, and the corresponding fault recognition rate is 99.33%. The above analysis results show that when the number of selected fault features is small (2 or 3), all the information related with the failure cannot be completely reflected, but when the number of selected fault features is excessively large, although the recognition rate will increase, the training process will require much more time. Therefore, in general, we set the number of selected sensitive fault features to the range from 5 to 7. In the following, the MDE and MPE are used to extract the fault feature from rolling bearing vibration signals for comparison to illustrate the superiority of the RTSMNDE. The MDE and MPE of all training and testing samples are calculated, where m = 3, c = 6, and d = 1 in the MDE calculation, and m = 6 and time delay λ = 1 in the MPE calculation [6]. The mean and standard deviation curves of the MDE and MPE of all rolling bearing experimental data are shown in Fig. 8 when the same training and testing samples as in the RTSMNDE case are used, along with the parameter selection in the PSOSVM-based multi-fault classifier. We also take different numbers of MDE and MPE values over different scale factors as the sensitive fault features for training and testing, and the corresponding fault recognition rates of the PSOSVM-based classifiers are shown in Fig. 9 for comparison with the proposed RTSMNDE-based method. From Fig. 9 it can be found that the highest fault identification rates of the methods using MPE and MDE for feature extraction are 87.33% and 98.67% for different numbers of feature vectors. Moreover, it can be found that the MDE and RTSMNDE-based fault diagnosis methods are not affected significantly by the number of sensitive fault features, and the fault identification Please cite this article as: J. Zheng, H. Pan, Q. Liu et al., Refined time-shift multiscale normalised dispersion entropy and its application to fault diagnosis of rolling bearing, Physica A (2019) 123641, https://doi.org/10.1016/j.physa.2019.123641.

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Fig. 8. MDE and MPE of vibration signals of a rolling bearing under different states. (a) MDE of vibration signals of a rolling bearing under different states; and (b) MPE of vibration signals of a rolling bearing under different states.

rates are all higher than 95% when the number of features is greater than three. However, the MPE-based fault diagnosis method for feature extraction is affected by the number of sensitive fault features, and the identification rates are all lower than 90% for different numbers of features for the ten-class problem. In summary, the proposed method has higher fault identification rates than those of the MDE and MPE-based fault diagnosis methods for different numbers of input features, and the results of the comparison verified the superiority of the RTSMNDE-based fault diagnosis method for rolling bearings. 5. Conclusions The RTSMNDE is proposed to overcome the shortcomings of the MSE and MDE for complexity measure of time series. Further, the RTSMNDE is compared with the MDE by analysing two types of noise signals, and the results show that the RTSMNDE has much more stability than the MDE. As the RTSMNDE is very suitable for the vibration signal analysis of faulty rolling bearings and can be utilised to extract the failure complexity features distributed in multiple time scales from vibration signals, based on the RTSMNDE and PSOSVM, an intelligent fault diagnosis method was proposed and applied to rolling bearing experimental data analysis. The proposed fault diagnosis method is compared with the MDE and MPEbased fault diagnosis approaches. In addition, the number of features input to the multi-fault classifier is investigated. The Please cite this article as: J. Zheng, H. Pan, Q. Liu et al., Refined time-shift multiscale normalised dispersion entropy and its application to fault diagnosis of rolling bearing, Physica A (2019) 123641, https://doi.org/10.1016/j.physa.2019.123641.

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Fig. 9. Comparison of identification rates of fault diagnosis methods based on MPE, MDE, and RTSMNDE.

comparison results show that the RTSMNDE can effectively distinguish the rolling bearing failure locations and degrees, and the identification rate of the proposed method reaches 100%, which is higher than that of the MDE and MPE-based fault diagnosis methods for different numbers of fault features. In summary, the proposed RTSMNDE is an effective nonlinear dynamic method for complexity measure of time series, and we will study its related theories in the future for determining broader application areas. Acknowledgements This work was supported by the National Key Technologies Research & Development Program of China (No. 2017YFC0805100) and the National Natural Science Foundation of China (Nos. 51975004, 51505002). The first author also thanks the Education Department of Anhui Province of China for supporting the academic visit to the UNSW, Sydney. References [1] R. Yan, R.X. Gao, Approximate Entropy as a diagnostic tool for machine health monitoring, Mech. Syst. Signal Process. 21 (2) (2007) 824–839. [2] S.M. Pincus, Approximate entropy as a measure of system complexity, Proc. Natl. Acad. Sci. USA 88 (6) (1991) 2297. [3] J.S. Richman, J.R. Moorman, Physiological time-series analysis using approximate entropy and sample entropy, Amer. J. Physiol. Heart Circ. Physiol. 278 (6) (2000) H2039. [4] C. Bandt, B. Pompe, Permutation entropy: a natural complexity measure for time series, Phys. Rev. Lett. 88 (17) (2002) 174102. [5] M. Costa, A.L. Goldberger, C.K. Peng, Multiscale entropy analysis of biological signals, Phys. Rev. E 71 (2Pt 1) (2005) 021906. [6] S.D. Wu, C.W. Wu, P.H. Wu, et al., Bearing fault diagnosis based on multiscale permutation entropy and support vector machine, in: Third International Conference on Mechanic Automation and Control Engineering, IEEE Computer Society, 2012, pp. 2650–2654. [7] H.B. Xie, B. Sivakumar, T.W. Boonstra, et al., Fuzzy entropy and its application for enhanced subspace filtering, IEEE Trans. Fuzzy Syst. 99 (2017) 1970–1982. [8] R. Yan, Y. Liu, R.X. Gao, Permutation entropy: A nonlinear statistical measure for status characterization of rotary machines, Mech. Syst. Signal Process. 29 (5) (2012) 474–484. [9] L. Zhang, G. Xiong, H. Liu, et al., An intelligent fault diagnosis method based on multiscale entropy and SVMs, in: Advances in Neural Networks – ISNN 2009, Springer Berlin Heidelberg, 2009, pp. 724–732. [10] Y. Li, M. Xu, Y. Wei, et al., A new rolling bearing fault diagnosis method based on multiscale permutation entropy and improved support vector machine based binary tree, Measurement 77 (2016) 80–94. [11] J. Zheng, J. Cheng, Y. Yang, et al., A rolling bearing fault diagnosis method based on multi-scale fuzzy entropy and variable predictive model-based class discrimination, Mech. Mach. Theory 78 (16) (2014) 187–200. [12] Y. Li, M. Xu, R. Wang, et al., A fault diagnosis scheme for rolling bearing based on local mean decomposition and improved multiscale fuzzy entropy, J. Sound Vib. 360 (2016) 277–299. [13] A. Humeau-Heurtier, The multiscale entropy algorithm and its variants: A review, Entropy 17 (2015) 3110–3123. [14] M. Rostaghi, H. Azami, Dispersion entropy: a measure for time-series analysis, IEEE Signal Process. Lett. 23 (5) (2016) 610–614. [15] H. Azami, J. Escudero, Coarse-graining approaches in univariate multiscale sample and dispersion entropy, Entropy 20 (2) (2018) 138. [16] H. Azami, M. Rostaghi, D. Abasolo, et al., Refined composite multiscale dispersion entropy and its application to biomedical signals, IEEE Trans. Biomed. Eng. 99 (2017) 1–8. [17] T.D. Pham, Time-shift multiscale entropy analysis of physiological signals, Entropy 19 (6) (2017) 257. [18] Q. Xiong, W. Zhang, T. Lu, et al., A fault diagnosis method for rolling bearings based on feature fusion of multifractal detrended fluctuation analysis and alpha stable distribution, Shock Vib. 2016 (3) (2015) 1–12. [19] K. Zhu, X. Song, D. Xue, A roller bearing fault diagnosis method based on hierarchical entropy and support vector machine with particle swarm optimization algorithm, Measurement 47 (1) (2014) 669–675. [20] Z. Liu, H. Cao, X. Chen, et al., Multi-fault classification based on wavelet SVM with PSO algorithm to analyze vibration signals from rolling element bearings, Neurocomputing 99 (1) (2013) 399–410. [21] Bearing data center, case western reserve university. http://csegroups.case.edu/bearingdatacenter/pages/download-data-file. [22] W.A. Smith, R.B. Randall, Rolling element bearing diagnostics using the Case Western Reserve University data: A benchmark study, Mech. Syst. Signal Process. (64–65) (2015) 100–131.

Please cite this article as: J. Zheng, H. Pan, Q. Liu et al., Refined time-shift multiscale normalised dispersion entropy and its application to fault diagnosis of rolling bearing, Physica A (2019) 123641, https://doi.org/10.1016/j.physa.2019.123641.