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Time-Shift Multiscale Fuzzy Entropy and Laplacian Support Vector Machine Based Rolling Bearing Fault Diagnosis Xiaolong Zhu 1,2 , Jinde Zheng 1,2, *, Haiyang Pan 1 , Jiahan Bao 1,2 and Yifang Zhang 1,2 1

2

*

School of Mechanical Engineering, Anhui University of Technology, Maanshan 243032, China; [email protected] (X.Z.); [email protected] (H.P.); [email protected] (J.B.); [email protected] (Y.Z.) Engineering Research Center of Hydraulic Vibration and Control, Ministry of Education, Maanshan 243032, China Correspondence: [email protected]

Received: 24 July 2018; Accepted: 9 August 2018; Published: 13 August 2018







Abstract: Multiscale entropy (MSE), as a complexity measurement method of time series, has been widely used to extract the fault information hidden in machinery vibration signals. However, the insufficient coarse graining in MSE will result in fault pattern information missing and the sample entropy used in MSE at larger factors will fluctuate heavily. Combining fractal theory and fuzzy entropy, the time shift multiscale fuzzy entropy (TSMFE) is put forward and applied to the complexity analysis of time series for enhancing the performance of MSE. Then TSMFE is used to extract the nonlinear fault features from vibration signals of rolling bearing. By combining TSMFE with the Laplacian support vector machine (LapSVM), which only needs very few marked samples for classification training, a new intelligent fault diagnosis method for rolling bearing is proposed. Also the proposed method is applied to the experiment data analysis of rolling bearing by comparing with the existing methods and the analysis results show that the proposed fault diagnosis method can effectively identify different states of rolling bearing and get the highest recognition rate among the existing methods. Keywords: multiscale entropy; fuzzy entropy; time-shift multiscale fuzzy entropy; rolling bearing; fault diagnosis; Laplacian support vector machine

1. Introduction Rolling bearing is a typical part for realizing the rotational motion and also prone to broken in the rotating machinery due to the complexity of working conditions, which will result in the machinery operation unstable or even unexpected accidents. Hence, it is of significance to timely recognize the faults for avoiding the loss of machinery life and factory finance. Generally, when the rolling bearing works with local faults, the non-linearity and non-stationarity are obviously expressed in vibration signals, which makes the traditional linear analysis methods loss their functions in analyzing these kinds of signals [1–3]. At present, many nonlinear dynamic analysis methods, such as approximate entropy (ApEn) [4,5], sample entropy (SampEn) [6,7], fuzzy entropy (FuzzyEn) [8,9] and multiscale entropy (MSE) [10,11] have been proposed and applied to the complexity measurement of the time series from different domains. Compared with traditional linear analysis methods, the nonlinear dynamic methods can effectively extract the nonlinear fault features from vibration signals of rolling bearings and promote the performance on fault identification and detection of rolling bearing [12]. For example, based on lifting wavelet packet transform and sample entropy as the method for extracting feature as well Entropy 2018, 20, 602; doi:10.3390/e20080602

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as binary tree system based classifier ensemble as the classification method, a new fault diagnosis method for bearing was proposed to automatically identify different fault types and severity levels in [13] and the experimental data analysis validated the more accuracy and stability performance of the proposed method on the fault recognition for rolling bearing. By combining empirical mode decomposition and sample entropy, a fault diagnosis method for rolling bearings was put forward by Zhong et al. in [14] and the results indicated that the feature information of intrinsic mode functions (IMFs) can be effectively extracted by calculating sample entropy and applied to fault diagnosis of rolling bearing. Chen et al. developed FuzzyEn as an enhanced algorithm of SampEn and ApEn and the analysis on electromyography (EMG) experimental data indicated that compared with SampEn and ApEn, FuzzyEn has more stronger relation consistence and little dependence on data length by using fuzzy function to measure the similarity of two vectors [15]. However, the entropy under unitary scale calculated from the original series is often insufficient for indicating the complexity of time series. To overcome this, MSE was put forward and developed in [16] by combing the coarse-grained method of time series. In [17], the MSE and adaptive neuro-fuzzy inference based fault diagnosis method for rolling bearing was proposed and the researches indicated that comparing the single scale-based entropy, MSE can be utilized to effectively extract the nonlinear, interaction and coupling characteristic contained in the mechanical signals and the proposed method could get a better fault recognition performance on bearing incipient fault diagnosis. Also, the anomaly detection method based on MSE and principal component analysis was proposed [18] and the simulation results suggested that MSE can be used to detect gear tooth pitting. However, the MSE method still has some intrinsic drawbacks. Firstly, the coarse-grained method in nature is a linear filter and the insufficient coarse-grained procedure will result in missing of pattern information. Secondly, with the increasing of time scale factors, the SampEns at larger factors will fluctuate heavily and even have no definitions. By combing the FuzzyEn proposed by Chen et al. [19] and inspired by the idea of time shift way [20], a new complexity measuring method of time series called time shift multiscale fuzzy entropy (TSMFE) is proposed in this paper to overcome the above limitations of MSE. Then TSMFE is compared with MSE and time shift sample entropy (TSME) by simulation signal analysis and is applied to the complexity feature extraction of vibration signals from faulty rolling bearings. After obtaining the fault features by using TSMFE, it is necessary to select an intelligent classifier to automatically identify the fault types and degrees. Support vector machines (SVM) have been widely used in various pattern classification researches, such as face and figure recognition, fault classification and so on [21–23]. However, SVM requires that the class labels of all training samples should be known. In the real world, lots of fault samples are easily collected but hardly marked. Combining the manifold learning and SVM, a semi-supervised learning method called Laplacian support vector machine (LapSVM) was developed [24,25]. Compared with SVM, LapSVM requires less marked samples for learning [26]. A new fault diagnosis method for rolling bearings was proposed by combining TSMFE with LapSVM and then was compared with the existing methods by analyzing the experiment data of rolling bearing and the results verified its effectiveness. The contents of this paper are constructed as follows. In Section 2, MSE and TSME algorithms are reviewed and then TSMFE is proposed for complexity measure of time series. The comparison analysis for 1/f noise and Gaussian white noise is given in Section 3. In Section 4, the new fault diagnosis method for rolling bearing is proposed by combining TSMFE with LapSVM and applied to the experimental data analysis of rolling bearing. Finally, the conclusions are given in Section 5. 2. Time Shift Multiscale Fuzzy Entropy and Related Theories 2.1. Multiscale Entropy Method The MSE method can be calculated as follows [27–29].

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For a determined number τmax , the initial time series X : x1 , x2 , · · · , x N can be reconstructed as jτ

y j (τ ) =

1 xi , (1 ≤ j ≤ N/τ, τ = 1, 2, · · · , τmax ), τ i=( j−∑ 1) τ +1

(1)

where τ is positive integer termed scale factor. The reconstructed time series is the original time series when τ = 1 in particular and in case τ ≥ 2, the reconstructed time series y j (τ ) is called coarse-grained time series with length no more than N/τ. For each scale factor τ, the SampEn of y j (τ ) is respectively calculated and MSE is expressed as  B m +1 ( r ) MSE(m, r, N, τ ) = SampEn (m, r, N ) = − ln m , B (r ) τ =1,2,··· ,τmax 

(2)

Actually, the coarse-grained process in MSE is regarded as a linear filter. The insufficient coarse-grained procedure will lead to the loss of pattern information hidden in time series. To surmount this problem, based on Higuchi’s fractal dimension (HFD) theory, time shift multiscale sample entropy was proposed by Tuan [20,30] by redefining the coarse-grained procedure and was applied to process physiological signals. 2.2. Time Shift Multiscale Sample Entropy Based on HFD theory, time shift multiscale sample entropy [20] is described as follows. (1) For the given time series X = { x1 , x2 , · · · , x N } with length N, the new time series can be obtained by β Yk = ( x β , x β+k , x β+2k , · · · , x β+∆(β,k) k ), (3) where k and β( β = 1, 2, · · · , k) are positive integer, standing for the initial time point and interval time, and in particular, ∆( β,k) is a rounding integer for ( N − β)/k, and represents the upper boundary. (2) For each τ, the average of the SampEns of all time shift time series under 1 ≤ β ≤ k is β expressed as TSMEk under single scale (namely k = τ), and for each k (1 ≤ k ≤ τ) the TSMEk with multiple scale can be described as TSME = 1≤ k ≤ τ k

1 k β TSMEk . k β∑ =1

(4)

TSME has overcame the drawbacks of MSE that the insufficient coarse graining will lead to the loss of patter information and however, some shortcomings remain to exist. In sample entropy used in MSE and TSME, the function for measuring similarity of two vectors cannot effectively recognize the two vectors with fuzzy boundaries in real world and the entropies will fluctuate heavily and even more have no definition with the increase of scale factor. To overcome this, TSMFE is proposed by combing TSME with FuzzyEn, by which the similarity of two vectors can be measured more robust than SampEn used in TSME and MSE. 2.3. Time Shift Multiscale Fuzzy Entropy According to TSME and FuzzyEn [19], the detailed procedures of the proposed TSMFE method can be described as follows. (1) For the given time series X = { x1 , x2 , · · · , x N }, it can be generated by using Equation (3). Particularly, when time interval k = 3, the new time series is illustrated in Figure 1.

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(2) For For aa determined determined time  ,the theaverage averageof ofFuzzyEns FuzzyEns of of all all time-shift time-shift time time series series under under (2) time scale scale τ,  β  β ≤k k is (namelyk k=τ),  ),and andfor foreach eachkk(1 ( 1≤  kk ≤ τ) ) the with 11 ≤ is expressed expressed as as TSMFE TSMFEkk (namely theTSMFE TSMFEkk with multiple multiple scale scale can can be be described described as as k

1 1k β TSMFE TSMFE=  ∑ TSMFE TSMFE k .k . 1≤1k≤kτ k k k k β =1



(5)

 1

TSMFE is is aa nonlinear nonlinear dynamic dynamic method method for for complexity complexity measurement measurement ofof time timeseries. series. The The comparison will will be be made made in in the the following following sections sections to to verify verify the the superiority superiorityof ofTSMFE. TSMFE. comparison

Figure 1. 1. The procedure for for reconstructing reconstructing the the time time series. series. Figure The procedure

3. Comparison of TSME and TSMFE 3. Comparison of TSME and TSMFE 3.1. Parameter Parameter Selection Selection 3.1. As shown shown in inEquation Equation(5), (5),the thecalculation calculationofofTSMFE TSMFEis is relative with embedding dimension As relative with thethe embedding dimension m, m, similarity tolerance r, gradient parameter n and time series length Firstly, with increase of similarity tolerance r, gradient parameter n and time series length N. N. Firstly, with thethe increase of m, m, the more feature information will be extracted from the reconstructed time series, however, the more feature information will be extracted from the reconstructed time series, however, a mucha much longer timeisseries is (generally needed (generally generally, setSecondly, m as 2. N 3010mm). ~Hence, 30m ). Hence, longer time series needed N = 10m ∼ generally, we set mwe as 2. Secondly, the tolerance similarityr tolerance stands for thewidth boundary width of fuzzy and ra will too the similarity stands forrthe boundary of fuzzy function and function a too smaller smaller will result in tooinformation much useless information entropy will be sensitive to result in rtoo much useless counted and thecounted entropyand willthe be sensitive to noise, while too noise, rwhile too larger r will the loss of much statistical characteristics. Thus from generally larger will lead to the loss of lead muchtostatistical characteristics. Thus generally r ranges 0.1SDr ranges from 0.1SD 0.25SD (SD is the standard deviation ofseries) the original time series) 0.15SD is to 0.25SD (SD is thetostandard deviation of the original time and 0.15SD is set and in this paper. set in this paper.parameter Thirdly, gradient parameter n controls similarity of twon tends vectors when Thirdly, gradient n controls the similarity of twothe vectors and when toand infinite, then tends to infinite, the exponential function becomes themakes unit much step function makes much exponential function becomes the unit step function which statistical which information missing. statistical information Hence, a smaller integer andboth in this n is set of as Hence, a smaller integermissing. is suggested [19] and in this paperisnsuggested is set as 2.[19] Lastly, thepaper computation 2. Lastly,and both the computation of SampEn and FuzzyEn scales are relative data SampEn FuzzyEn over different scales are relative withover data different length and the literature [17]with suggests length and the with literature [17] FuzzyEn suggests need that compared withlength SampEn, need a shorter that compared SampEn, a shorter data and FuzzyEn thus in this paper N is setdata no length and thus in this paper N is set no less than 2000. less than 2000. 3.2. Simulation Analysis To the influence influenceof ofdata datalength lengthononTSMFE, TSMFE, Gaussian white noise 1/f noise To illustrate the thethe Gaussian white noise andand 1/f noise with with length 1500, 25,000 respectivelyused usedasasexamples exampleswith with comparison comparison with data data length 1000,1000, 1500, 20002000 andand 25,000 areare respectively TSME, and TSMFEs TSMFEsof ofthe theGaussian Gaussianwhite whitenoises noisesand and1/f1/f noises with different length TSME, i.e., TSMEs and noises with different length are are calculated the results are shown in Figure 2 where 2, r = 0.15SD, time internal k = 20 calculated andand the results are shown in Figure 2 where m = 2,mr == 0.15SD, time internal k = 20 (that is (that = 20). as Firstly, as illustrated in 2, Figure 2, both TSMEs and TSMFEs of 1/f noise gradually illustrated in Figure both TSMEs and TSMFEs of 1/f noise gradually increase   20is).τ Firstly, increase the increase scale factor, TSMEs and TSMFEs of white noise almost remain with thewith increase of scaleoffactor, while while TSMEs and TSMFEs of white noise almost remain to to bebea aconstant constantvalue valuewhich whichgenerally generallyisislarger largerthan thanthat thatofof1/f 1/f noise most scales. This indicates that it noise at at most scales. This indicates that it is is importance conductmultiscale multiscaleanalysis analysisfor fortime timeseries. series.Secondly, Secondly,for for the the time time series under a ofof importance totoconduct larger have nono definition. ByBy contrast, TSMFEs of larger time timescale, scale,i.e., i.e.,with withaashorter shorterdata datalength, length,TSME TSMEwill will have definition. contrast, TSMFEs 1/f noise andand white noises varyvary slightly withwith the increase of data and the change of dataof length of 1/f noise white noises slightly the increase of length data length and the change data cannot lead to lead no definition of all time scales, that compared with TSME, data length length cannot to no definition of all time which scales, indicates which indicates that compared with TSME, data has little influence on TSMFE. The above indicates that both TSMFE and TSME reflect length has little influence on TSMFE. The analysis above analysis indicates that both TSMFE and can TSME can reflect the pattern information and complexity of the two kinds of random signals. By observing the

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the pattern information and complexity of the two kinds of random signals. By observing the TSME and TSMFE curves of Gaussian noise andnoise 1/f noise, can we findcan that TSMFE is moreis stable Entropy 2018, 20, x FOR PEER REVIEW 5 of 16 than TSME and TSMFE curves of white Gaussian white and 1/fwe noise, find that TSMFE more TMSE especially in analyzing time series.time series. stable than TMSE especially ainshorter analyzing a shorter TSME and TSMFE curves of Gaussian white noise and 1/f noise, we can find that TSMFE is more stable than TMSE especially inN=1000 analyzing N=1500 a shorter time series. N=2500 N=2000 N=3000

2.0 2.5

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5 10 15 20 10 15 20 Figure 2. TSMFE and TSME of Gaussian white noises 0and 1/f 5noises under different lengths: (a)

 Figure 2. TSMFE and TSME of Gaussian white noises and 1/f noises under  different lengths: (a) TSMFE TSMFE of 1/f noise; (b) TSME of 1/f noise; (c) TSMFE of Gaussian white noise and (d) TSME of of 1/fFigure noise;2. (b) TSME of 1/f noise; (c) TSMFE of Gaussian white noise anddifferent (d) TSME of Gaussian TSMFE and TSME of Gaussian white noises and 1/f noises under lengths: (a) Gaussian white noise. whiteTSMFE noise. of 1/f noise; (b) TSME of 1/f noise; (c) TSMFE of Gaussian white noise and (d) TSME of

Gaussian white the noise. To investigate influence of similarity tolerance on TSMFE and TSME, the TSME and TSMFE To investigate the influence of similarity tolerance on TSMFE and TSME, the TSME and TSMFE of of Gaussian white noises and 1/f noises with N = 2048 are calculated under different similarity Gaussian white and 1/f noises with N =and 20480.25SD are calculated under similarity tolerances To investigate the 0.1SD, influence of similarity tolerance on(where TSMFEmand TSME TSMFE tolerances r noises = 0.05SD, 0.15SD, 0.2SD = 2,TSME, ) and theand results are  different  20the of Gaussian white noises and 1/f noises with N = 2048 are calculated under different similarity r = 0.05SD, 0.1SD, 0.15SD, 0.2SD and 0.25SD (where m = 2, τ = 20) and the results are in shown in Figure 3. From the Figure 3, it can be found that for the same similarity tolerance r, withshown the tolerances r = 0.05SD, 0.1SD, 0.15SD, 0.2SD and 0.25SD (where m = 2, ) and the results are   20 Figure 3. From the Figure 3, it can be found that for the same similarity tolerance r, with the increase increase of scale factor, the TSMFEs of 1/f noise and white noise vary slightly and even tend to a shown in while Figure 3. Fromofthe 3, it can found that for the same similarity tolerance r,to with the constant, TSMEs theFigure twonoise kinds of be noises fluctuate obviously. Besides, with the increase of of scale factor, the TSMFEs of 1/f and white noise vary slightly and even tend a constant, increase of scale factor, the TSMFEs of 1/f noise and white noise vary slightly and even tend to a r, the TSMFE and TSME values under the Besides, same timewith scales decrease gradually. whilesimilarity TSMEs tolerance of the two kinds of noises fluctuate obviously. the increase of similarity constant, while TSMEs of the two kinds of noises fluctuate obviously. Besides, with the increase of However, a smaller tolerance, TSMEs have no definition for a larger scale factor while for tolerance r, thefor TSMFE andsimilarity TSME values under the same time scales decrease gradually. However, similarityare tolerance r, the TSMFESince and TSME the same timeinformation scales decrease gradually. TSMFEs always meaningful. a largervalues r will under cause much statistic missing while a smaller similarity tolerance, TSMEs have no definition a largerfor scale factor while TSMFEs a smaller tolerance, TSMEs haveinformation, nofor definition a larger while are aHowever, smaller rforwill result similarity in more unexpected statistical generally, wescale set factor r = 0.15SD. always meaningful. Since a larger r will cause much statistic information missing while a smaller TSMFEs are Sincethat a larger r will cause much statistic whiler will Therefore, thealways abovemeaningful. analysis indicate the TSMFE curves of two kinds information of noises aremissing more smooth result moresmaller statistical information, generally, we set r = 0.15SD. the above a in smaller runexpected will result in more unexpected information, generally, weTherefore, set r = 0.15SD. and have standard deviation than thatstatistical of TSME at different similar tolerance. analysis indicate TSMFE curves ofthe two kindscurves of noises more smooth smaller Therefore, the that abovethe analysis indicate that TSMFE of twoare kinds of noises areand morehave smooth and have smallerthan standard deviation than that of TSME at different similar tolerance. standard deviation that of TSME at different similar tolerance. r=0.05 r=0.1 r=0.15 r=0.2 r=0.25 5.0

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Figure 3. TSMFE and TSME ofGaussian white noise and 1/f noise under different similar tolerance:  (a) TSMFE of 1/f noise; (b) TSME of 1/f noise; (c) TSMFE of Gaussian white noise and (d) TSME of Figure 3. white TSMFE and TSME Gaussianwhite white noise noise and under different similar tolerance: Figure 3. TSMFE and TSME of of Gaussian and1/f 1/fnoise noise under different similar tolerance: Gaussian noise. (a) TSMFE of 1/f noise; TSMEofof1/f 1/f noise; Gaussian white noise and and (d) TSME of (a) TSMFE of 1/f noise; (b)(b) TSME noise;(c) (c)TSMFE TSMFEofof Gaussian white noise (d) TSME of Gaussian white noise. Gaussian white noise.

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4. TSMFE and LapSVM Based Fault Diagnosis Method for Rolling Bearing 4.1. LapSVM Algorithm A better classifier is needed to fulfill an intelligent identification and classification. SVM, as a supervised learning method, has been used in many areas [31–33]. However, in SVM too many samples called training data should be labeled, which is difficult to obtain for the real world data. To overcome this drawback of SVM, Combining the manifold regularization, Laplacian support vector machine (LapSVM), a semi-supervised learning, has been proposed and wildly applied on feature recognition [34–36]. In LapSVM, the recognition performance can be promoted by manifold regularization and thus much unmarked data are used to estimate the internal data manifold structure that will be applied to design the classifier. When a set of marked samples ( xi , yiE ) (i = 1, 2, · · · , l ) and kernel function K are determined, and the hinge loss function is selected as loss function, the manifold regularization framework can be expressed as 1 l γI f ∗ = argmin ∑ (1 − yiE f ( xi )) + γ A k f k2k + F T LF, (6) l i −1 ( u + l )2 where F = [ f ( x1 ), f ( x2 ), · · · , f ( xl +u )] T , l and u respectively stand for the number of given marked samples ( xi , yiE ) and unmarked samples x j , ( j = l + 1, l + 2, · · · , l + u), manifold regularization item

is denoted as k f k2I , both γ I and γ A are manifold regularization parameter, L represents Laplacian matrix, hinge loss function V ( xi , yiE , f ) = (1 − yiE f ( xi ))+ = max(0, 1 − yiE f ( xi )) is used to measure the deviation of the desired outputs yiE ∈ {−1, 1} and real outputs f ( xi ) of training samples xi . Combining express theorem, the Equation (6) is solved as f∗ =

l +u

∑ ai∗ K(x, xi ),

(7)

i =1

when α∗ = [α1 , α2 , · · · , αl +u ] T , α∗ = [2γ A I + 2

γ1

(u + l )

2

LK ]

−1

J T Yβ∗ ,

(8)

in which I(l +u)(l +u) and L(l +u)(l +u) are respectively unit matrix and Laplacian matrix, nuclear matrix is denoted as K(l +u)(l +u) , Y = diag(y1E , y2E , · · · , ylE ) is used to balance the complexity of the experience loss and function, by quadratic planning β∗ is expressed as l 1 β∗ = max ∑ β i − β T Qβ, 2 β ∈ R l i =1

(9)

l

when ∑ β i yiE = 0(0 ≤ β i ≤ 1/l, i = 1, · · · , l ), i =1

Q = Y JK (2γ A I + 2

γ1

( I + u)

2

LK )

−1

JTY

(10)

Based on the above analysis, the unmarked samples used in LapSVM can be employed to estimate the internal manifold structure used to the aided learning of marked samples, which obviously promotes the recognition performance, while in SVM all the samples should be labeled. Considering the difficulty of sample to be labeled in the real world, LapSVM is applied in this paper.

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4.2. The Proposed Fault Diagnosis Method

The above analysis indicate that TSMFE, as a new complexity and irregularity measuring 4.2. The Proposed Fault Diagnosis Method method of time series, can get much better performance than TSME. Meanwhile, the data length and similar tolerance also have less influence than TSME the entropies, obtained by The above analysis indicate that TSMFE,on as TSMFE a new complexity andand irregularity measuring method TSMFE, are also more stable. Hence, TSMFEthan is employed to the fault feature extraction and of time series, can get much better performance TSME. Meanwhile, the data length and similar diagnosis for rolling bearing. tolerance also have less influence on TSMFE than TSME and the entropies, obtained by TSMFE, Themore proposed fault diagnosis method for rolling described as follows: are also stable. Hence, TSMFE is employed to bearing the faultisfeature extraction and diagnosis for rolling (1) Forbearing. given p kinds of states of rolling bearing, each state has mp samples and thus the number of The proposed fault diagnosis method for rolling bearing is described as follows: p whole samples is M = (  m i ) ; i  1 of rolling bearing, each state has m samples and thus the number of (1) For given p kinds of states p p

whole samples is M ( ∑ mi )are ; calculated and the feature sets (2) TSMFE of all the M=samples p

i =1

(Tp ,p) T p  Rmp  max ,

are

thecalculated p-th feature sets; (2) obtained, TSMFE ofwhere all theTM represents samples are and the feature sets p ), ∈ R are p (3) The m p samples of the p-th state are randomly divided into h as marked sample sets, i.e., obtained, where T represents the p-th feature sets; p ( m  h ) max (3) T The m hsamples of the p-th state are randomly divided T2p into R p h asmaxmarked sample sets, i.e., 1  Rp and (m p - h) as unmarked sample sets, i.e., ; p p T1 ∈ Rh×τmax and (mp − h) as unmarked sample sets, i.e., T2 ∈ R(m p −h)×τmax ; ( m  h ) h max p Tp  Rh×τ and T T2pp  R (pm p −hmax)×τmax (4) The sensitive sensitive fault fault features features sets (4) The sets of of training training samples: samples: both both T11 ∈ R max and are 2 ∈ R are input to LapSVM the LapSVM classifier for training, learning testing. input to the classifier for training, learning andand testing.

(T p ,

Tp

m p ×τmax

The procedure procedure of of the the proposed proposed method method can can be be described described as asfollows follows(Figure (Figure4). 4).

Figure 4. The fault diagnosis diagnosis procedure procedure for for rolling rolling bearing. bearing. Figure 4. The proposed proposed fault

4.3. Experimental Data 4.3. Experimental Data Analysis Analysis In this byby Case Western Reserve University [37][37] are this subsection, subsection,the theexperimental experimentaldata datasupported supported Case Western Reserve University employed to verify the effectiveness of the proposed method. As shown in Figure 5, the experiment are employed to verify the effectiveness of the proposed method. As shown in Figure 5, the stand is composed fan end bearing, end bearing, transducer andtransducer dynamometer. experiment stand isofcomposed of fan drive end bearing, drive torque end bearing, torque and The type of tested is 6205-2RS JEM SKF (SKF,JEM Göteborg, Sweden), in which single dynamometer. Therolling type ofbearing tested rolling bearing is 6205-2RS SKF (SKF, Göteborg, Sweden), in point faults have been machined by electro-discharge. Particularly, the fault diameters are respectively which single point faults have been machined by electro-discharge. Particularly, the fault diameters 0.1778 mm, 0.3556 mm and mm. depth There 0.2794 are fourmm. states for rolling bearing, are respectively 0.1778 mm,0.5334 0.3556mm mmwith anddepth 0.53340.2794 mm with There are four states i.e., Normal (Norm), Ball element default (BE), Inner race default (IR) and Outer race default (OR). for rolling bearing, i.e., Normal (Norm), Ball element default (BE), Inner race default (IR) and Outer The vibration acceleration signals of rolling bearing under four states have been respectively collected race default (OR). The vibration acceleration signals of rolling bearing under four states have been when the motor speeds are 1797 r/min, 1772 r/min, 1750 r/min, and 1730 r/min. The sensors are

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respectively collected when the motor speeds are 1797 r/min, 1772 r/min, 1750 r/min, and 1730 r/min. respectively collected when the motor speeds are 1797 r/min, 1772 r/min, 1750 r/min, and 1730 r/min. The sensors are fixed at fan end with sampling frequency 12 kHz and drive end with sampling The sensors are with fixedsampling at fan end with sampling frequency 12 kHz and drive end with sampling fixed at fan end frequency 12 kHz and is drive with sampling 12 and frequency 12 and 48 kHz. In this paper, the motor speed 1797end r/min without load,frequency the acceleration frequency 12 and 48 kHz. In this paper, the motor speed is 1797 r/min without load, the acceleration 48 kHz. In this paper, the motor speed is 1797 r/min without load, the acceleration signals of rolling signals of rolling bearings on normal state and three fault states under two fault diameters: 0.1778 signals of rolling bearings on normal state and three fault states under two fault diameters: 0.1778 bearings on normal three fault two fault diameters: 0.1778 12 mmkHz. and 0.5334 mm mm and 0.5334 mmstate are and collected fromstates driveunder end with sampling frequency They are mm and 0.5334 mm are collected from drive end with sampling frequency 12 kHz. They are are collected from drive end with sampling frequency 12 kHz. They are successively denoted as: successively denoted as: (a)normal state (Norm), (b) ball element fault with fault diameters: 0.1778 successively denoted as: (a)normal state (Norm), (b) ball element fault with fault diameters: 0.1778 (a)normal state (Norm), (b) ball element fault with fault diameters: 0.1778 mm(BEI), (c) ball element mm(BEI), (c) ball element fault with fault diameters: 0.5334 mm (BEII), (d) inner ring fault with fault mm(BEI), (c) ball element fault with fault diameters: 0.5334 mmwith (BEII), (d)diameters: inner ring fault with fault fault with 0.1778 fault diameters: mmring (BEII), (d)with inner ringdiameters: fault fault mmring (IRI), diameters: mm (IRI),0.5334 (e) inner fault fault 0.5334 mm (IRII),0.1778 (e) outer diameters: 0.1778 mm (IRI), (e) inner ring fault with fault diameters: 0.5334 mm (IRII), (e) outer ring (e) inner with fault diameters: 0.5334 mm (IRII), outer ring fault with fault0.5334 diameters: fault with ring fault fault diameters: 0.1778 mm (ORI) and (f) outer ring(e)fault with fault diameters: mm fault with fault diameters: 0.1778 mm (ORI) and (f) outer ring fault with fault diameters: 0.5334 mm 0.1778 All mmthe (ORI) and (f) outer ring fault with fault diameters: 0.5334 (ORII). All the time-domain (ORII). time-domain vibration signals of the seven states of mm rolling bearings are shown in (ORII). All the time-domain signalsbearings of the seven statesinofFigure rolling6 and bearings shown in vibration signals thedifficulty seven vibration states rolling are the shown it is ofare difficulty Figure 6 and it isof of to of entirely distinguish states of rolling bearing from theto Figure 6 and it is of difficulty to entirely distinguish the states of rolling bearing from the entirely distinguish time-domain signals. the states of rolling bearing from the time-domain signals. time-domain signals.

Figure 5. The test stand of CWRU. Figure Figure 5. 5. The test stand of CWRU.

-0.5 0 -0.5

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Acceleration Acceleration a/(ms-2) a/(ms-2)

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Figure 6. Time-domain vibration signals of seven states of rolling bearing. Figure 6. 6. Time-domain vibration signals signals of of seven seven states states of of rolling rolling bearing. bearing. Figure Time-domain vibration

Next, 50 samples with length 2048 of each state of rolling bearing were used to test the Next, 50 50samples samples with length of state eachofstate ofbearing rollingwere bearing to test the Next, with length 20482048 of each rolling usedwere to testused the performance performance of TSMFE in vibration signal analysis of rolling bearing. TSMFEs and TSMEs of all the performance of TSMFE in vibration analysis of rolling bearing. TSMFEs all the of TSMFE in vibration signal analysis signal of rolling bearing. TSMFEs and TSMEs of alland the TSMEs selectedof samples selected samples were respectively calculated and exhibited in Figure 7. It can be seen that with the selected samples were respectively calculated and exhibited in Figure 7. It can be seen that with the were respectively calculated and exhibited in Figure 7. It can be seen that with the increase of scale increase of scale factor, TSMEs of vibration signals of rolling bearing under several categories have increase of scale factor, TSMEs of vibration signals of rolling bearing under several categories have

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no definition at vibration the largersignals scale factor and bearing this indicates TSMFE is more stable and effective to factor, TSMEs of of rolling under that several categories have no definition at the represent the pattern features than TSME. Besides, at the same time scale, the TSMFEs of vibration larger scale factor and this indicates that TSMFE is more stable and effective to represent the pattern signals of rolling bearing fluctuate slighter of TSMEofand it is beneficial promote the features than TSME. Besides, at the same time than scale,that the TSMFEs vibration signals of to rolling bearing identification performance. Finally, as shown in Figure 7, TSMFEs and TSMEs of vibration signals of fluctuate slighter than that of TSME and it is beneficial to promote the identification performance. rolling bearing with ball element fault are generally larger than that of rolling bearing with inner Finally, as shown in Figure 7, TSMFEs and TSMEs of vibration signals of rolling bearing with ball ring fault, which are largerlarger than than that of rolling bearing withwith outer ringring faults at the most scale element fault are generally that of rolling bearing inner fault, which areoflarger factors. The above analysis indicates that compared with TSME of vibration signals of rolling than that of rolling bearing with outer ring faults at the most of scale factors. The above analysis bearing, TSMFE can represent the fault feature more effectively and get much better performance on indicates that compared with TSME of vibration signals of rolling bearing, TSMFE can represent the the fault diagnosis of rolling bearing. fault feature more effectively and get much better performance on the fault diagnosis of rolling bearing. BEI

IRI

IRII

ORI

3

(a)

2.4

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TSME

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

Figure7.7.Mean Meanand and standard deviation of TSMFE and TSME: (a) and mean and deviation square deviation of Figure standard deviation of TSMFE and TSME: (a) mean square of TSMFE; TSMFE; b) mean and square deviation (b) mean (and square deviation of TSME. of TSME.

Next, Next, 50 50 samples samples of of each each states states are arerandomly randomly divided divided into into 10 10marked marked training training sample sample feature feature sets and 40 unmarked testing ones. The sensitive fault features subset of samples are correspondingly sets and 40 unmarked testing ones. The sensitive fault features subset of samples are input to the LapSVM classifier, where classifier, the radialwhere basis function selected as theis correspondingly inputbased to themulti-fault LapSVM based multi-fault the radialisbasis function kernel function of LapSVM, andofthe parameter theparameter kernel function set tofunction 0.35. Theisbinary tree selected as the kernel function LapSVM, andofthe of the is kernel set to 0.35. theory is adopted to construct the multi-fault classifier. The detailed results are shown in Table 1 The binary tree theory is adopted to construct the multi-fault classifier. The detailed results are and Figure 8. As shown in Table 1 and Figure LapSVM1, LapSVM2, shown in Table 1 and Figure 8. As shown in Table8,1 in and Figure 8, in LapSVM1,LapSVM3, LapSVM2,LapSVM4, LapSVM3, LapSVM5 LapSVM6, all samples accurately identifiedidentified and the total identification rate is LapSVM4,and LapSVM5 and LapSVM6, allare samples are accurately and the total identification 100%, that thethat proposed method for fault of rolling bearing has a has wella rate iswhich 100%, validates which validates the proposed method fordiagnosis fault diagnosis of rolling bearing recognition performance. well recognition performance. Table Table1.1.Output Outputresults resultsof ofLapSVM LapSVMclassifier. classifier. Sample LapSVM1 LapSVM2 LapSVM3LapSVM4 LapSVM4 LapSVM5 LapSVM5 LapSVM6 Sample SetsSetsFaults Faults LapSVM1 LapSVM2 LapSVM3 LapSVM6 T1~T50 T1~T50 Norm T51~T100 T51~T100 BEI T101~T150 T101~T150 BEII T151~T200 T151~T200 T201~T250 IRI T201~T250 T251~T300 IRII T251~T300 T301~T350 ORI T301~T350 ORII

Norm +1(50) BEI −1(50) BEII IRI −1(50) IRII−1(50) ORI−1(50) ORII−1(50) −1(50)

+1(50) −1(50) +1(50) +1(50) −1(50) −1(50) −1(50) −1(50)−1(50) −1(50) −1(50)−1(50) −1(50) −1(50)−1(50) −1(50) −1(50)−1(50) −1(50)

+1(50) +1(50) −1(50) −1(50) −1(50) −1(50) −1(50) −1(50) −1(50) −1(50)

+1(50) +1(50) −1(50) −1(50) −1(50) −1(50) −1(50) −1(50)

+1(50) −+1(50) 1(50) −−1(50) 1(50) −1(50)

+1(50) +1(50) −1(50) −1(50)

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8. Output results ofthe theLapSVM LapSVM based multi-classifier samples. 0 Figure 50 100of 150 200 250 of test 300samples.350 Figure 8. Output results based multi-classifier of test Sample number

The above obtained marked samples are input to SVM the SVM multiple The above obtained and and marked samples alsoalso are input to the basedbased multiple faultfault classifier Figure 8. Output resultsset of in theLapSVM. LapSVM based multi-classifier of testinsamples. classifier with the same parameters The outputs are shown Table 2 and Figure 9. with the same parameters set in LapSVM. The outputs are shown in Table 2 and Figure 9. As shown As shown in Table 2 and Figure 9, the recognition rate is 98.57% in which four samples with BEII are in Table 2The andabove Figureobtained 9, the recognition is 98.57% in which withmultiple BEII arefault wrongly and markedrate samples also are input four to thesamples SVM based wrongly divided into that with IRI and IRII respectively and equally in SVM3. Eventually, classifier withwith the same parameters set in LapSVM. The outputs are shown in Tablecomparing 2 and FigureTables 9. divided into that IRI and IRII respectively and equally in SVM3. Eventually, 1 comparing Table 1 and Table 2, the fault type of rolling bearing can be effectively classified by As shown in Table 2 and Figure 9, the recognition rate is 98.57% in which four samples with BEII are and 2,LapSVM the fault typemuch of rolling bearing can beare effectively by LapSVM much unmarked when unmarked samples adopted, classified which indicates that thewhen proposed method wrongly divided which into that with IRI and IRII respectively and equally inrecognition SVM3. Eventually, samples are adopted, indicates that the proposed method has a better performance has a better recognition performance on fault diagnosis of rolling bearing. comparing Table 1 and Table 2, the fault type of rolling bearing can be effectively classified by on fault diagnosis of rolling bearing. LapSVM when much unmarked samples are results adopted, which indicates that the proposed method Table 2. Output of SVM classifier. has a better recognition performance on fault diagnosis of rolling bearing. results SVM3 of SVM classifier. Sample Sets Faults Table SVM12. Output SVM2 SVM4 SVM5 SVM6 T1~T50

Norm

Sample Sets T51~T100

Sample Sets

T101~T150 T1~T50 T1~T50 T151~T200 T51~T100 T51~T100 T201~T250 T101~T150 T101~T150 T251~T300 T151~T200 T151~T200 T301~T350 T201~T250 T201~T250 T251~T300 T251~T300 T301~T350 T301~T350

+1(40) Table 2. Output results of SVM classifier. SVM1 SVM2 SVM3 SVM4 SVM5 SVM6 −1(40) +1(40) SVM1 SVM2 SVM3 SVM4 SVM5 SVM6 −1(40) −1(40) +1(36) +1(40) +1(40) −1(40) +1(40) −1(40) −1(40) +1(40) −1(42) −1(40) +1(40) −1(40) −1(40) −1(42) +1(40) −1(40) −1(40) +1(36) −1(40) −1(40) −1(40) +1(36) −1(40) +1(40) −1(40) −1(40) −1(40) −1(40) −1(42) −1(40)+1(40) −1(40) −1(40) −1(40) −1(42) +1(40) −1(40) −1(40) −1(40) −1(40) −1(40) −1(40) −1(40) −1(42) −1(40) +1(40)−1(40) −1(40) −1(40) −1(42) −1(40) +1(40) −1(40) −1(40) −1(40) −1(40) −1(40) +1(40) −1(40) −1(40) −1(40) −1(40) −1(40) +1(40) −1(40) −1(40) −1(40) −1(40) −1(40) −1(40) −1(40) −1(40) −1(40) −1(40) −1(40) Real output −1(40) Ideal output

Faults BEI

Faults

BEII Norm Norm IRI BEI BEI IRII BEII BEII ORI IRI IRI ORII IRII IRII ORI 8 ORI ORII 7

ORII

8 6

Real output

Output Output

7 5

Ideal output

6 4 5 3 4 2 3 1 2 0 0 1

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Figure 9. Output results of the SVM based multi-classifier of test samples. 0 0

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Figure 9. Output resultsofofthe theSVM SVM based based multi-classifier of of testtest samples. Figure 9. Output results multi-classifier samples.

To illustrate the superiority of TSMFE, for comparison, TSME and MSE methods are also used to extract the feature information by analyzing the above experimental data of rolling bearing. TSMFEs,

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To illustrate the superiority of TSMFE, for comparison, TSME and MSE methods are also used 11 of 15 to extract the feature information by analyzing the above experimental data of rolling bearing. TSMFEs, TSMEs and MSEs of all selected vibration signals of rolling bearing are calculated and input toand theMSEs LapSVM multi-classifier for multi-fault of rolling bearing.and Theinput recognition TSMEs of all selected vibration signals of recognition rolling bearing are calculated to the results aremulti-classifier shown in Figure where therecognition number of marked samples is ten the number of input LapSVM for10, multi-fault of rolling bearing. Theand recognition results are features are varying from 2 to 20. Firstly, when the number of marked samples is determined, with shown in Figure 10, where the number of marked samples is ten and the number of input features are the increase number selected features,of the recognition rateisof the MSE, TSME andincrease TSMFE varying fromof2 the to 20. Firstly,ofwhen the number marked samples determined, with the based fault diagnosis methods all increase and thenrate decrease someTSME degree, which means thatfault not of the number of selected features, the recognition of thetoMSE, and TSMFE based all of entropies in MSE, TSMEand andthen TSMFE are suitable reflectwhich the fault feature of rolling bearing diagnosis methods all increase decrease to some to degree, means that not all of entropies and some of them are redundant information. compared with the existing it is in MSE, TSME and TSMFE are suitable to reflectSecondly, the fault feature of rolling bearing and methods, some of them obvious that the rate recognition of the proposed method fluctuates slightly and tend to be 100%, are redundant information. Secondly, compared with the existing methods, it is obvious that the rate which are always larger than that fluctuates of the TSME and and LapSVM faultwhich methods for different recognition of the proposed method slightly tend tobased be 100%, are always larger numbers ofthe features, as well as the MSEfault andmethods LapSVMforbased faultnumbers diagnosis method as when than that of TSME and LapSVM based different of features, wellthe as number of input features is smaller than 14. Particularly, when the number of input features is larger the MSE and LapSVM based fault diagnosis method when the number of input features is smaller than than 14, the recognition of the TSME LapSVM is decreasing heavily 14. Particularly, when the performance number of input features is and larger than 14, method the recognition performance ofand the this isand because thatmethod the MSE has no definition larger scale that factor cannot TSME LapSVM is decreasing heavily at andthe this is because theand MSEthe hasLapSVM no definition at classify the faultfactor feature Although the obstacles of MSE that with the the larger scale andeffectively. the LapSVM cannot TSME classifyovercomes the fault feature effectively. Although TSME increase of the timeobstacles scales, the entropies in TSME tend to of notime definition some degree, drawback overcomes of MSE that with the increase scales,to the entropies in this TSME tend to stilldefinition exist. Finally, thedegree, above analysis has validated superiority of TSMFE extracting no to some this drawback still exist.the Finally, the above analysisfor has validatedfault the feature of rolling bearing. Compared with TSMEofand MSE, the number marked samples little superiority of TSMFE for extracting fault feature rolling bearing. Compared with TSME have and MSE, influence onmarked the performance of TSMFE, which expresses much more stability. When theexpresses numbers the number samples have little influence on the performance of TSMFE, which are suitable, the identification rate of the proposed method for extracting fault features by TSMFE much more stability. When the numbers are suitable, the identification rate of the proposed method tends to 100%fault in particular. for extracting features by TSMFE tends to 100% in particular. Entropy 2018, 20, 602

100 90

Identification rate(%)

80 70

LapSVM-TSMFE LapSVM-TSME

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30 20

2

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8

10

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Number of eigenvalues

Figure 10. 10. Identification Identification rate rate comparison comparison of of the the TSMFE-, TSMFE-, TSMETSME- and and MSE-based MSE-based methods. methods. Figure

illustrate the the superiority superiority of LapSVM, LapSVM, the the above above calculated calculated entropies entropies of of TSMFE, TSMFE, TSME TSME Next, to illustrate MSE are are input input to to SVM SVM based based multi-classifier multi-classifier for for identification identification and the the recognition recognition results are and MSE shown in in Figure Figure11, 11,where wherethe thenumber numberofofinput inputfeatures featuresare areset setasas1010 and that the marked samples shown and that ofof the marked samples is is varying from to 30. With increase of the number of marked samples, recognition of varying from 2 to2 30. With thethe increase of the number of marked samples, the the recognition ratesrates of the the TSMFE based diagnosis method increase andto tend be a constant 100% and TSMFE and and SVMSVM based faultfault diagnosis method increase and tend be atoconstant 100% and that ofthat the of the and TSME and SVM based fault diagnosis methodwith increase with little fluctuation, while the TSME SVM based fault diagnosis method increase little fluctuation, while the recognition recognition ratesand of the MSE andfault SVMdiagnosis based fault diagnosis method fluctuate severely tending84%. to a rates of the MSE SVM based method fluctuate severely tending to a constant constant 84%. particular,rate theofrecognition of the TSMFE SVM method based fault diagnosis In particular, theInrecognition the TSMFE rate and SVM based faultand diagnosis is much larger method is of much larger (or than those ofSVM the TSME MSE) and SVM based methods. with than those the TSME MSE) and based (or methods. Secondly, with the increaseSecondly, of the number of marked samples, the recognition rates of the TSMFE and LapSVM based fault diagnosis method

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the increase of the number of marked samples, the recognition rates of the TSMFE and LapSVM based fault diagnosis method for rolling bearing remains 100%, which is larger than that of the TSME (or MSE) andremains SVM based thus thethat above analysis superiority of for rolling bearing 100%,methods. which isAnd larger than of the TSMEvalidates (or MSE)the and SVM based TSMFE. with increase of the numberthe of marked samples, the recognition performance of methods.Thirdly, And thus thethe above analysis validates superiority of TSMFE. Thirdly, with the increase LapSVM is always better than that of SVM for the same input features. Finally, by observing Figures of the number of marked samples, the recognition performance of LapSVM is always better than that 10 and 11 conclude that the proposed and Figures LapSVM fault method for of SVM forwe thecan same input features. Finally, byTSMFE observing 10based and 11 we diagnosis can conclude that the rolling bearing much better performance for rolling than fault the proposed TSMFE has and LapSVM basedfault fault recognition diagnosis method for rolling bearing hasbearing much better contrasting recognition methods. performance for rolling bearing than the contrasting methods. 100 96

Identification rate(%)

92 88 84 80

LapSVM-TSMFE LapSVM-TSME

76

LapSVM-MSE

72

SVM-TSMFE SVM-TSME SVM-MSE

68 64 60

2

4

6

8

10

12

14

16

18

20

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24

26

28

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Number of marked samples

Figure 11. Identification Identification rate comparison of the mentioned methods: TSMFE (or TSME and MSE) and LapSVM (or SVM).

method, wewe also investigate thethe influence of the Next, to illustrate illustratethe thesuperiority superiorityofofthe theproposed proposed method, also investigate influence of number of marked samples and input features on the fault recognition rates of the proposed method, as the number of marked samples and input features on the fault recognition rates of the proposed well as the The recognition for different are shown in Tables 3 andin4 method, asexisting well as methods. the existing methods. Theresults recognition resultsmethods for different methods are shown in detail, where data where in the first line stands for the samples and thesamples first column Tables 3 and 4 inthe detail, the data in the first linenumber stands of formarked the number of marked and represents the number of input features. Table 3 provides the recognition results of the LapSVM based the first column represents the number of input features. Table 3 provides the recognition results of faultLapSVM diagnosis methods. When themethods. number of marked with is thedetermined, increase of the based fault diagnosis When the samples number is ofdetermined, marked samples the number of extracted features,ofthe recognition ratesthe of the different rates methods i.e., MSE, TSME and with the increase of the number extracted features, recognition of the different methods TSMFE fluctuate obviously which indicates that the number of extracted features have some influence i.e., MSE, TSME and TSMFE fluctuate obviously which indicates that the number of extracted on the recognition performance. When number of extracted eigenvalues determined, with the features have some influence on the the recognition performance. When theis number of extracted increase of the number of marked samples, the recognition rate of the proposed method almost eigenvalues is determined, with the increase of the number of marked samples, the recognitiontends rate to the be aproposed constant method that are almost larger than ofathe MSE and based fault method with of tendsthat to be constant thatTSME are larger than thatdiagnosis of the MSE and TSME more obvious fluctuation, whichwith means that obvious the number of markedwhich samples has little on the based fault diagnosis method more fluctuation, means that influence the number of identification results of the proposed methods comparing the existing methods. The recognition results marked samples has little influence on the identification results of the proposed methods comparing of the SVM based methods shown in results Table 4.of Comparing MSE, TSME and SVM the existing methods. The are recognition the SVM with basedthemethods are shown in methods, Table 4. the TSMFE and method for fault diagnosis of SVM rollingbased bearing hasdiagnosis a higher Comparing withSVM the based MSE, fault TSMEdiagnosis and SVM methods, the TSMFE and fault recognition rater with little fluctuation than the existing methods which also validates the superiority method for fault diagnosis of rolling bearing has a higher recognition rater with little fluctuation of TSMFE to MSE methods and TSME. The comparing of Tables 3 and 4 indicates the LapSVM based methods than the existing which also validates the superiority of TSMFE to MSE and TSME. The have higher identification performance than the SVM based methods. From the two tables, we can comparing of Tables 3 and 4 indicates the LapSVM based methods have higher identification find that for the proposed generally,From we suggest numbers of marked input performance than the SVMmethod based methods. the twothe tables, we can find thatsamples for the and proposed features generally, should bewe ranging from to 20 andoffrom 8 to samples 12, respectively to features obtain a should better recognition method suggest the4numbers marked and input be ranging performance. These two8tables indicate superiority the proposed methods. These two from 4 to 20 and from to 12,further respectively to the obtain a betterof recognition performance. tables further indicate the superiority of the proposed methods.

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Table 3. Identification results with LapSVM (%). Methods 4

8

12

16

20

MSE TSME TSMFE MSE TSME TSMFE MSE TSME TSMFE MSE TSME TSMFE MSE TSME TSMFE

4

6

8

10

12

14

16

18

20

22

86.85 86 99.14 88.28 96.29 100 87.14 92.85 100 87.71 57.14 98.86 87.14 28.57 98.86

96.57 91.43 99.14 92.85 96.57 100 89.14 93.42 100 89.71 57.14 98.86 90 28.57 98.86

96.85 92 99.14 92.85 96.29 100 87.42 93.42 100 88.57 57.14 98.86 88.28 28.57 98.86

96.85 92.57 99.14 92.85 96.86 100 88 94 100 89.14 57.14 98.86 88 28.57 98.86

97.42 93.43 99.14 93.71 97.14 100 89.14 94.28 100 89.14 57.14 98.86 88.85 28.57 98.86

97.42 96 99.14 94.57 97.43 100 92.85 94.57 100 90.57 57.14 98.86 89.42 28.57 98.86

97.14 95.71 99.14 95.42 97.43 100 94.57 95.14 100 92.85 57.14 98.86 91.14 28.57 98.86

96.85 96.29 99.14 95.42 97.71 100 95.42 95.71 100 92.85 57.14 98.86 91.71 28.57 98.86

97.42 96.29 99.14 94.85 97.71 100 95.71 96.57 100 94 57.14 98.86 95.14 28.57 98.86

97.42 96.87 99.14 94.85 97.71 100 95.71 96.28 100 95.42 57.14 98.86 95.42 28.57 98.86

Table 4. Identification results with SVM (%). Methods 4

8

12

16

20

MSE TSME TSMFE MSE TSME TSMFE MSE TSME TSMFE MSE TSME TSMFE MSE TSME TSMFE

4

6

8

10

12

14

16

18

20

22

78.26 86.95 97.51 79.81 95.03 98.44 77.32 93.16 98.13 77.32 79.81 97.82 77.01 47.20 97.82

87.01 93.18 97.72 84.41 95.45 98.70 84.74 94.15 98.70 83.11 57.14 98.37 78.57 41.88 98.37

87.07 93.87 97.61 83.33 94.21 98.29 85.37 94.55 98.63 81.63 57.14 98.29 79.59 41.83 98.29

87.14 92.50 97.85 85.00 92.85 98.21 84.28 93.57 98.57 80.00 57.14 98.21 80.71 41.78 98.21

86.84 90.97 98.12 84.21 95.48 98.49 85.71 94.73 98.49 81.95 57.14 98.49 81.57 47.74 98.49

85.31 92.85 98.41 84.52 96.03 98.41 83.73 95.23 98.41 83.73 57.14 98.41 79.36 41.66 98.01

84.87 91.59 98.31 84.45 95.79 98.31 84.03 94.95 98.31 84.45 68.48 98.31 81.09 41.59 98.31

84.82 92.41 98.21 83.92 96.42 98.21 83.48 95.53 98.21 83.48 57.14 98.21 80.35 28.57 98.21

85.23 93.80 98.09 85.23 96.66 98.02 83.80 96.19 98.09 83.33 57.14 98.09 82.85 28.57 98.09

87.24 93.36 97.95 85.20 96.42 97.88 84.18 95.91 97.95 82.65 57.14 97.95 81.12 36.22 97.95

5. Conclusions In this paper, an improved algorithm of multiscale sample entropy (MSE) called TSMFE is proposed to measure the complexity of time series. The influence of similar tolerance and data length on TSMFE are investigated by analyzing Gaussian white noise and 1/f noise signals. The results indicate that compared with TSME, TSMFE obviously expresses more anti-noise property, more stable entropies even in shorter time series and slight fluctuation with the increase of time scale factors. Combining TSMFE and LapSVM, a new fault diagnosis method for rolling bearing is put forward and applied to analyze experimental data of rolling bearing. The comparison analysis of the proposed method with the existing methods, i.e., TSMFE with MSE and TSME, LapSVM with SVM are made and the analysis results validate that the proposed method has much better performance on fault recognition rate than the used comparison method. At present, the entropy theories have not been wildly used in mechanical signal processing for digging the nonlinear fault features hidden in the vibration signals. In this paper, we tried to apply the proposed TSMFE method to multi-fault diagnosis for rolling bearing and the simulation and experimental data analysis were also conducted. The following work will be emphasized on optimized feature selection and on-line multi-fault diagnosis. Author Contributions: Methodology, X.Z. and J.Z.; Software, X.Z.; Validation, Y.Z.; Data Curation, H.P.; Writing—Original Draft Preparation, X.Z.; Writing—Review & Editing, J.B.; Supervision, J.Z.

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Acknowledgments: This work was supported by the National Natural Science Foundation of China (Grant No. 51505002), the National Key Research and Development Program of China (No. 2017YFC0805100) and the Major Program of Natural Science Research of Higher Education in Anhui Province, China (Grant No. KJ2018ZD005). Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations AppEn SampEn MSE SVM LapSVM FuzzyEn TSMFE TSME Norm ORI BEI IRI ORII BEII IRII SD L V

Nomenclature Approximate entropy Sample entropy Multiscale sample entropy Support vector machine Laplace support vector machine Fuzzy entropy Time shift multiscale fuzzy entropy Time shift multiscale entropy Normal rolling bearing Outer race fault under fault diameters 0.1778 mm Ball element under fault diameters 0.1778 mm Inner race fault under fault diameters 0.1778 mm Outer race fault under fault diameters 0.5334 mm Ball element under fault diameters 0.5334 mm Inner race fault under fault diameters 0.5334 mm Standard deviation Laplacian Hinge loss function

τ X N m r k β ∆( β,k) l u k f k2I f∗ β∗ p M Tp p T1 p T2

Time scale Initial time series Length of data X Embedding dimension Similar tolerances Initial time point Interval time Upper rounding boundary Number of given marked samples Number of given unmarked samples Manifold regularization item Manifold regularization framework Quadratic planning Number of rolling bearing states Total number of samples p-th feature sets Marked sample sets Unmarked sample sets

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