Wavelet leaders multifractal features based fault diagnosis of rotating mechanism

Mechanical Systems and Signal Processing 43 (2014) 57–75

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Wavelet leaders multifractal features based fault diagnosis of rotating mechanism Wenliao Du a,b,n, Jianfeng Tao b, Yanming Li b, Chengliang Liu b,nn a School of Mechanical and Electronic Engineering, Zhengzhou University of Light Industry, No. 5 Dongfeng Road, Zhengzhou 450002, China b State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, No. 800 Dongchuan Road, Shanghai 200240, China

a r t i c l e i n f o

abstract

Article history: Received 11 November 2011 Received in revised form 21 August 2013 Accepted 12 September 2013 Available online 24 October 2013

A novel method based on wavelet leaders multifractal features for rolling element bearing fault diagnosis is proposed. The multifractal features, combined with scaling exponents, multifractal spectrum, and log cumulants, are utilized to classify various fault types and severities of rolling element bearing, and the classification performance of each type features and their combinations are evaluated by using SVMs. Eight wavelet packet energy features are introduced to train the SVMs together with multifractal features. Experiments on 11 fault data sets indicate that a promising classification performance is achieved. Meanwhile, the experimental results demonstrate that the classification performance of the SVMs trained with eight wavelet packet energy features in tandem with multifractal features outperforms that of the SVMs trained only with wavelet packet energy features, time domain features, or multifractal features, and it is also superior to that of wavelet packet energy features in tandem with time domain features, or multifractal features combined with time domain features. The feature selection method based on distance evaluation technique is exploited to select the most relevant features and discard the redundant features, and therefore the reliability of the diagnosis performance is further improved. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Multifractal features Wavelet leaders Rolling element bearing Fault diagnosis Support vector machines

1. Introduction With the rapid development of science and technology, modern mechanical equipment is controlled automatically and with high precision. Dealing with the equipment breakdown promptly and accurately is very helpful in terms of enhancing its reliability and decreasing downtime. Rolling element bearings constitute the key parts in rotating machinery, and their fault detection and diagnosis are of great importance. To this end, researches focused on extracting the features relevant to the bearing conditions from mechanical vibration signals which contain abundant running information [1–3]. Mechanical vibration signal is a typical non-linear signal. The traditional feature extraction methods based on linear system cannot work effectively. In the last few years, fractal theory has been employed to depict the complex non-linear dynamic behavior of the mechanical fault signal. In Logan and Mathew's seminal papers [4,5], the correlation dimension of vibration signal was used for rolling element bearing fault diagnosis. Then, the application of capacity dimension and fractal dimension were discussed in the fault diagnosis [6–9]. Furthermore, Yang et al. proposed a fault diagnosis method using all

n Corresponding author at: School of Mechanical and Electronic Engineering, Zhengzhou University of Light Industry, No. 5 Dongfeng Road, Zhengzhou 450002, China. Tel.: þ86 371 63556785; fax: þ 86 371 63556718. nn Corresponding author at: State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, No. 800 Dongchuan Road, Shanghai 200240, China. E-mail addresses: [email protected] (W. Du), [email protected] (C. Liu).

0888-3270/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ymssp.2013.09.003

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of the three fractal dimensions for rolling element bearing [10]. All of the above methods utilize single fractal features, which only reflect the overall irregularity of signals; but fail to describe the local scaling properties. However, different local conditions and fluctuations of features are important indicators of mechanical faults [11]. Multifractal features can fully display the distribution of signal's singularities, while the geometric characters and the local scaling behaviors are described more precisely [12,13]. As an effective tool for unstable and non-linear signals, wavelet transform is widely used. The wavelet coefficients are utilized to calculate multifractal spectrum directly in some situations. But wavelet decompositions necessarily yield a large number of coefficients closing to 0, which implies that the computation of structure functions for negative orders qs will be numerically instable [14]. The wavelet transform modulus maxima (WTMM) method, which is based on continuous wavelet transform (CWT), is an alternative approach for multifractal analysis [15–21]. It overcomes the shortcoming of the origin wavelet coefficients method and some progress is made in fault diagnosis [11,20,21]. But no mathematical results have been proved to hold in a sense for this technique even now, such as the Legendre transform of the scaling function yields an upper bound for the spectrum of singularities is not available, and the continuous transform implies more computational complexity [22]. Therefore, the WTMM method is hardly applied to online real time diagnosis. Recently, a new multifractal analysis method based on wavelet leaders was proposed by Lashermes et al. [23–25]. Based on discrete wavelet transform (DWT), the new approach not only describes the characters of spectrum on a full domain (for the negative orders qs and positive orders qs), but also has a solid theoretical mathematical supports. Furthermore, the complicated calculation is avoided. So, the wavelet leaders based method is an effective tool for extracting multifractal features, which has been successfully used in texture classification [26–28], analysis of heart rate variability [29], turbulent velocity [30], fMRI time series [31], and ECG signal [32], etc. Essentially, fault diagnosis can be considered as a pattern recognition problem. Some intelligent classification technologies, such as artificial neural networks (ANNs) and support vector machines (SVMs) have been successfully applied to the fault diagnosis of rotating machinery [33–35]. The most difference between them is the rule of risk control. The former only implements the experience risk minimization (ERM) principle, and it leads to poor generalization ability; however, SVMs, a universal learning algorithm, is established on the theory of the structural risk minimization (SRM) principle, and seems to prevail in the field of intelligence fault diagnosis for its favorable generalization ability. In many practical applications, the classification performance of SVMs outperforms many traditional classification technologies [10,34]. In this study, the classification performance of multifractal features, including scaling exponents, multifractal spectrum, log cumulants and their combinations, on various fault data sets is studied, and the intelligent classification technology of SVMs is adopted [33–35]. In order to improve the performance of the classifier, eight wavelet packet energy (WPE) features [35] are introduced to train the SVMs together with multifractal features, and the classification performance of multifractal features in tandem with eight WPE features is investigated. The schematic diagram of fault diagnosis is shown in Fig. 1. This paper is organized as follows. The experimental system and data sets are introduced in Section 2. Section 3 displays a brief description of extracting multifractal features. In Section 4, the multifractal analysis for simulation and real vibration

Data Acquisition

Feature Extraction Diagnosis Result

Test Feature Selection

Trained SVMs

Train

Training SVMs

No

Yes Good

Fig. 1. Flowchart of fault diagnosis system.

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signal is proposed. The classification performance of multifractal features and their combinations on various fault data sets is studied. The classification performance of multifractal features in tandem with eight WPE features is also discussed in this section. In addition, the performance comparison of features with different combination forms is implemented. A conclusion is given in Section 5. 2. Vibration data The main objective of the study is to diagnose the different fault conditions of rolling element bearing with wavelet leaders multifractal features. The vibration data used in this paper have been obtained from the data sets of the rolling element bearings [36]. Fig. 2 shows the schematic of the experimental system. A 2 hp, three-phase induction motor is connected to a dynamometer and a torque sensor by a self-aligning coupling. The dynamometer is controlled to obtain the desired torque load levels. The bearings are installed in the motor driven mechanical system and support the motor shaft. An accelerometer is mounted on the motor housing at the drive end of the motor to acquire the vibration signals from the test bearings. The vibration data are acquired by a 16 channel DAT recorder with a sample frequency of 12 kHz, and each data set is made up of 480,000 points. Single point faults are introduced to the test bearings using electro-discharge machining with fault diameters of 0.18 mm, 0.36 mm, 0.53 mm, and the fault depth is 0.28 mm. Drive end bearing type is 6205-2RS JEM SKF, deep groove ball bearing. The bearing is tested under four different loads (0, 1, 2 and 3 hp), four different fault locations (normal condition, ball fault, inner race fault and outer race fault), and three different fault degrees (the different fault diameters correspond to the different fault degrees: slight, medium and serious). The experimental rotating frequency is about 30 Hz. In order to evaluate the classification performance of different multifractal features and their combinations, we separate the experiment data sets with load 0 hp into 11 fault data sets, which include different fault condition combinations, and each sample in the fault data sets includes 2048 points. These data sets statistics are presented in Table 1. For DALL① data set, the cases with different fault locations or different fault degrees are regarded as having different types, and for DALL② data set, each fault type includes three different fault conditions. By using the FFT spectrum or the envelope spectrum, if the component of fault character frequency is observed, the corresponding fault can be recognized. But in some situations, for example, for the contaminated signal, this character frequency is not easy to be extracted. Fig. 3 shows a typical raw data sample for each condition, its spectrum and Hilbert envelope spectrum for each condition. It shows the 10 conditions of

Accelerometer Drive End Bearing

Induction Motor

Load

Fig. 2. Schematic of the experimental system.

Table 1 Data sets statistics. Data set

Training data

Testing data

Fault diameter (mil)

Fault type

Classification label

D070707 D141414 D212121 D071421 D142107 D210714 DINN DOUT DBALL DALL①

150

50

0 0 0 0 0 0 0

NIOB

1234

NIII NOOO NBBB N I O B N I O B

1234

375

125

DALL②

N – normal, B – ball fault, I – inner race fault, O – outer race fault

0 7 7 7 0 7 7 7

777 14 14 14 21 21 21 7 14 21 14 21 7 21 7 14 7 14 21

14 21 14 21 14 21 14 21 14 21 14 21

1 234 567 8 9 10 1 2 3 4

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bearing: normal condition, slight fault, medium fault, and serious fault in the inner race, respectively, slight fault, medium fault, and serious fault in the ball, respectively, slight fault, medium fault, and serious fault in the outer race, respectively. From the raw vibration signals and the corresponding FFT spectrums, it is not easy to identify the different fault categories. With the envelope spectrums, some faults, such as normal station, fault in inner race may be recognized. However, it is extremely difficult to recognize all of the 10 different conditions only using the envelope spectrums. For more clearly

Envelope spectrum

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Fig. 3. Typical raw data sample, its spectrum, and envelope spectrum for 10 conditions.

500

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illustrating the details, the original vibration signal with inner race fault and the corresponding envelope spectrum are shown in the following Fig. 4(a) and (b), and the original vibration signal with outer race fault and the corresponding envelope spectrum are shown in Fig. 4(c) and (d). For the former, the fault character frequency (162 Hz), its multiple frequencies, and the axial frequency are apparent. However, for the latter, both the fault frequency (107 Hz) and the axial frequency in the background noise are not reflected in the envelope spectrum of the original signal. Furthermore, for the different severities of the same fault categories occurring in the bearings, the character frequency is coincident, so it is difficult to distinguish the fault condition with the envelope method directly. For the published references [1,2], in a majority of the cases, the authors extract the fault frequency component after the signal is processed by some preconditioning technologies, and it often only analyzes two or three fault categories of inner race fault, outer race fault,

3

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0.2 0.15 29

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6

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Frequency(Hz) Fig. 4. Original signals with inner race defect, outer race defect, respectively, and their envelope spectrums. (a) The original vibration signal with inner race defect, (b) The envelope spectrum of the original signal with inner race defect, (c) The original vibration signal with outer race defect and (d) The envelope spectrum of the original signal with outer race defect.

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and ball fault. In order to identify the 10 different conditions of bearing accurately, more efficient features and classification method are necessary to apply. 3. Wavelet leaders based multifractal features 3.1. Definition of wavelet leaders Suppose a function ψ 0 ðtÞ with a compact time support, for 8 k ¼ 0; 1; :::; N ψ  1, where, the positive integer Nψ Z1, satisfies R ψ 0 ðtÞ dt  0 and R t Nψ ψ 0 ðtÞ dt a 0, then ψ 0 ðtÞ can be selected as mother-wavelet and the vanishing moment is N ψ . Let  j=2 j fψ j;k ðtÞ ¼ 2 ψ 0 ð2 t  kÞ; jA Z; k A Zg denote templates of ψ 0 ðtÞ dilated to scales 2j and translated to time positions 2j k, and they form an orthonormal basis of L2 ðRÞ. For signal X ¼ fxk ; k A Zg, its discrete wavelet transform is defined in the following way Z dX ðj; kÞ ¼ XðtÞ2  j ψ 0 ð2  j t kÞ dt: ð1Þ

R

Rt

k

R

Define the dyadic interval λ ¼ λj;k ¼ ½k U2j ; ðk þ 1Þ U2j , and let 3λ denote the union of the interval λ with its two adjacent dyadic intervals: 3λj;k ¼ λj;k  1 [ λj;k [ λj;k þ 1 . Then, the wavelet leader is defined as the local supremum of wavelet coefficients taken within a spatial neighborhood over all finer scales [23,24]: LX ðj; kÞ  Lλ ¼ supλ′  3λ jdX;λ′ j. Hence, LX ðj; kÞ consists of the largest wavelet coefficient dX ðj′; k′Þ, which is calculated at all finer scales 2j′ r2j within a narrow time neighborhood ðk  1Þ2j r 2j′ k′o ðk þ 2Þ2j . 3.2. Wavelet leaders based features extraction For signal X, let SL ðq; jÞ denote the structure functions, and the corresponding scaling exponents are denoted as ζ L ðqÞ [22], where, q is the order of these multiresolution quantities SL ðq; jÞ ¼

1 nj ∑ jLX ðj; kÞjq ; nj k ¼ 1

ζ L ðqÞ ¼ lim inf j-0

  log2 SL ðq; jÞ : j

ð2Þ

ð3Þ

The multifractal spectrum DðhÞ can be obtained by the Legendre transform of the scaling exponents DðhÞ ¼ inf ð1 þqh ζ L ðqÞÞ: qa0

ð4Þ

In addition, the structure functions SL ðq; jÞ can be read as sample mean estimators for the ensemble averages ELX ðj; U Þq , so, the scaling exponents are expanded as [24] 1

ζ L ðqÞ ¼ ∑ cp p¼1

qp ; p!

ð5Þ

where the log cumulants cp satisfy 8 p Z 1; C L ðj; pÞ ¼ c0;p þ cp ln 2j ; where C L ðj; pÞ stands for the cumulant of order p Z1 of the random variable ln LX ðj; U Þ. The log cumulants cp have specific meanings: c1 mostly characterizes the location of the maximum of DðhÞ, c2 corresponding to its width and c3 corresponding to its asymmetry. So, the triplet ðc1 ; c2 ; c3 Þ thus gathers most of the multifractal information of the signal X. ζ L ðqÞ and cp can be estimated via log2 2j ¼ j versus log2 SL ðj; qÞ and ln 2j versus C L ðj; pÞ j2

^ ζðqÞ ¼ ∑ wj log2 SL ðj; qÞ;

ð6Þ

j ¼ j1

j2

L c^ p ¼ ðlog2 e ∑ wj C^ ðj; pÞ:

ð7Þ

j ¼ j1

To estimate the multifractal spectrum, a parametric formulation of Chhabra is used to avoid the complicated Legendre transform [37]. In this research, the multifractal features consist of ζ L ðqÞ, DðqÞ, hðqÞ and cp , p ¼ 1; 2; 3 and the total number of multifractal features is 33. 3.3. Advantages over the WTMM based approach It is well known that the WTMM based multifractal formalism is the popular expression yielding correct results for negative orders qs, and has been widely used in real systems [15–22]. However, as mentioned by Lashermes, Wendt, and Abry et al. [23,24], the wavelet leaders based approach has some significant advantages over the WTMM based approach.

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From a mathematical point of view, the wavelet leader based multifractal formalism benefits from well-established theoretical mathematical results, while, for the case of the WTMM, the spacing between local maxima need not be of the order of magnitude of the scale or even be regularly spaced [23]. So, the WTMM scaling exponents may differ from those acquired with wavelet leaders, and no mathematical result is expected to hold for the formula (4) (In reference [38], counterexamples are constructed). On a more practical perspective, the WTMM approach, which is based on a full continuous wavelet transform followed by maxima tracking and chaining operations, involves a high computational cost. However, the leaders approach is based on the discrete wavelet transform, and the use of fast pyramidal algorithm can reduce computational effort greatly. This implies that the wavelet leaders approach can be used for signals of arbitrary length while the WTMM is often restricted to much shorter ones. Furthermore, the wavelet leader approach can be theoretically and practically extended to arbitrary higher dimensions. But, this is far less the case for the MMWT method, which needs a significant modification with more complex procedure. For specific details of the wavelet leaders, see the references [23–32].

3.4. Selection of related parameters 3.4.1. Mother wavelet and vanishing moments The choice of mother wavelet and its vanishing moment is mainly based on the trade-off between the relief of frequency aliasing and the preservation of spatial characterization. A localized impact occurs instantaneously when a fault appears in a rolling element bearing, and the impact demonstrates that transient impulses are added on a periodical signal. If these transient components of vibration signals resemble the shape of wavelet, larger wavelet coefficients will be obtained and the fault characters are more evident [35]. As for multifractal analysis, the vanishing moment should be larger than the largest singularity exponent, because, for the negative qs, the larger vanishing moment stabilizes the estimates of the structure functions and is capable of suppressing the potentially superimposed smooth trends [27]. On the other hand, an overlarge vanishing moment will introduce border effects. A reasonable smallest vanishing moment is obtained when the vanishing moment increase while the estimated multifractal attributes do not significantly change. For the rolling element bearing data sets, the mother wavelet is selected as Daubechies 3 (Db3) function.

3.4.2. Range of qs The orders qs are chosen in a scope that avoids the linearization effect in scaling exponents estimation procedure [23]. To obtain the entire curve DðhÞ by the Legendre transform with scaling exponents, both positive and negative values of q are needed. When p qA ½qn ;ffi qnþ , both for q Z qnþ and q rqn , ζ L ðqÞ is a linear function of q. For the parabolic approximation, qn7 is ffiffiffiffiffiffiffiffiffiffiffiffi obtained by 7 2=jc2 j [29]. In this study, c2 is estimated on a data set of inner race fault with fault diameters of 0.18 mm by bootstrap technology. As shown in Fig. 5, c2 is about 0.1, so qn7 ¼ 75.

3.4.3. Scaling range of regression ζ L ðqÞ, DðqÞ, hðqÞ and cp can be estimated with SL ðj; qÞ by linear regression over a certain scaling range ½j1 ; j2 . The range of linear scaling regression is resolved by means of statistical procedures [29]. That is, the plot of ζ L ðqÞ versus log2 2j ¼ j decides a posteriori range in which the data actually shows a scaling behavior. Certainly, the range of scales must be the same for all parameters. Take the inner race fault data set (with fault diameters of 0.18 mm) for instance, the relation between ζ L ðqÞ and j is shown in Fig. 6, so the scaling range is from 3 to 7.

LWT c2 -0.09

-0.11615

-0.1 -0.11 -0.12 -0.13 -0.14 Fig. 5. Estimation of c2 of the example signal based on bootstrap technology, the number of resample is 49.

W. Du et al. / Mechanical Systems and Signal Processing 43 (2014) 57–75 LWT ζ(q) LogScale, q=-5

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0 -5

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Slope=0.33067

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10

Slope=-0.44425 ζ(q)

64

0 -10

10

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10

j

j

Fig. 6. Relationship of scaling exponents ζL(q) and scale j.

4. Simulation and experiment 4.1. Analysis of “devil staircase” function For the popular fractal series, such as “devil staircase”, Fractional Brownian motions, etc., the efficiency of classic multifractal analysis methods, including multifractal fluctuation detrended analysis (MFDFA) and WTMM method, has been validated [22,23]. In this section, we further evaluate the performance of the wavelet leaders approach with “devil staircase” function by comparing with the result of the analytical method [19,39]. For the triadic Cantor set measure μ, define f : ½0; 1-½0; 1 as the distribution function of μ, Z x f ðxÞ ¼ dμ; μA ½0; 1; ð8Þ 0

which is an increasing continuous function. As the measure of Cantor set is constructed recursively with the two distribution weights pi ði ¼ 1; 2Þ and it looks like a staircase, the function is called the “devil staircase” and the graph is selfsimilar 8 p1 f ð3xÞ x A ½0; 1=3 > < x A ½1=3; 2=3 p1 ð9Þ f ðxÞ ¼ > : p þ p f ð3x  2Þ x A ½2=3; 1 1

2

For this function, the scaling exponents can be calculated analytically by ζðqÞ ¼

lnðpq1 þ pq2 Þ : lnð1=3Þ

ð10Þ

The multifractal spectrum can be obtained by ( n htemp ¼ ðlogðp2 Þ  logðp1 ÞÞ=ðlogðp2 Þ þh logð3ÞÞ n

n

DðhÞ ¼ ððhtemp  1Þn logðhtemp 1Þ  htemp logðhtemp ÞÞ=ðhtemp logð1=3ÞÞ

:

ð11Þ

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Fig. 7(a) shows the scaling exponents of the “devil staircase” function with p1 ¼ 0:4 and p2 ¼ 0:6 obtained by the analytical method and wavelet leaders method, respectively. It can be seen that the difference is very small, especially for q 4 0. The multifractal spectrums with the above two approaches are shown in Fig. 7(b). In addition, the spectrum obtained by the wavelet leaders with Legendre transform method almost coincides with the theoretical spectrum, which further validates that the wavelet leaders based scaling exponents are effective. But the multifractal spectrum obtained by the wavelet leaders with Chhabra regression method seems to deviate from the analytical result distinctly. This may be induced by the regression with limited scales data and the scale weight distribution. However the obtained spectrum still reflects the multifractal character of the “devil staircase” function. This attributes to the two spectrums have a similar shape. Furthermore, the three of four parameters used to characterize the spectrum [40], which are the minimum and the maximum of the Hölder index corresponding to the two spectrums, and the range of the Hölder index, are similar; the forth n parameter, which is the h of the wavelet leaders with Chhabra regression method that corresponds to the Hölder index of the maximum Hausdorff dimension, is about 0.65, and agrees with the analytical method. 4.2. Analysis of simulated vibration In this section, a simulated vibration signal, which is modulated by two harmonic frequencies with an exponential decay, is used to simulate the vibration signal of a faulty bearing [41]. This signal is presented to test the proposed multifractl features extraction method, and it is generated by the function as follows 0

with

xðkÞ ¼ e  αt ð sin 2πf 1 kT þ1:2 sin 2πf 2 kTÞ

ð12Þ

  1 t 0 ¼ mod kT; fm

ð13Þ

where T ¼ 1=25; 000 s is the sampling interval, and α ¼ 800, f m ¼ 100 Hz, f 1 ¼ 3000 Hz and f 2 ¼ 8000 Hz are the exponential frequency, the modulation frequency and two carrier frequencies, respectively. Gaussian noise is the most popular noise in the vibration signal of mechanical system. The noise is added to the original signal to obtain the corrupted signal with a signal-to-noise ratio (SNR) of 10 dB. The multifractal character of this signal is analyzed with the MFDFA, WTMM, and wavelet leaders approaches, respectively. Fig. 8 shows the simulated vibration signal, CWT and DWT of the signal, and their corresponding maximum modulus and wavelet leaders. The obtained multifractal spectrums of this signal with the three methods are presented in Fig. 9(a). The signal has an evident multifractal character. Compared with the WTMM, the wavelet leaders is much sparser. The spectrum estimation corresponding to WTMM method is reasonable for the right side, but at the left side the estimation is not right. Although the WTMM method does not work correctly, the MFDFA method and the

10 5

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h Fig. 7. Scaling exponents and multifractal spectrum of the devil staircase. (a) Scaling exponents obtained by analytical method and wavelet leaders method, respectively and (b) Multifractal spectrum obtained by analytical method, wavelet leaders with Legendre transform method (wavelet leaders -1), and wavelet leaders with Chhabza regression method (wavelet leaders -2), respectively.

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1.4 WTMM MFDFA wavelet leaders

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Fig. 9. Multifractal spectrums of the simulated vibration signal. (a) The multifractal spectrums obtained by MFDFA, WTMM, and wavelet leaders method, respectively and (b) The multifractal spectrums obtained by wavelet leaders with different vanishing moments.

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wavelet leaders method obtain the multifractal spectrums with a similar shape. The range of Hölder exponents h estimated by MFDFA method is larger than the one estimated by wavelet leaders method, but the other three parameters, hmin and hmax that correspond to the minimum and the maximum of the Hölder index, and Δh ¼ hmax  hmin that corresponds to the range of the Hölder index, are similar, while the value of the Hölder exponent corresponding to the maximum DðhÞ of each spectrum is almost coincident. Therefore, as mentioned by Serrano [22] and Figliola [40], for the natural series provided by systems whose properties are unknown, though the obtained spectrums with different method have variance, the efficiency

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of the MFDFA and that of the wavelet leaders are similar. However, as mentioned in the published paper [22], the algorithm of wavelet leaders method is notably faster than MFDFA. In this case, the MFDFA method spends about 1222.96 s, while the wavelet leaders method only spends about 11.22 s. The maximum h estimated by wavelet leaders method in Fig. 9(a) is less than 1, so the wavelet vanishing moments of a value just greater than 1 satisfies the analysis requirement. Fig. 9(b) depicts the multifractal spectrums estimated by the wavelet leaders method with different wavelet vanishing moments, and the mother wavelet is Db1, Db2, and Db3, respectively. It shows that the obtained spectrums are matched correctly, which supports the argument that the wavelet leaders method is a stable multifractal estimator. 4.3. Analysis of rolling element bearing To evaluate the proposed method, 11 data sets covering seven different fault conditions (normal condition, ball fault, outer race fault, inner race fault, slight fault, medium fault, severe fault) obtained from the experiment system are conducted to demonstrate the effectiveness of the proposed algorithm. Each data set comprises different number of data samples which are divided into training samples and testing samples, and each sample in the fault data sets includes 2048 points. The detailed description of the data is shown in Table 1. For the raw vibration signals in Fig. 3, the corresponding multifractal spectrums are shown in Fig. 10. It is not easy to identify the different fault categories with these features, so the support vector machines are utilized to realize the automatic recognition for these different conditions. Support vector machine is an intelligent learning algorithm and derived from the work of Vapnik et al. [42,43]. The RBF kernel is adopted in this study, since in many practical applications, the RBF kernel always obtains a superior performance to other kernel functions [44]. The original SVM is designed for binary classification, which is not suitable for fault diagnosis, because there are other kinds of fault in addition to health condition. However, multi-classification can be obtained by the combination of binary classification. Currently, several methods have been proposed for this problem, such as “one-againstone”, “one-against-all”, and directed acyclic graph SVMs [45]. Hsu and Lin researched these methods and point out that the “one-against-one” method is more suitable for practical use than other methods [46]. In this study, we adopt “one-againstone” method to identify the different fault categories. The features extracted from the 11 data sets are used to train and test SVMs. In order to give fair and objective evaluation for each multifractal feature, the parameters of SVMs are not optimized. Both the kernel width parameter and penalty parameter are set as 2, and the CV (cross-validation) strategy is applied to select training and testing samples. The obtained classification accuracy is averaged over 10 runs for each data set. 4.3.1. Performance of simple multifractal features Multifractal features can fully display the distribution of signal's singularities, while the geometric characters and the local scaling behaviors are described more precisely than single fractal features. When the condition of mechanical equipment changed, the distribution of signal's singularities will be different from the normal condition. Thus the multifractal features contain the fault information and can be used for diagnosis. Fig. 11 illustrates the scaling exponents and multifractal spectrums of data set D070707. For q A ½ 5;  1, the whole four conditions of the bearing can be clearly divided, but for q A ½1; 5, these four conditions are close to each other; for multifractal spectrums, the whole four conditions are also obvious distinguished. LWT ζ(q) 2

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0.8 normal inner race fault ball fault outer race fault

0.6 0.4 0.2 0

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Fig. 11. Scaling exponent features and multifractal features of example in data set D070707.

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4.3.1.1. Performance for different fault categories. In this section, the data sets correspond to D070707, D141414, D212121 in Table 1. Each data set consists of four data subsets, which are acquired under normal condition, outer race fault, inner race fault and ball fault of bearings, respectively, and they are applied to evaluate the performance of the proposed fault diagnosis scheme. Each of the subsets contains 50 samples, and therefore the whole data sets corresponding to the four conditions contain altogether 200 samples. Table 2 gives the classification performance of the multifractal features and their combinations on these data sets. For visualization, principal component analysis (PCA) method is implemented on each of the clustering results. Take D070707 for instance, the plots of the first three principal components (PCs) of their results are shown in Fig. 12. From Table 2 and Fig. 12, utilize the features of scaling exponents or multifractal spectrums (including DðqÞ and hðqÞ), almost all of the samples can be classified correctly. But for the combination of these features, the performance is not improved significantly.

4.3.1.2. Performance for different fault severities. To examine the effect of the multifractal features in identifying different severities occurring in bearings, the proposed algorithm is applied on the data sets corresponding to DINN, DOUT, DBALL in Table 1. Each data set consists of 200 samples: 50 samples being normal condition, 50 samples for slight fault (0.18 mm), 50 samples for medium fault (0.36 mm), and 50 samples for serious fault (0.53 mm). The classification performance of the multifractal features and their combinations on these data sets is listed in Table 3. For visualization, take DINN for instance, the first three PCs of their results are plotted in Fig. 13. From Table 3 and Fig. 13, for different fault severities of outer or inner race, almost all of the samples can be classified correctly. But for the fault occurring in ball, the classification performance is poor and the recognition rate is only 76.2%. Here, the multifractal spectrum features perform more effectively than the scaling exponent features. Furthermore, the combined features cannot improve the recognition rate remarkably, which indicates that some redundant and irrelevant features exist in the features set.

4.3.1.3. Performance for mixture conditions with different fault categories and severities. In rotating machinery, besides the different categories of a single fault and the different severities of a fault, a more general situation is the mixture condition with different fault categories and severities. Thus, it is of considerably practical value to address this situation in rotating machinery correctly.

Table 2 Classification performance of multifractal features for different fault categories. Data set

Feature

Rate (%)

Feature

Rate (%)

Feature

Rate (%)

Feature

Rate (%)

D070707 D141414 D212121

ζ L ðqÞ

83.6 85.4 92.0

DðqÞ

81.2 87.4 90.4

hðqÞ

87.2 84.2 92.6

ζ L ðqÞ DðqÞ hðqÞ CðpÞ

86.6 86.6 93.4

Rate represents the classification accuracy rate of SVMs averaged over 10 runs for each data set. The best classification accuracy rate on each data set is denoted in bold face.

D070707

normal inner race fault ball fault outer race fault

0.5

3rd PC

0 -0.5 -1 -1.5 -2 1

0

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-1 -2

0

-3 -0.5

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1st PC

-5

Fig. 12. Scatter plots of the first three principal components for clustering results of data set D070707.

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Table 3 Classification performance of multifractal features for different fault severities. Data set

Feature

Rate (%)

Feature

Rate (%)

Feature

Rate (%)

Feature

Rate (%)

DINN DOUT DBALL

ζ L ðqÞ

100 97.4 66.6

DðqÞ

99.0 95.6 70.4

hðqÞ

99.8 95.4 69.8

ζ L ðqÞ DðqÞ hðqÞ CðpÞ

95.6 90.2 76.2

Rate represents the classification accuracy rate of SVMs averaged over 10 runs for each data set. The best classification accuracy rate on each data set is denoted in bold face.

DINN

1.2 normal slight fault medium fault serious fault

3rd PC

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0

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0

-3 -1

1st PC

-4

Fig. 13. Scatter plots of the first three principal components for clustering results of data set DINN.

Table 4 Classification performance of multifractal features for mixture conditions with different fault categories and severities. Data set

Feature

Rate (%)

Feature

Rate (%)

Feature

Rate (%)

Feature

Rate (%)

D071421 D142107 D210714 DALL① DALL②

ζ L ðqÞ

86.2 88.8 94.0 50.9 63.6

DðqÞ

88.0 85.2 94.6 60.7 69.1

hðqÞ

87.6 82.6 91.8 58.1 67.4

ζ L ðqÞ

90.4 85.8 94.8 67.4 81.4

DðqÞ hðqÞ CðpÞ

Rate represents the classification accuracy rate of SVMs averaged over 10 runs for each data set. The best classification accuracy rate on each data set is denoted in bold face.

In this experiment, the data sets not only contain different fault categories and severities but also consist of the mixture faults. The data sets correspond to D071421, D142107, D210714, DALL①, DALL② in Table 1. Here, DALL① and DALL② have the same samples, but the former regards the samples with same fault category but different fault severities as an identical category. The data sets D071421, D142107, D210714 consist of 200 samples, respectively, while DALL② and DALL① consist of 500 samples, respectively. Each of the data subsets contains 50 samples. Table 4 lists the classification performance of the multifractal features and the combined features on these data sets. For visualization, take D071421 for instance, the first three PCs of their results are plotted in Fig. 14. From Table 4 and Fig. 14, compared with D070707, D141414, D212121, the recognition rate has no obvious changes. For almost all of the data sets, a highly recognition rate is obtained by multifractal features, and the scaling exponent features seem more effective than the multifractal spectrum features in this case. But for DALL① and DALL②, the classification performance is very poor, especially for DALL①. Also, with combined features, the improvement of recognition rate seems obvious, but is still unacceptable. In order to improve the classification performance, the multifractal features should be combined with other features. The features selection method based on the distance evaluation criteria is applied for the data sets [47]. The classification accuracy is listed in Table 5 with different features selection thresholds. When the threshold is set as 0.4, for the data sets DALL① and DALL②, 21 and 25 sensitive features are selected from the original 33 features, and the recognition rates

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D071421

2 normal inner race fault outer race fault ball fault

3rd PC

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Table 5 Classification performance of multifractal features with features selection method for mixture conditions with different fault categories and severities. DALL①

Threshold Number Rate (%)

0.1 33 68.7

0.2 26 70.1

0.3 24 70.1

0.4 21 70.2

0.5 18 69.1

0.6 14 65.7

0.7 11 66.3

0.8 7 51.2

DALL②

Number Rate (%)

33 81.8

30 83.0

26 84.7

25 86.3

25 82.7

19 86.2

9 79.6

5 74

Rate represents the classification accuracy rate of SVMs averaged over 10 runs for each data set and number represents the selected feature number with feature selection method. The best classification accuracy rate on each data set is denoted in bold face.

increase from 67.4% and 81.4% to 70.2% and 86.3%, respectively. It shows that after discarding redundant features, a better performance is achieved. 4.3.2. Performance of multifractal features in tandem with eight WPE features In this study, we mainly consider the classification performance of multifractal features of vibration signal. In the literature [10], the fractal dimensions tandem with 11 time-domain features are applied to improve the classification performance. For the information complementarity of features, we utilize multifractal features in tandem with eight WPE features. With the 3 levels wavelet packet decomposition, the characteristic frequency of the rolling element bearing is extracted adequately [35]. A wavelet function of Db3 and 3 levels decomposition are used in this case study, and eight WPE features are extracted. Eight WPE features in tandem with multifractal features, are used to recognize various fault types and evaluate various fault severities of rolling element bearing. Table 6 gives the classification performance of combination features on various data sets. Experimental results show that the classification performance is greatly improved, especially for data sets DBALL, DALL① and DALL②. For DBALL data set, the recognition rate increases from 76.2% to 94.6%. In addition, for DALL① and DALL② data sets, the recognition rates increase from 67.4% and 81.4% to 88.9% and 96.1%, respectively. For these two data sets, shown in Table 6 and Table 7, both WPE features and time domain features (including mean, standard, deviation, root mean square, shape factor, skewness, kurtosis, crest factor, K factor and pulse index), even time domain features in tandem with multifractal features, behave badly. Because the information complementarity of features, multifractal features in tandem with eight WPE features achieve a promising classification accuracy, which is superior to the accuracies of 82.2% and 94.2% of the WPE features combined with the time features. For data sets DALL① and DALL②, the features selection method based on the distance evaluation criteria is applied on the 41 features, and the threshold is set as 0.3. As shown in Fig. 15, 26 and 15 features are selected for the two data sets, and the classification accuracies reach 86.0% and 99.2%, respectively. If the features selection method is applied on 10 scaling exponent features and eight WPE features, and the thresholds are set as 0.35 and 0.2, 11 and 10 features are selected for the two data sets, and the classification accuracies of 89.10% and 97.3% are obtained, respectively. 4.3.3. Performance comparison of the proposed method and the previously established ones It is very helpful to accurately detecting and diagnosing the existence and severity of the faults occurring in bearings in terms of reliability for mechanical equipment. The rolling element bearing data used in this paper has been widely used to verify the performance of the diagnosis algorithm [3,10,35,47]. Various features extraction technologies and classification

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Table 6 Classification performance of multifractal features in tandem with eight wavelet packet energy (WPE) features on various data sets. Data set

Feature

Rate (%)

Feature

Rate (%)

Feature

Rate (%)

Feature

Rate (%)

D070707 D141414 D212121 D071421 D142107 D210714 DINN DOUT DBALL DALL① DALL②

ζL ðqÞ WPE

99.0 97.4 97.8 99.2 97.6 98.0 99.6 100 92.8 83.8 96.1

DðqÞ WPE

98.2 94.4 96.6 97.2 97.0 94.0 98.2 97.6 92.4 88.9 94.5

hðqÞ WPE

98.6 96.8 98.6 97.0 95.2 96.8 98.8 97.0 94.6 83.1 95.8

ζL ðqÞ DðqÞ hðqÞ CðpÞ WPE

96.2 87.2 95.4 95.4 93.6 91.2 93.6 95.2 86.2 82.2 89.4

Rate represents the classification accuracy rate of SVMs averaged over 10 runs for each data set. The best classification accuracy rate on each data set is denoted in bold face.

Table 7 Classification performance comparison of eight wavelet packet energy (WPE) features, time domain features, time domain features combining with WPE features, multifractal features in tandem with time features on data sets DALL① and DALL②. Data set

WPE features

Time features

Time & WPE feature

Time & multifractal features

DALL① DALL②

57.2 85.0

67.2 68.4

82.2 94.2

67.0 89.0

Performance is described by the classification accuracy rate (%) of SVMs averaged over 10 runs for each data set.

algorithms are presented in these literatures. As the main objective of this paper is to investigate the classification performance of the multifractal features and the multifractal combination features, for the purpose of assessing the performance of different features fairly, the ensemble classification approach is not considered in this comparison. As mentioned above, for the bearing data sets DBALL, DALL① and DALL②, the states are the most difficult indicators to identify. Table 8 gives the best performance of the proposed method (MF-SVMs) and the previously established methods, which include the frequency-domain feature based ANFIS (F-ANFIS) [3] and the fractal dimensions features based SVMs (F-SVMs) [10]. Though the parameters of F-SVMs are optimized, the recognition rate of MF-SVMs is also superior to that of F-SVMs for the data set DALL②, and for the data set DBALL, the two methods are evenly matched. For the data set DALL①, the recognition rate of MF-SVMs is superior to that of F-ANFIS. It indicates that the MF-SVMs has more excellent diagnostic performance than F-ANFIS and F-SVMs in diagnosis of rolling element bearing. The classification accuracy of the improved wavelet package energy features based SVMs (IWPE-SVMs) [35] for the data set including seven states (normal, slight and medium fault occurring in inner race, outer race, and ball, respectively) is only about 75%. As for the modified fuzzy ARTMAP method (M-FAM) [47], the recognition rate for the data set including seven states (normal, ball fault, outer race fault, slight, medium, severe, and very severe inner race fault) is 87.3%, which is also inferior to that of the proposed MF-SVMs method.

5. Conclusions In this paper, the wavelet leaders multifractal features are applied to classify various fault types and evaluate various fault severities of rolling element bearing. For the vibration simulation signal, the wavelet leaders method, MFDFA, and WTMM method are utilized, respectively. It validates the advantage of the wavelet leaders over the other two methods. Due to better generalization ability than traditional classification techniques, the SVMs classifier is employed to evaluate the classification performance of multifractal features on 11 fault data sets of rolling element bearing. For the scaling exponent features and multifractal spectrum features, we obtain high classification accuracy for almost all of the data sets, however, for some data sets, such as DALL① and DALL②, the performance is poor. In addition, with the use of the combination features, including scaling exponent features, multifractal spectrum features and log cumulant features, the classification performance is not obviously improved. It shows that some redundant and irrelative features exist and they disturb the classification. Using the features selection method with an appropriate threshold, the redundant features are discarded, and the efficiency of training and testing is increased. In addition, the classification performance is improved, for example, for the above DALL① and DALL② data sets, the classification accuracy rates increase about 10%, respectively. In order to improve the classification performance of the multifractal features further, taking the information complementarity of features into account, eight WPE features are introduced to combine with multifractal features for fault diagnosis, and

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Distance evaluation criteria

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 threshold

0.1 0

5

10

15

20

25

30

35

40

25

30

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40

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Distance evaluation criteria

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10

15

20

Feature Fig. 15. Distance evaluation factor of all 41 features. (a) Corresponding to data sets DALL① and (b) Corresponding to data sets DALL②.

Table 8 Classification performance comparison of the proposed method and the previously established methods. Data set

MF-SVMs

F-ANFIS

F-SVMs

DBALL DALL① DALL②

94.6 88.9 96.1

– 87.7 –

96.3 – 93.3

Performance is described by the classification accuracy rate (%).

experiments demonstrate that the classification performance of the SVMs is greatly improved. Especially, for the data set DBALL, the classification accuracy rate increases from 76.2% to 94.6%; for the data sets DALL① and DALL②, the classification accuracies increase from 67.4% and 81.4% to 88.9% and 96.1%, respectively; moreover, with the use of features selection technology and an appropriate feature selection threshold, the classification accuracy rates reach 89.1% and 99.2%, respectively. By all accounts, the classification performance of multifractal features in tandem with WPE features gets a remarkable improvement, which is superior to other combination features, such as multifractal features and time domain features, time domain features and WPE features. Furthermore, the experimental results are also compared with the previously published results using the same bearing data set, and it verifies the performance of the proposed approach is superior to that of F-ANFIS, F-SVMs, IWPE-SVMs, and M-FAM. Furthermore, this method can provide a reference for the fault diagnosis of other rotating machinery.

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Acknowledgments This work is supported by the National Nature Science Foundation of China (Grant nos. 51205371, 61175038), National Basic Research Program of China (“973” Program) (Grant no. 2013CB035403), National High Technology Research and Development Program of China (“863” Program) (Grant nos. 2012AA041804,2012AA041803),Innovation Program of Shanghai Committee of Science and Technology (Grant no. 11JC1405800). References [1] B. Li, P.L. Zhang, D.S. Liu, S.S. Mi, G.Q. Ren, H. Tian, Feature extraction for rolling element bearing fault diagnosis utilizing generalized S transform and two-dimensional non-negative matrix factorization, Journal of Sound and Vibration 330 (10) (2011) 2388–2399. [2] Y. Ming, J. Chen, G. Dong, Weak fault feature extraction of rolling bearing based on cyclic Wiener filter and envelope spectrum, Mechanical Systems and Signal Processing 25 (5) (2011) 1773–1785. [3] Y. Lei, Z. He, Y. Zi, Q. Hu, Fault diagnosis of rotating machinery based on multiple ANFIS combination with GAs, Mechanical Systems and Signal Processing 21 (5) (2007) 2280–2294. [4] D. Logan, J. Mathew, Using the correlation dimension for vibration fault diagnosis of rolling element bearings—I. Basic concepts, Mechanical Systems and Signal Processing 10 (3) (1996) 241–250. [5] D.B. Logan, J. Mathew, Using the correlation dimension for vibration fault diagnosis of rolling element bearings—II. Selection of experimental parameters, Mechanical Systems and Signal Processing 10 (3) (1996) 251–264. [6] O. Castillo, P. Melin, A hybrid fuzzy‐fractal approach for time series analysis and plant monitoring, International journal of intelligent systems 17 (8) (2002) 751–765. [7] P. Purkait, S. Chakravorti, Impulse fault classification in transformers by fractal analysis, IEEE Transactions on Dielectrics and Electrical Insulation 10 (1) (2003) 109–116. [8] C.H. Chen, R.J. Shyu, C.K. Ma, A new fault diagnosis method of rotating machinery, Shock and Vibration 15 (6) (2008) 585–598. [9] E.P. de Moura, A.P. Vieira, M.A.S. Irmao, A.A. Silva, Applications of detrended-fluctuation analysis to gearbox fault diagnosis, Mechanical Systems and Signal Processing 23 (3) (2009) 682–689. [10] J. Yang, Y. Zhang, Y. Zhu, Intelligent fault diagnosis of rolling element bearing based on SVMs and fractal dimension, Mechanical Systems and Signal Processing 21 (5) (2007) 2012–2024. [11] S. Loutridis, Self-similarity in vibration time series: application to gear fault diagnostics, Journal of Vibration and Acoustics 130 (3) (2008) 0310041–0310049. [12] E. Bacry, J. Muzy, A. Arnéodo, Singularity spectrum of fractal signals from wavelet analysis: exact results, Journal of Statistical Physics 70 (3) (1993) 635–674. [13] R. Lopes, N. Betrouni, Fractal and multifractal analysis: a review, Medical Image Analysis 13 (4) (2009) 634–649. [14] A. Arnéodo, B. Audit, N. Decoster, J.F. Muzy, C. Vaillant, Wavelet based multifractal formalism: applications to DNA sequences, satellite images of the cloud structure, and stock market data, in: A. Bunde, J. Kropp, H.J. Schellnhuber (Eds.), The Science of Disasters, Springer, 2002, pp. 27–102. [15] A. Arneodo, G. Grasseau, M. Holschneider, Wavelet transform of multifractals, Physical Review Letters 61 (20) (1988) 2281–2284. [16] C. Rodrigues Neto, A. Zanandrea, F.M. Ramos, R.R. Rosa, M.J.A. Bolzan, L.D.A. Sa, Multiscale analysis from turbulent time series with wavelet transform, Physica A: Statistical Mechanics and its Applications 295 (1–2) (2001) 215–218. [17] A. Arnéodo, Y. d'Aubenton-Carafa, E. Bacry, P.V. Graves, J.F. Muzy, C. Thermes, Wavelet based fractal analysis of DNA sequences, Physica D: Nonlinear Phenomena 96 (1) (1996) 291–320. [18] J.F. Muzy, E. Bacry, A. Arnéodo, Multifractal formalism for fractal signals: the structure-function approach versus the wavelet-transform modulusmaxima method, Physical review E. 47 (2) (1993) 875–884. [19] J. Muzy, E. Bacry, A. Arnéodo, The multifractal formalism revisited with wavelets, International Journal of Bifurcation and Chaos 4 (2) (1994) 245–302. [20] Z. Peng, F. Chu, P.W. Tse, Singularity analysis of the vibration signals by means of wavelet modulus maximal method, Mechanical Systems and Signal Processing 21 (2) (2007) 780–794. [21] S. Yuhai, K. Yuzhe, C. Xiangguo, Fault pattern recognition of turbine-generator set based on wavelet network and fractal theory, in: Proceedings of the 8th International Conference on Electronic Measurement and Instruments, ICEMI, 2007, pp. 3414–3417. [22] E. Serrano, A. Figliola, Wavelet Leaders: a new method to estimate the multifractal singularity spectra, Physica A: Statistical Mechanics and its Applications 388 (14) (2009) 2793–2805. [23] B. Lashermes, S. Jaffard, P. Abry, Wavelet leader based multifractal analysis, in: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP, 2005, pp. 161–164. [24] H. Wendt, P. Abry, S. Jaffard, Bootstrap for empirical multifractal analysis, IEEE Signal Processing Magazine 24 (4) (2007) 38–48. [25] H. Wendt, P. Abry, Multifractality tests using bootstrapped wavelet leaders, IEEE Transactions on Signal Processing 55 (10) (2007) 4811–4820. [26] H. Wendt, P. Abry, S. Jaffard, H. Ji, Z. Shen, Wavelet leader multifractal analysis for texture classification, in: Proceedings of the International Conference on Image Processing, ICIP, 2009, pp. 3829–3832. [27] H. Wendt, SG. Roux, S. Jaffard, P. Abry, Wavelet leaders and bootstrap for multifractal analysis of images, Signal Processing 89 (6) (2009) 1100–1114. [28] Y. Xu, X. Yang, H. Ling, H. Ji, A new texture descriptor using multifractal analysis in multi-orientation wavelet pyramid, in: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2010, pp. 161–168. [29] P. Abry, H. Wendt, S. Jaffard, H. Helgason, P. Goncalves, E. Pereira, C. Gharib, P. Gaucherand, M. Doret, Methodology for multifractal analysis of heart rate variability: from LF/HF ratio to wavelet leaders, in: Proceedings of the Annual International Conference of the IEEE Engineering in Medicine and Biology Society, EMBC'10, 2010, pp. 106–109. [30] B. Lashermes, S.G. Roux, P. Abry, S. Jaffard, Comprehensive multifractal analysis of turbulent velocity using the wavelet leaders, European Physical Journal B. 61 (2) (2008) 201–215. [31] P. Ciuciu, P. Abry, C. Rabrait, H. Wendt, Log wavelet leaders cumulant based multifractal analysis of EVI fMRI time series: evidence of scaling in ongoing and evoked brain activity, IEEE Journal on Selected Topics in Signal Processing 2 (6) (2008) 929–943. [32] P. Abry, H. Helgason, P. Goncalves, E. Pereira, P. Gaucherand, M. Doret, Multifractal analysis of ECG for intrapartum diagnosis of fetal asphyxia, in: Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP, 2010, pp. 566–569. [33] B.A. Paya, I.I. Esat, M.N.M. Badi, Artificial neural network based fault diagnostics of rotating machinery using wavelet transforms as a preprocessor, Mechanical Systems and Signal Processing 11 (5) (1997) 751–765. [34] B. Samanta, Gear fault detection using artificial neural networks and support vector machines with genetic algorithms, Mechanical Systems and Signal Processing 18 (3) (2004) 625–644. [35] Q. Hu, Z. He, Z. Zhang, Y. Zi, Fault diagnosis of rotating machinery based on improved wavelet package transform and SVMs ensemble, Mechanical Systems and Signal Processing 21 (2) (2007) 688–705. [36] K.A. Loparo, Bearings vibration data set. Case Western Reserve University 〈http://www.eecs.cwru.edu/laboratory/bearing/download.html〉. [37] A.B. Chhabra, C. Meneveau, R.V. Jensen, K. Sreenivasan, Direct determination of the f(α) singularity spectrum and its application to fully developed turbulence, Physical Review A. 40 (9) (1989) 52–84.

W. Du et al. / Mechanical Systems and Signal Processing 43 (2014) 57–75

75

[38] S. Jaffard, Multifractal formalism for functions. Part I: results valid for all functions, SIAM Journal on Mathematical Analysis 28 (4) (1997) 944–970. [39] S. Mallat, A Wavelet Tour of Signal Processing, Academic press, San Diego, 1999. [40] A. Figliola, E. Serrano, G. Paccosi, M. Rosenblatt, About the effectiveness of different methods for the estimation of the multifractal spectrum of natural series, International Journal of Bifurcation and Chaos 20 (2) (2010) 331–339. [41] W. Su, F. Wang, H. Zhu, Z. Zhang, Z. Guo, Rolling element bearing faults diagnosis based on optimal Morlet wavelet filter and autocorrelation enhancement, Mechanical Systems and Signal Processing 24 (5) (2010) 1458–1472. [42] V.N. Vapnik, Statistical Learning Theory, John Wiley & Sons, New York, 1998. [43] B.E. Boser, I.M. Guyon, V.N. Vapnik, A training algorithm for optimal margin classifiers, in: Proceeding of the 1992 Fifth Annual ACM Workshop, 1992, pp. 144–152. [44] J. Yang, Y. Zhang, Application research of support vector machines in condition trend prediction of mechanical equipment, Lecture Notes in Computer Science 3498 (3) (2005) 857–864. [45] J. Cui, Y. Wang, A novel approach of analog circuit fault diagnosis using support vector machines classifier, Measurement 44 (1) (2011) 281–289. [46] C.W. Hsu, C.J. Lin, A comparison of methods for multiclass support vector machines, IEEE Transactions on Neural Networks 13 (2) (2002) 415–425. [47] Z. Xu, J. Xuan, T. Shi, B. Wu, Y. Hu, Application of a modified fuzzy ARTMAP with feature-weight learning for the fault diagnosis of bearing, Expert Systems with Applications 36 (6) (2009) 9961–9968.