Rolling bearing fault diagnosis using generalized refined composite multiscale sample entropy and optimized support vector machine

Journal Pre-proofs Rolling bearing fault diagnosis using generalized refined composite multiscale sample entropy and optimized support vector machine Zhenya Wang, Ligang Yao, Yongwu Cai PII: DOI: Reference:

S0263-2241(20)30111-1 https://doi.org/10.1016/j.measurement.2020.107574 MEASUR 107574

To appear in:

Measurement

Received Date: Accepted Date:

4 November 2019 2 February 2020

Please cite this article as: Z. Wang, L. Yao, Y. Cai, Rolling bearing fault diagnosis using generalized refined composite multiscale sample entropy and optimized support vector machine, Measurement (2020), doi: https:// doi.org/10.1016/j.measurement.2020.107574

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2020 Published by Elsevier Ltd.

Rolling bearing fault diagnosis using generalized refined composite multiscale sample entropy and optimized support vector machine Zhenya Wang, Ligang Yao∗, Yongwu Cai School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, PR China

Abstract Rolling bearing fault diagnosis is an important and time sensitive task, to ensure the normal operation of rotating machinery. This paper proposes a fault diagnosis for rolling bearings, based on Generalized Refined Composite Multiscale Sample Entropy (GRCMSE), Supervised Isometric Mapping (S-Isomap) and Grasshopper Optimization Algorithm based Support Vector Machine (GOA-SVM). First, GRCMSE is utilized to characterize the complexity of vibration signals, at different scales. Furthermore, an effective manifold learning algorithm, named S-Isomap, is applied, to compress the high-dimensional feature set into a low-dimensional space. Subsequently, GOA-SVM classifier is proposed for pattern recognition, having higher recognition accuracy than other classifiers. The performance of the proposed method has been verified by its successful application in a rolling bearing fault diagnosis experiment. Compared with the existing methods, this approach improves the classification accuracy to 100%. The produced results indicate that the proposed method can effectively detect bearing faults, maintaining high accuracy. Keywords: generalized refined composite multiscale sample entropy, supervised isometric mapping, grasshopper optimization algorithm, support vector machine, rolling bearing, fault diagnosis

1. Introduction As connecting and fixed parts, rolling bearings are widely used in rotating machinery, due to their advantages of high running precision, good substitutability, low price and easy mass production [1]. However, rolling bearings will be damaged, during the working process, due to the effects of alternating load, machining error, improper installation, etc. As a result, rotating machines will not work properly, while even catastrophic accidents may occur [2]. Moreover, on account of various nonlinear factors (e.g., stiffness and friction), the vibration signals of rolling bearings are usually nonlinear and non-stationary, thus increasing the difficulty of fault diagnosis [3]. Hence, a new or effective method for feature extraction and pattern recognition of rolling bearings has become an important task, requiring immediate attention. In recent years, machine learning-based fault diagnosis methods have attracted much attention, in the field of rolling bearing fault diagnosis [4]. These methods include mainly three steps: feature extraction, dimensionality reduction, and pattern recognition. First of all, useful features need to be extracted from the vibration signal. As nonlinear technologies evolve, many nonlinear dynamic methods, based on estimations of statistical parameters, have been applied, to extract fault characteristics [5–8]. Various techniques mainly include approximate entropy [5], Sample Entropy (SE) [6], permutation entropy [7], and fuzzy entropy [8]. Among them, sample entropy, as one of the most commonly used entropy methods, can effectively measure the regularity of signals and detect the weak changes. When failure occurs in rolling bearings, the nonlinear dynamic complexity will change accordingly. Thus, SE is quite suitable for feature extraction of rolling bearings. However, SE analyzes the time series, only in one scale. To address this ∗ Corresponding

author. Email address: [email protected] (Ligang Yao)

Preprint submitted to Measurement

January 22, 2020

issue, Costa et al. proposed the Multiscale Sample Entropy (MSE) method, which can quantify the complexity of the system at multiple scales [9]. According to the MSE algorithm, the original time series is initially divided into non-overlapping segments, with length s (called scale factor). Next, the coarse-grained time series is obtained, by calculating the average value of each segment. Finally, the sample entropy of the coarse-grained time series, at each scale, is computed. MSE has also been successfully in features extraction in mechanical vibration signals. For example, Zhang et al. utilized MSE and adaptive neuro-fuzzy inference to detect rolling bearing faults and to determine their severity [10]. Li and Jiang proposed a rolling bearing feature extraction method, based on local mean decomposition and MSE, while the experimental results showed that MSE can effectively extract bearing feature vectors from selected components [11].Hsieh et al. applied the MSE curve to identify a number of characteristic defects of the high-speed spindle [12]. Despite its popular uses, following the coarse-grained process, to estimate the average of each segment, the dynamic mutation behavior of the time series is neutralized and therefore, the computed MSE entropy values are biased. To overcome this shortcoming, Costa and Goldberger proposed the Generalized Multiscale Sample Entropy (GMSE) algorithm, by using the second moment (variance), rather than the first moment (average), in the coarse-grained process [13]. In [13], the GMSE was utilized to analyze the heartbeat time series of healthy young and older people and patients with congestive heart failure. The results showed that the healthy young people exhibit the highest complexity, while healthy older people show complexity, significantly higher than that of patients with congestive heart failure. The above analysis validated that the GMSE can effectively assess the physiological complexity of heart signals. Although GMSE is a powerful algorithm, when applied to short time series, the estimated entropy value will exhibit large variance. According to the GMSE algorithm, for a time series with N points, the length of the coarse-grained time series, at the scale factor s, is N/s. That is to say, if the scale factor has a higher value, the length of the coarse-grained time series will be shorter, thus the variance of the GMSE entropy values will increase [14]. In order to reduce the variance of the GMSE entropy values, at the high scale factor, a Generalized Refined Composite Multiscale Sample Entropy (GRCMSE) algorithm is proposed, in this paper. The effectiveness of the proposed GRCMSE algorithm is demonstrated experimentally, by analyzing noise signals and rolling bearing vibration signals. The second step of intelligent fault diagnosis is to reduce the dimension of the original features vector. Since the fault features, extracted by GRCMSE, have characteristics of high dimensionality and nonlinearity, if directly input to the classifier, for training and testing, the recognition accuracy may be reduced. In order to avoid this outcome, dimensionality reduction methods are required to reduce the dimension of the derived high-dimensional feature set, obtaining a low-dimensional and fault sensitive feature set. Classic dimensionality reduction methods mainly include Principal Component Analysis (PCA) [15, 16], Linear Discriminant Analysis (LDA) [16, 17] and Canonical Correlation Analysis (CCA) [18]. However, these algorithms are of linear nature, which may not be suitable for the high-dimensional and nonlinear feature sets of rolling bearings. Recently, manifold learning, as a classical nonlinear dimensionality reduction method, has provided the possibility of dimensionality reduction of a GRCMSE feature set. The classic manifold learning algorithms mainly include Isometric Mapping (Isomap) [19], Local Tangent Space Alignment (LTSA) [20], Laplacian Eigenmap (LE) [21], Local Linear Embedding (LLE) [22] and t-distributed Stochastic Neighbor Embedding (t-SNE) [23]. However, the above methods are all unsupervised dimensionality reduction methods, where the sample labels information is not considered in the dimensionality reduction process [24]. At the same time, many studies indicate that using sample labels information, in the dimensionality reduction process, can improve the visualization and classification abilities [24–26]. Therefore, in this paper, a supervised manifold learning algorithm, called Supervised Isometric Mapping (S-Isomap) [26], is introduced, to reduce the dimension of the nonlinear feature sets. Last but not least, it is necessary to use a classifier to perform pattern recognition in the low-dimensional feature

2

set. Many classifiers have been presented and applied to fault detection processes in rotating machines, such as expert systems [27], artificial neural networks [28], and fuzzy logic classifier [29]. However, these classifiers have some drawbacks (e.g., the local optimal solutions, low convergence rates and significant over-fitting), which limit their applications in rolling bearing fault detection cases of pattern recognition. Moreover, when the training samples are inadequate, these methods demonstrate poor generalization [30]. A Support Vector Machine (SVM) classifier, based on statistical learning theory, can solve the small sample problem and exhibits good generalization performance. Thus, many scholars have applied SVM to the pattern recognition, in fault detection cases of various mechanical parts, such as bearings [31], gears [32], and rotors [33]. Nevertheless, the performance of the SVM is closely related to the parameters (i.e., the penalty factor and the kernel parameter) selection. Namely, incorrect selection of these parameters will result in a negative impact on the SVM recognition performance [34]. Grasshopper Optimization Algorithm (GOA) is a neoteric meta-heuristic algorithm, recently proposed by Saremi et al. [35]. The GOA mimics the social behavior of grasshoppers in nature. Moreover, the authors in [35] confirmed that GOA can effectively solve a series of complex mathematical test functions and actual structural design problems, while also outperform other well-designed meta-heuristic algorithms. In view of the advantages of GOA and SVM, the Grasshopper Optimization Algorithm based Support Vector Machine (GOA-SVM) classifier is presented, in this paper. The proposed method is applied to the pattern recognition of wine data set and rolling bearing feature sets, while it is also compared to the SVM, the Particle Swarm Optimization based Support Vector Machine (PSO-SVM) [34], the Firefly Algorithm based Support Vector Machine (FA-SVM) [36], the Chicken Swarm Optimization based Support Vector Machine (CSO-SVM) [37], and the Gray Wolf Optimization based Support Vector Machine (GWO-SVM) [38]. In summary, the contribution of this paper is the proposal of a novel rolling bearing fault diagnosis model, based on GRCMSE, S-Isomap and GOA-SVM, which mainly contains three steps (i.e., feature extraction, dimensionality reduction and pattern recognition). First, the GRCMSE algorithm is used to calculate the entropy values of the rolling bearing signals, while the original high-dimensional feature set is constructed. Next, the dimension of the original feature set is reduced, by using the S-Isomap algorithm, resulting in a low-dimensional and fault sensitive feature set. Last, this low-dimensional feature set serves as input to the GOA-SVM classifier, to automatically identify the fault types. Furthermore, the rolling bearings fault diagnosis experiment showed that the proposed method has higher recognition accuracy in cases of bearings under various working conditions. The paper is organized as follows: Section 2 introduces the theory of the GRCMSE algorithm and verifies its effectiveness, by analyzing noise signals experiment. Section 3 provides a brief review of the S-Isomap dimensionality reduction method. The grasshopper optimization algorithm based support vector machine is presented in Section 4. A novel rolling bearing fault diagnosis method is proposed in Section 5, while the experimental evaluation is presented in Section 6. Finally, conclusions are listed in Section 7.

2. Feature extraction by GRCMSE MSE can measure the complexity of time series and effectively detect the slight changes. When failure occurs in rolling bearings, the nonlinear dynamic complexity will change accordingly. Thus, MSE is quite suitable for feature extraction in cases of rolling bearings failure. However, this algorithm has a specific drawback. That is, using the coarse-grained process, to estimate the average of each segment, the dynamic mutation behavior of the time series is neutralized and therefore, the computed MSE entropy value is biased. To overcome this shortcoming, the GMSE algorithm is proposed, by using the second moment (variance), rather than the first moment (average), in the coarsegrained step. Nevertheless, GMSE still shows some drawbacks. First, GMSE is heavily relying on the length of the time series, and the GMSE entropy values may be uncertain or unreliable. Second, the probability of the invalid 3

entropy value appearing may increase. In order to address these drawbacks, the GRCMSE algorithm is proposed. 2.1. Multiscale Sample Entropy (MSE) MSE was proposed to measure the complexity of time series at different scale factors, while the specific processes are as follows:  (1) For a time series X = {x(i), i = 1, 2, . . . , N }, the coarse-grained series y (s) , at a scale factor s is defined as: (s)

yj

=

1 s

js X

xi , 1 ≤ j ≤

i=(j−1)s+1

N s

(1)

 (2) The coarse-grained series y (s) is formed as: h i N (s) ym (i) = y (s) (i), y (s) (i + 1), . . . , y (s) (i + m − 1) , i = 1, . . . , −m+1 s

(2)

where, m is embedding dimension.   (s) (s) (s) (s) (3) The maximum norm between the two vectors ym (i) and ym (j) is defined as the distance dm ym (i), ym (j) between two such vectors.     (s) (s) (s) (s) dm ym (i), ym (j) = max |ym (i + k) − ym (j + k)| , 0 ≤ k ≤ m − 1

(3)

(4) A function nm s,i (r) is defined as: 1 N um (i), i = 1, . . . , −m+1 N/s − m + 1 s   (s) (s) where, r represents the similar tolerance, and um denotes the number of dm ym (i), ym (j) ≤ r. nm s,i (r) =

(4)

(5) A function nm+1 s,i (r) is defined as: 1 N um+1 (i), i = 1, . . . , −m+1 N/s − m + 1 s   (s) (s) denotes the number of dm+1 ym+1 (i), ym+1 (j) ≤ r. nm+1 s,i (r) =

where, um+1

(6) Then, the probability of m and m + 1 matched points is calculated, respectively, by:  PN/s−m+1 m 1  nm (r) = ns,i (r) s i=1 N/s−m+1 PN/s−m m+1 1  nm+1 (r) = n (r) s

N/s−m

(5)

(6)

s,i

i=1

m+1 where, nm (r) represents the s (r) represents the total number of m-dimensional matched vector pairs, and ns

total number of (m + 1)-dimensional matched vector pairs. (7) Finally, the MSE entropy value, at the scale factor s , is computed as:   m+1  ns (r) MSE(X, s, m, r, N ) = lim − ln nm N/s→∞ s (r)

(7)

When the data length is finite, Eq. (7) can be replaced by:  MSE(X, s, m, r, N ) = − ln

nm+1 (r) s nm (r) s

 (8)

2.2. Generalized Multiscale Sample Entropy (GMSE) In MSE, using the coarse-grained process, to estimate the average of each segment, the dynamic mutation behavior of the time series is neutralized and therefore, the computed MSE entropy value is biased. To overcome this shortcoming, GMSE is proposed, by using the second moment (variance), rather than the first moment (average), in the coarse-grained step. 4

Consequently, for a time series X = {x(i), i = 1, 2, . . . , N }, the generalized coarse-grained time series

n o (s) yG of

GMSE, at the scale factor s, is calculated as: (s)

yG (j) =

1≤j≤

1 s

js X

2

(xi − xi )

i=(j−1)s+1

s−1 N 1X , 2 ≤ s, xi = xi+h s s

(9)

h=0

According to Eq. (9), we can easily observe that, when the scale factor s = 1, the GMSE entropy value is zero, therefore, the focus is on the scale factor s > 1, in the following study. 2.3. Generalized Refined Composite Multiscale Sample Entropy (GRCMSE) GMSE has been successfully applied in quantifying the complex fluctuations of human heartbeat time series, nevertheless, this algorithm still shows some drawbacks. First of all, GMSE is heavily relying on the length of the time series. Notably, when the scale factor s is increased, the length of the coarse-grained series will become shorter, thus the entropy value of the GMSE may be uncertain or unreliable. Second, the entropy value of GMSE will be undefined, m+1 if the nm (r) is zero in Eq. (8). To address these drawbacks, in this paper, the GRCMSE algorithm is i (r) or ni

proposed, with specific procedures being as follows: (s)

(1) Given a time series X = {x(i), i = 1, 2, . . . , N }, the generalized composite coarse-grained time series yG,h = n o (s) (s) (s) yG,h,δ1 , yG,h,δ2 , . . . , yG,h,δs is constructed as: (s)

yG,h,δ (j) =

1≤j≤

1 s

js+h−1 X

2

(xi − xi )

i=(j−1)s+h

s−1 N 1X , 1 ≤ h ≤ s, xi = xi+h s s

(10)

h=0

(s)

where, yG,h,δ (j) represents the δ-th generalized multi-scale coarse-grained time series under the scale factor s. m+1 (2) For the scale factor s, the number of matched points (i.e., nm G,h,s and nG,h,s ) is calculated for all s generalized

composite coarse-grained series. m+1 m+1 m (3) Let the nm G,h,s and nG,h,s denote the mean values of the nG,h,s and nG,h,s , respectively. Then, the GRCMSE

entropy value, at the scale factor s, is computed as: GRCMSE(X, s, m, r, N ) = − ln where, nm G,h,s =

1 s

Ps

h=1

m+1 nm G,h,s and nG,h,s =

1 s

Ps

h=1

nm+1 G,h,s

! (11)

nm G,h,s

nm+1 G,h,s . Eq. (11) can be replaced by:

GRCMSE(X, s, m, r, N ) = − ln

nm G,h,s nm+1 G,h,s Ps

= − ln

!

Ph=1 s h=1

= − ln nm+1 G,h,s nm G,h,s

1 s 1 s

Ps

h=1 Ps h=1

nm+1 G,h,s

!

nm G,h,s

! (12)

Eq. (10) shows that, GRCMSE considers all s different coarse-grained series, at the scale factor s, therefore, the entropy values, computed by this algorithm, are more accurate and stable than those computed by GMSE. In addition, m+1 Eq. (12) indicates that the RCMSE value is undefined, only when all nm G,h,s or nG,h,s are zero. Hence, compared

to the GMSE, GRCMSE can significantly reduce the probability of invalid entropy values. The respective GRCMSE flowchart is shown in Fig. 1.

5

Time series

Construct the generalized compound S coarse-grained time series YG,h h=h+1 m m+1 and nG,h,s Calculate the nG,h,s

Yes h
Calculate the entropy value

Yes s
Fig. 1. The flowchart of GRCMSE.

2.4. Experimental analysis of noise signals Four parameters in the GRCMSE need to be preset, namely embedding dimension m, data length N , similar tolerance r and scale factor s. According to the findings in [13, 14], m and s are set to 2 and 25, respectively, during the entire study. In order to investigate the influence of N on GMSE and GRCMSE, they are both studied, on 30 groups of white noise (wn) and 1/f noise with different length, while the results are plotted in Figs. 2 and 3, respectively. It becomes evident that, for the uncorrelated white noise, the entropy values of GMSE and GRCMSE decrease, as the scale factor increases. Conversely, for the correlated 1/f noise, the entropy values of GMSE and GRCMSE are first reduced and then tend to be constant, over higher scale factors. These are consistent with the results in [39], while they indicate that the 1/f noise is more complex than white noise [14, 39]. Moreover, the GMSE mean entropy curves of white noise, or 1/f noise, with different length, are relatively close. Similarly, the GRCMSE produces the same experimental results, confirming that N has less impact on the GMSE and GRCMSE, therefore, N is set to 3000.

Entropy value

2.0

wn N=2000 wn N=3000 wn N=4000 wn N=5000 1/f N=2000 1/f N=3000 1/f N=4000 1/f N=5000

1.5

1.0

0.5 0

5

10

15

20

25

No. of scale

Fig. 2. GMSE analysis results for white noise and 1/f noise with different length N .

6

Entropy value

2.0

wn N=2000 wn N=3000 wn N=4000 wn N=5000 1/f N=2000 1/f N=3000 1/f N=4000 1/f N=5000

1.5

1.0

0.5 0

5

10

15

20

25

No. of scale

Fig. 3. GRCMSE analysis results for white noise and 1/f noise with different length N .

In addition, Fig. 4 illustrates the mean entropy curves and error values (i.e., the standard deviation) of the GMSE and GRCMSE, for 30 groups white noise or 1/f noise, with different lengths. According to Fig. 4, the GRCMSE mean entropy curves of white noise or 1/f noise, compared to the GMSE, are smoother, with lower error values, thus verifying its stability.

10

15

20

10

15

20

-1

0.5

0.0

25

Error value × 10

0.9

5

10

15

20

error value of GMSE

1.5

1.0

1.0

0.5

0.5

0.0

0.0

25

5

10

15

20

No. of scale

(a)

(b)

(c)

(d)

10

15

20

25

0.5

5

10

15

20

1.7

error value of GRCMSE

0.6 1.3

0.4 0.9

0.2

0.0

0.0

-1

-1

0.5

error value of GMSE

Error value × 10

-1

0.5

0.0

Error value × 10

0.9

0.2

1.0

0.8

0.5

0.0

25

5

10

15

20

25

entropy value of GRCMSE error value of GMSE

1.5

error value of GRCMSE

1.0

1.0

0.5 0.5

Entropy value

1.3

0.4

1.5

error value of GRCMSE

2.0 entropy value of GMSE

Entropy value

0.6

entropy value of GRCMSE

25

1.5

2.1

entropy value of GMSE

error value of GMSE

1.0

1.0

entropy value of GRCMSE

Entropy value

1.7

Entropy value

error value of GMSE error value of GRCMSE

2.0 entropy value of GMSE

Error value × 10

1.5

2.1

1.5

error value of GRCMSE

No. of scale

entropy value of GRCMSE

Error value × 10

-1

5

1.3

0.5

0.0

0.0

1.7

error value of GMSE

No. of scale

entropy value of GMSE

5

0.5

error value of GRCMSE

1.0

entropy value of GRCMSE

No. of scale

1.0

0.8

1.0

0.5

25

-1

-1

Error value × 10

5

1.5

error value of GRCMSE

1.0

0.5

0.0

error value of GMSE

1.5

entropy value of GMSE

Entropy value

0.9

entropy value of GRCMSE

Entropy value

1.3

0.5

entropy value of GMSE

entropy value of GRCMSE

2.0

2.0

2.1

entropy value of GMSE

Entropy value

1.7

error value of GRCMSE

Entropy value

error value of GMSE

1.0

Error value × 10

-1

entropy value of GRCMSE

1.5

2.0

2.0

2.1

entropy value of GMSE

Error value × 10

1.5

0.0

0.0 5

10

15

20

No. of scale

No. of scale

No. of scale

No. of scale

(e)

(f)

(g)

(h)

25

Fig. 4. The mean entropy curves and error values of the GMSE and GRCMSE for white noise or 1/f noise with different N : (a) white noise with N =2000; (b) 1/f noise with N =2000; (c) white noise with N =3000; (d) 1/f noise with N =3000; (e) white noise with N =4000; (f) 1/f noise with N =4000; (g) white noise with N =5000 and (h) 1/f noise with N =5000.

Following, GMSE and GRCMSE, with different r, are employed, to analyze 30 groups of white noise or 1/f noise, in order to study the influence of r, while the results are illustrated in Figs. 5 (a) and (b), respectively. It is evident that the mean entropy curves of GRCMSE or GMSE decrease, along the increase of r. This occurs due to the fact that, when r has a lower value, the number of matched templates will increase, thus, the entropy values will rise. Conversely, when r has a higher value, the number of matched templates will decrease, therefore, the entropy value will be lower. Furthermore, considering the same value r, the trends of the GMSE or GRCMSE entropy curves of the two of noises are approximately the same. Generally, r is set to 0.1 − 0.2SD (SD represents the Standard Deviation of the original time series), while it is set r = 0.15SD, in the whole study.

7

2.5 GRCMSE r=0.1SD GRCMSE r=0.15SD GRCMSE r=0.2SD GRCMSE r=0.25SD GMSE r=0.1SD GMSE r=0.15SD GMSE r=0.2SD GMSE r=0.25SD

1.5

Entropy value

Entropy value

2.0

1.65

1.0

GRCMSE r=0.1SD GRCMSE r=0.15SD GRCMSE r=0.2SD GRCMSE r=0.25SD GMSE r=0.1SD GMSE r=0.15SD GMSE r=0.2SD GMSE r=0.25SD

1.15

0.65

0.5

0

5

10

15

20

0.15 0

25

5

10

15

20

25

No. of scale

No. of scale

(b)

(a)

Fig. 5. The comparisons of the mean entropy curves of the two noises by using GMSE and GRCMSE with different r: (a) the GMSE and GRCMSE analysis results for white noise with different r; (b) the GMSE and GRCMSE analysis results for 1/f noise with different r.

3. Dimensionality reduction by S-Isomap However, the fault features, extracted by GRCMSE, with high-dimensional and nonlinear characteristics, are directly input to the classifier for training and testing, which may reduce the recognition accuracy. S-Isomap, as a supervised manifold learning algorithm, has strong visualization and classification capabilities, therefore, it is very suitable for the dimensionality reduction of a high-dimensional features set. Given an input data set U = {ui , i = 1, 2, . . . , N }, the steps of S-Isomap are briefly outlined as follows: (1) The distance matrix Ds = {ds (ui , uj )} of the data set is calculated. The distance between ui and uj is computed as:

 q  1 − exp −d2 (ui ,uj ) , Lu = Lu i j ψ q ds (ui , uj ) =  exp d2 (ui ,uj ) − τ, L 6= L ψ

ui

(13)

uj

where, d(ui , uj ) represents the Euclidean distance between the ui and uj , Lui denotes the label of ui , parameter ψ is set to the average Euclidean distance of all pairs of data points, while parameter τ is applied, to adjust the similarity between the sample points with different labels. (2) A neighborhood graph G is constructed. If ui is the K nearest neighbor of uj , then the pair of ui and uj has edge connection, while the edge length is ds (ui , uj ); otherwise, there is no edge connection. (3) The geodesic distance matrix Dg is computed. The shortest path of any pair of data points, on the graph G, is calculated by using the Dijkstra method, while this path is approximated as the geodesic distance. (4) The d-dimensional embedding is constructed. The geodesic distance matrix Dg is mapped into a d-dimensional space, achieved by using the multidimensional scaling (MDS) algorithm, then the d-dimensional embedding result Y is obtained.

4. Pattern recognition by GOA-SVM In order to realize intelligent fault diagnosis of rolling bearings, it is necessary to identify the type of fault by using a classifier. Support Vector Machine (SVM) classifier, based on statistical learning theory, can solve the small sample problem and exhibit good generalization. For these reasons, it is widely employed in rolling bearing pattern recognition. However, there are two important parameters (i.e. penalty factor c and kernel function σ) that need to be preset first, before using SVM. Parameter c is utilized to adjust the trade-off between the proportion of the misclassified samples and the complexity of the algorithm, while parameter σ, which is the width of the kernel function, 8

controls the complexity of the feature subspace distribution. Obviously, these two parameters have a great influence on the final pattern recognition results of SVM. So, in order to improve the learning ability and generalization ability of the SVM, GOA-SVM is proposed, where GOA is used to determine the best parameters. 4.1. Grasshopper Optimization Algorithm (GOA) The GOA is a novel meta-heuristic algorithm, which simulates the social behavior of the grasshoppers in nature. The flight path of each grasshopper is mainly affected by three factors: social interaction (Si ), gravity force (Gi ) and wind advection (Ai ). Therefore, the mathematical model of grasshopper swarm behavior can be given as: Pi = Si + Gi + Ai

(14)

The calculation expression of social interaction (Si ), in Eq. (14), is computed by: Si =

n X

so (dij )dc ij

j=1,j6=i

pj − pi dij = |pj − pi | , dc , so (dij ) = f e−dij /l − e−dij ij = dij

(15)

where, dij and dc ij represent the distance and unit vector from i-th grasshopper to the j-th grasshopper, respectively. n is the number of grasshoppers, so (r) is the social strength function, f is the attraction, and l indicates the attraction length scale. The calculation expression of gravity force (Gi ), in Eq. (14), is calculated by: Gi = −g ebg

(16)

where, g is the gravitational constant, and ebg represents a uniform vector towards the center of the Earth. The calculation expression of wind advection (Ai ), in Eq. (14), is computed as: Ai = qc ew

(17)

where, q and ec w represent the constant drift and the unity vector of wind direction, respectively. Taking the Eqs. (15) - (17) into Eq. (14), the mathematical model of each grasshopper is derived as: Pi =

n X

so (|pj − pi |)

j=1,j6=i

pj − pi − g ebg + qc ew dij

(18)

This mathematical model is effective in simulating the social behavior of grasshopper swarm. However, to solve the optimization problems, some adjustments need to be made to this model. Pi = C(

n X j=1,j6=i

C

ubd − lbd pj − pi cd so (|pj − pi |) )+T 2 dij

C = Cmax − l

Cmax − Cmin L

(19)

cd is the value of the where, ubd and lbd represent the upper and lower bounds in the d-th dimension, respectively. T target, in the d-th dimension (the current best solution). C denotes the reduction coefficient in narrowing down the comfort, exclusion, and attraction zones. Cmax and Cmin the maximum and minimum values of C, respectively. l and L represent the current iteration number and the maximum number of iterations, respectively. In this paper, Cmax and Cmin are defined as 1 and 0.00004, respectively. The pseudo codes of the GOA are shown in Fig. 6.

9

Initialize the swarm Pi (i=1,2,...,n) Initialize Cmax, Cmin, and maximum number of iterations Calculate the fitness of each search agent T = the best search agent While (l
Fig. 6. Pseudo codes of the GOA algorithm.

4.2. Support Vector Machine (SVM) SVM is a classifier that can effectively solve the small sample and nonlinear classification problems. The core of SVM is to map the input samples into the high-dimensional feature space and then implement linear regression. For a set V = {(vi , yi ) |yi ∈ {−1, 1},

i = 1, 2, . . . , N }, where vi represents sample data and yi represents sample

label of vi , SVM converts the solution of the best hyperplane into the following:  P   min 12 kwk2 + c N  i=1 ξi (ξi ≥ 0)    y (wv + b) ≥ 1 − ξ (i = 1, 2, . . . , N ) i i i  s.t.     c≥0

(20)

where, w denotes the weight vector, b means the partial vector, ξi is the relaxation factor and c represents the penalty factor. Therefore, the classification problem of SVM can be described as:   max W (a) = PN a − 1 PN a a y y K (v , v ) i j i=1 i i,j=1 i j i j 2  s.t. PN a y (0 ≤ a ≤ c; i = 1, 2, . . . , N ) i=1

i i

(21)

i

where, K(vi , vj ) represents a kernel function and satisfies K(vi , vj ) =< φT (vi ) · φ(vj ) >. Consequently, the final classification result of SVM can be discriminated by: N N X X f (x) = sign( ai yi < φT (vi ) · φ(vj ) > +b) = sign( ai yi K(vi , vj ) + b) i=1

(22)

i=1

Common kernel functions include: linear kernel, polynomial kernel, Gaussian kernel and Radial Basis Function kernel (RBF). In this paper, RBF is used as the kernel function in SVM, and its expression is as follows: K(vi , vj ) = exp(−||vi − vj ||2 /2σ 2 )

(23)

where, σ is the kernel parameter. The accuracy of the SVM prediction model is significantly affected by the penalty coefficient c and the kernel parameter σ. The parameter c is employed to balance the generalization and classification errors. If the value of c is too high, serious over-fitting problems will occur, while the generalization ability of SVM will be reduced. On the contrary, if the value of c is too low, the model of SVM will easily fall into under-fitting condition. In addition, the kernel parameter σ affects the distribution of input samples, in the kernel space. Therefore, the GOA is applied to find the optimal parameters values (i.e., cbest and σbest ) for the SVM and build the best SVM prediction model. 4.3. Grasshopper Optimization Algorithm based Support Vector Machine (GOA-SVM) In view of the advantages of GOA and SVM, a novel GOA-SVM classifier is presented, in this paper, which can be described as follows: 10

(1) Data preprocessing. The data set is divided into training set and test set. The training set and test set are normalized to [0, 1], using the following formula: v 0 = (v−min )/(max − min), where, v and v 0 represent the original and the normalized eigenvalue, respectively; max and min represent the maximum eigenvalue and minimum eigenvalue, respectively. (2) The parameters of the GOA are initialized. The number of populations is set to 20, the parameter Cmax to 1, the parameter Cmin to 0.00004 and the maximum number of iterations to 100. Since the two parameters (c, σ) in SVM need to be optimized, the position of each grasshopper is defined as (c, σ). Furthermore, the lower and upper bounds of each grasshopper position are defined as (0.001, 0.001) and (100, 100), respectively. (3) The fitness value of each grasshopper is calculated. The key to the GOA-SVM classifier is the selection of fitness function. To evaluate the quality of each grasshopper, the average error recognition rate of training samples, achieved by following a three-fold cross validation procedure, is defined as the fitness function. Therefore, the parameters optimization problem of SVM is formulated as a problem of minimizing the fitness function. (4) The grasshopper, with the best fitness value, at the current iteration, is selected, while its location is considered as the current target location T . (5) The reduction coefficient C is updated using Eq. (19), while the distance between the grasshoppers is normalized to [1, 4], in each iteration. Following, the position of each grasshopper is updated, according to Eq. (19), while the fitness value of each update grasshopper is computed. However, if the fitness value of the updated grasshopper is better than that of the target, this updated grasshopper replaces the previous grasshopper. Otherwise, the previous grasshopper continues to be updated. (6) It is determined whether the iteration stop condition is satisfied. Namely, if the maximum number of iterations is reached, this loop is terminated and the best target position (cbest , σbest ) is the output. Otherwise, the algorithm returns to step (2), until the iteration stop condition is satisfied. (7) The (cbest , σbest ) is applied, to establish the best SVM prediction model, followed by the identification of the test data set. The flowchart of GOA-SVM is illustrated in Fig. 7. 4.4. Experimental analysis of wine data set To verify the performance of the GOA-SVM classifier, the wine data set (specific description is given in Table 1), in the UCI database [40], is used in this experiment, while the GOA-SVM is also compared to the SVM, the Particle Swarm Optimization based Support Vector Machine (PSO-SVM), the Firefly Algorithm based Support Vector Machine (FA-SVM), the Chicken Swarm Optimization based Support Vector Machine (CSO-SVM) and the Gray Wolf Optimization based Support Vector Machine (GWO-SVM). In order to even things out, the fitness functions of all optimization based SVM classifiers are defined as a GOASVM classifier, while the experiment is repeated 50 times. Table 2 lists the parameters of all classifiers. Moreover, these classifiers are developed in the Matlab R2017a environment and executed on a PC with Intel Core i5-3230, 2.60 GHz, 12 GB RAM and Windows 10. The average fitness value curves and the average recognition rate of all classifiers,

11

Initialize the grasshoppers swarm (Number of search agents, Cmax, Cmin, Number of iterations)

Normalization

Training data set

Calculate the fitness of the location of search agent

T = the best search agent

l=1

l=l+1

Update the C

Update the position of the search agent according to the C

Stopping Criterion

Update the T if there is a better solution

Calculate the fitness of the update search agent

Data set

Optimized parameters (Cbest, sbest)

Normalization Test data set

SVM prediction model

Output the results

Fig. 7. The flowchart of GOA-SVM. Table 1 List of the wine data set. Number of

Features

Classes

samples 178

13

3

Number of samples per

Number of training

Number of test

class

samples per class

samples per class

59, 71, 48

20, 24, 16

39, 47, 32

after repeating the experiments on the wine data set, 50 times, are illustrated in Figs. 8 and 9, respectively. 10

2.5

PSO-SVM

Fitness value / %

CSO-SVM

2.0

FA-SVM 7

1.5

GW O-SVM GOA-SVM

1.0 60

70

80

90

100

4

1 0

20

40

60

80

100

No. of iterations

Fig. 8. The average fitness value curves of five optimization based SVM classifiers.

12

Table 2 Initial parameters of the classifiers. Classifier

Parameter

Value

SVM

c

1

σ

6

Acceleration constants

[2, 2]

PSO-SVM

FA-SVM

CSO-SVM

GWO-SVM

GOA-SVM

Inertia

[0.4, 0.9]

Number of search agents

20

Number of iterations

100

Alpha

0.25

Beta

0.2

Gamma

1

Number of search agents

20

Number of iterations

100

Number of groups

10

Number of roosters

4

Number of hens

12

Number of chicks

4

Number of mother hens

2

FL

[0.5, 0.9]

Number of search agents

20

Number of iterations

100

a

Min=0, max=2

Number of search agents

20

Number of iterations

100

Cmax

1

Cmin

0.00004

Number of search agents

20

Number of iterations

100

GOA-SVM GWO-SVM FA-SVM CSO-SVM PSO-SVM SVM 94

95

96

97

Accuracy / %

Fig. 9. The average recognition accuracy using six classifiers.

Fig. 8 shows that the final average fitness value of the GOA-SVM classifier (i.e., the average error recognition rate) is obviously lower than that of other optimization based SVM classifiers. This confirms that GOA is effective and feasible, for SVM parameters optimization. In addition, according to Fig. 9, the following results can be derived. First, the average recognition accuracy of the test samples, by the optimization based SVM classifiers, are significantly higher than that of the original SVM classifier, which indicates that the optimization based SVM classifiers can overcome the parameters selection problem of the original SVM classifier. Second, compared to other optimization based SVM classifiers, GOA-SVM classifier has the highest average recognition accuracy for the test samples, verifying 13

its superiority to other optimization based SVM classifiers. 5. Fault diagnosis process Based on the superiorities of GRCMSE, S-Isomap and GOA-SVM, a novel rolling bearing fault diagnosis method is presented (Fig. 10 is the flowchart of the proposed rolling bearing fault diagnosis method), as follows: Rolling bearing

Data collection unit

Experiment system

Raw vibration signals

Features extraction by using GRCMSE

Dimensionality reduction by using S-Isomap

Pattern recognition by using GOA-SVM

Fig. 10. The flowchart of the proposed rolling bearing fault diagnosis method.

14

(1) Acceleration sensors are used to collect the vibration signals of the rolling bearings, under different working conditions. (2) The GRCMSE algorithm is applied, to calculate the entropy values of the rolling bearing signals, while the original high-dimensional feature set is constructed. (3) The S-Isomap manifold learning algorithm is employed, to reduce the dimension of the GRCMSE features set, deriving the low-dimensional features set. (4) The low-dimensional fault features set, achieved by S-Isomap, is randomly divided into a training sample set and a test sample set. Next, the training samples are input to the GOA-SVM classifier, for establishing the best SVM prediction model, while the test samples are input to the SVM prediction model, to identify the working conditions.

6. Experimental study The experimental data of rolling bearings is collected from a Drivetrain Diagnostics Simulator (DDS) and the complete experimental platform is shown in Fig. 11. In this experiment, the working conditions of the speed system are set to 20 Hz - 0 V, while the sampling frequency is set to 2000 Hz. Therefore, four types of working condition data sets of rolling bearings are collected, including three fault conditions and one normal operation condition, as described in Table 3. In addition, the time domain waveforms of the four working condition types of rolling bearings are given in Fig. 12.

Fig. 11. The experimental platform.

Table 3 Description of four working conditions of rolling bearings. Working condition

Description

Class

Data

Number

points 1

3000

type Normal

Health

of

Number of training

Number of test

samples

samples

samples

100

10

90

Inner race crack

Crack occurs in the inner race

2

3000

100

10

90

Outer race crack

Crack occurs in the outer race

3

3000

100

10

90

Ball crack

Crack occurs in the ball

4

3000

100

10

90

15

Acceleration

0.01 0.00 -0.01 0.0

Acceleration

Acceleration

-0.04 0.0

Acceleration

Normal 0.04 0.00 0.5

1.0

1.5

1.0

1.5

1.0

1.5

1.0

1.5

Inner race crack

0.5 Outer race crack

0.01 0.00 -0.01 0.0

0.5

Ball crack 0.02 0.00 -0.02 0.0

0.5

time /s

Fig. 12. The time domain waveforms of the four working conditions of rolling bearings.

Obviously, it is not easy to distinguish the working conditions of rolling bearings, based on the time domain waveforms, therefore, the proposed method is applied to the fault diagnosis process. First, the entropy values of rolling bearing samples are extracted, by using GRCMSE algorithm (the parameters are set as follows: m = 2, r = 0.15SD, s = 25). The mean entropy curves of the four working conditions, at each scale, are plotted in Fig. 13. 2

25 24

outer race crack

5

1.6

22

inner race crack

4

1.8

23

normal

3

ball crack 6

1.4 1.2

21

7 1.0

20

8

0.8

19

9

18

10 17

11 16

12 15

14

13

Fig. 13. The entropy mean curves of four working conditions by using GRCMSE.

According to Fig. 13, for the rolling bearing samples under normal operating condition, the entropy values of GRCMSE decrease, as the scales factor increases. Conversely, for the rolling bearing samples under fault conditions, the entropy values of GRCMSE are first reduced and then tend to be constant, over a large value range of scale factor. These findings indicate that GRCMSE can effectively monitor the occurrence of faults. Also, this convergence to a constant value is due to the fact that, for the uncorrelated normal signals, the important information is included only on a low scale factor. However, for the correlated fault signals, the important information is also included on the high scale factor. In addition, it is noted that, the GRCMSE entropy values of the four working conditions are different, for different scales. For example, when the scale factor is 2, the relationship between the entropy values of the four operating conditions is: normal > inner race crack > ball crack > outer race crack. Nevertheless, when the scale factor is 20, the relationship between these entropy values of four operating conditions is: normal > ball crack > inner race crack > outer race crack. These observations show that the important fault characteristics information is contained also on other scales. However, the mean entropy curve of the inner race crack is close to that of the ball crack, therefore, it is difficult to distinguish between these two working conditions. Moreover, the fault features set, extracted using GRCMSE, contains redundant information, so that, if it is directly input to the classifier for pattern recognition, poor recognition results 16

Table 4 The 3-dimensional eigenvectors of dimensionality reduction by using S-Isomap. Working condition type

Sample serial number

3-dimensional eigenvector

Normal

5

(1.994, -0.317, -0.109)

10

(1.533, -0.226, -0.081)

15

(1.314, -0.177, -0.066)

Inner race crack

Outer race crack

Ball crack

5

(0.121, 0.421, 0.467)

10

(0.080, -0.153, 0.336)

15

(0.160, -0.007, 0.348)

5

(-1.119, -0.013, 0.002)

10

(-1.293, -0.169, 0.008)

15

(-1.452, -0.539, -0.106)

5

(-0.147, 0.149, -0.316)

10

(-0.149, 0.102, -0.337)

15

(-0.044, 0.297, -0.262)

may occur. Thus, the excellent manifold learning algorithm, called S-Isomap, is employed, to reduce the dimension of the original GRCMSE features set, while the result of the low-dimensional features set is shown in Fig. 14. The parameters of S-Isomap are set as follows: d = 3, K = 40 and τ = 0.4.

0.6

normal inner race crack outer race crack

3rd coordinates

0.4

ball crack

0.2

0.0

-0.2

-0.4 -1.5

-1.0

-4

1s

-2 t c oo rd in a

-0.5

0.0

0 te

s

0.5

2

1.0

2

n

d

co

o

rd

in

a

te

s

Fig. 14. The distribution of the 3-dimensional feature set after dimensionality reduction of S-Isomap.

Fig. 14 shows that, in the 3-dimensional space, the four kinds of samples are clearly separated from each other, while the aggregation of each working condition is better. Table 4 also lists several 3-dimensional eigenvectors, obtained using the S-Isomap algorithm. Next, the low-dimensional feature set is input to the GOA-SVM classifier, for pattern recognition, while the results are illustrated in Fig. 15 and Table 5. The parameters of GOA-SVM classifier are given in Table 2. Both, Fig. 15 and Table 5, show that the average recognition accuracy of 360 test samples reached 99.72%, while only one normal working condition sample is misclassified as outer race crack type. The following three fields are employed, to prove the advantages of the proposed rolling bearing fault diagnosis method, based on GRCMSE, S-Isomap and GOA-SVM. (1) The feature extraction effect of GRCMSE is compared to GMSE (the parameters are set as follows: m = 2, r = 0.15SD, s = 25), while the results are illustrated in Fig. 16. It is obvious that the mean entropy curves of the GMSE and GRCMSE, for each working condition, are relatively similar. However, compared to GMSE, the mean entropy curves, extracted using GRCMSE, are smoother. In addition, for each working condition, GRCMSE

17

Class label

4

3

2

GOA-SVM outputs

1

Desired outputs 0

90

180 No. of test samples

270

360

Fig. 15. The outputs of test samples of the proposed method using GOA-SVM. Table 5 The fault diagnosis result using the proposed method. Working

Normal

Inner

Outer

Ball

condition

(groups)

race crack

race crack

crack

accuracy

(groups)

(groups)

(groups)

(%)

type

Fault type

Recognition

Recognition

89

0

1

0

Normal

98.89

results

0

90

0

0

Inner race crack

100

0

0

90

0

Outer race crack

100

0

0

0

90

Ball crack

100

demonstrates lower error values (i.e., the standard deviation) than GMSE, at most scales. These phenomena show that GRCMSE, considering different coarse-grained time series at the same scale, can improve the stability of the entropy calculation.

1.2

0.08

1.2

0.07

1.1

0.06 1.0

0.05

10

15

20

25

1.0

0.03

error value of GRCMSE

0.09

1.0

0.08 0.07

0.8

entropy value of GRCMSE

1.25

error value of GMSE

0.10

error value of GRCMSE

0.09 1.20

0.08 0.07

1.15

0.06 0.05

0.04 5

0.10

Error value

Error value

Error value

0.09

1.3

error value of GRCMSE

1.30 entropy value of GMSE

0.11

error value of GMSE

error value of GMSE

0.09

0.12 1.2

entropy value of GRCMSE

5

10

15

No. of scale

No. of scale

(a)

(b)

20

25

0.9

0.06

0.8

0.05

Entropy value

0.10

0.11

Entropy value

1.4

0.11

0.12

1.4

entropy value of GMSE

entropy value of GRCMSE

Entropy value

0.12

Entropy value

1.6

error value of GRCMSE

1.5 entropy value of GMSE

0.10

error value of GMSE

0.13

0.08

0.11

entropy value of GRCMSE

Error value

1.8 entropy value of GMSE

0.14

1.10

0.04 5

10

15

No. of scale

(c)

20

25

0.6

0.03

5

10

15

20

25

1.05

No. of scale

(d)

Fig. 16. GRCMSE and GMSE analysis results: (a) normal; (b) inner race crack; (c) outer race crack and (d) ball crack.

Next, the features sets, extracted by using GMSE and GRCMSE, are input to the GOA-SVM classifier, for pattern recognition, while the results are depicted in Figs. 17 (a) and (b) and Table 6. It can be found that 50 test samples of the GMSE feature set are misclassified, while the average recognition accuracy is 86.11%. Nevertheless, only 34 test samples of the GRCMSE feature set result in misclassification, while the average recognition accuracy reaches 90.56%, which is 4.45% higher than that of the GMSE. The above analysis proves that GRCMSE provides superior feature extraction results than GMSE. (2) In order to verify the necessity of using the S-Isomap manifold learning algorithm, to reduce the dimension of the GRCMSE fault feature set, the fault recognition accuracy in the case of dimensionality reduction and the case of non-dimensionality reduction are compared (Table 7). Obviously, when the feature set, obtained by the GRCMSE, with high-dimensional and redundant characteristics, is directly input to the GOA-SVM classifier, for pattern recognition, the classifier cannot effectively determine the right fault types of test samples. However, 18

4

3

3

Class label

Class label

4

2

GOA-SVM outputs

1

90

180 No. of test samples

270

GOA-SVM outputs

1

Desired outputs 0

2

360

Desired outputs 0

90

180 No. of test samples

(a)

270

360

(b)

Fig. 17. The outputs of GMSE and GRCMSE using GOA-SVM. (a) the outputs of test samples of GMSE using GOA-SVM; (b) the outputs of test samples of GRCMSE using GOA-SVM. Table 6 Comparison of the fault diagnosis results achieved by feature extraction method with GMSE and GRCMSE. Feature

Normal

Inner

Outer

Ball

extraction

(groups)

race crack

race crack

crack

accuracy

(groups)

(groups)

(groups)

(%)

method

Fault type

Recognition

Recognition

88

0

2

0

Normal

97.78

results of

0

66

0

24

Inner race crack

73.33

GMSE

0

0

89

1

Outer race crack

98.89

0

22

1

67

Ball crack

74.44

Recognition

89

1

0

0

Normal

98.89

results of

0

70

0

20

Inner race crack

77.78

GRCMSE

0

0

89

1

Outer race crack

98.89

0

11

1

78

Ball crack

86.67

if the high-dimensional fault feature set is compacted into a low-dimensional fault feature set, as achieved by S-Isomap, the average fault recognition accuracy of test samples increases by 9.16%. This is due to the fact that, S-Isomap manifold learning algorithm can effectively mine the low-dimensional components, embedded in high-dimensional space. Therefore, S-Isomap can easily distinguish the working conditions of test samples and improve the recognition effect of GOA-SVM classifier. (3) In the following comparison, the pattern recognition performance of GOA-SVM is compared to that of SVM, PSO-SVM, FA-SVM, CSO-SVM and GWO-SVM. The parameters settings of each classifier are given in Table 2, while the identification results of combining different classifiers with different feature extraction methods (e.g., GMSE, GRCMSE, and GRCMSE+S-Isomap) are shown in Fig. 18. Fig. 18 shows that, the relationship between the fault recognition accuracy of each classifier, for GMSE, GRCMSE and GRCMSE+S-Isomap feature sets is: GRCMSE+S-Isomap > GRCMSE > GMSE. This proves the superiority of the feature extraction method, combining GRCMSE and S-Isomap. Second, the average fault recognition accuracy of the optimization based SVM classifiers, according to three feature extraction methods, is significantly higher than that of the original SVM classifier. Namely, compared to SVM classifier, the average recognition accuracy of PSO-SVM, FA-SVM, CSO-SVM, GWO-SVM and GOA-SVM classifier is increased by 1.95%, 3.34%, 3.89%, 2.87% and 5.93%, respectively. This verifies that the optimization based SVM classifier can solve the

19

Table 7 Results comparison by using S-Isomap dimensionality reduction method versus not using S-Isomap dimensionality reduction method. Whether

Normal

Inner

Outer

Ball

reducing

(groups)

race crack

race crack

crack

accuracy

(groups)

(groups)

(groups)

(%)

dimension Yes

No

Fault type

Recognition

89

0

1

0

Normal

98.89

0

90

0

0

Inner race crack

100

0

0

90

0

Outer race crack

100

0

0

0

90

Ball crack

100

89

1

0

0

Normal

98.89

0

70

0

20

Inner race crack

77.78

0

0

89

1

Outer race crack

98.89

0

11

1

78

Ball crack

86.67

parameters selection problem of the SVM classifier. In addition, compared to other parameters optimization based SVM classifiers, GOA-SVM classifier has the highest fault recognition accuracy, according to all feature extraction methods, while it has the highest average fault recognition accuracy. This confirms that the performance of GOA is superior to PSO, FA, CSO and GWO, in SVM parameters optimization.

100 SVM

Accuracy / %

PSO-SVM 90 FA-SVM CSO-SVM 80 GWO-SVM GOA-SVM 70 GMSE

GRCMSE

GRCMSE+ S-Isomap

Average

Fig. 18. Recognition accuracy (%) of combining different classifiers with different feature extraction methods.

The above-mentioned rolling bearing fault diagnosis experimental results fully confirm the superiority of the proposed fault diagnosis method, based on GRCMSE, S-Isomap and GOA-SVM. They highlight that the feature extraction effect of GRCMSE is significantly better than that of GMSE. The fault pattern recognition, in the low-dimensional feature set, obtained by S-Isomap, performs clearly better than in the original GRCMSE feature set. It also becomes obvious that the pattern recognition effect of GOA-SVM is obviously better than that of SVM, PSO-SVM, FA-SVM, CSO-SVM and GWO-SVM classifiers.

7. Conclusions In this paper, a novel rolling bearing fault diagnosis method, based on GRCMSE, S-Isomap and GOA-SVM, is proposed. According to the presented method, the GRCMSE is used to extract the features of rolling bearing vibration signals, at different scales. Moreover, in order to obtain the low-dimensional and fault sensitive feature set, a superior supervised manifold learning algorithm, named S-Isomap, is applied, to reduce the dimension of the original GRCMSE

20

feature set. Finally, the low-dimensional feature set is input to the GOA-SVM classifier, to automatically determine the various working condition types. Based on rolling bearing experimental data, for fault diagnosis, the results show that the proposed method can appropriately and effectively diagnose the different working conditions. The novelties and contributions of this paper mainly include: (1) GRCMSE, overcoming the uncertain or unreliable estimations of GMSE entropy values, in the coarse-grained process, is proposed, to measure the complexity of time series. The experimental analysis of the noise signals and the rolling bearing vibration signals confirm that, compared to GMSE, the mean entropy curves, obtained by GRCMSE, are smoother, while entropy values are more accurate and stable. (2) S-Isomap is introduced, to compress high-dimensional feature sets into low-dimensional feature space. Rolling bearing experiment shows that, compared to the full dimensionality case, S-Isomap can easily distinguish the working condition types, thus improving the recognition accuracy. (3) In view of the advantages of GOA and SVM, the GOA-SVM is presented, for pattern recognition. Experimental analysis of wine data set and rolling bearings feature sets shows that GOA-SVM classifier is superior to SVM as well as some existing optimization based SVM classifiers. In this study, the proposed method proves suitable for the fault diagnosis of rolling bearings. Further work will be carried out to demonstrate the effectiveness of the proposed method for other equipment and datasets.

Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant No. 51775114, 51875105 and 51275092) and Fujian Provincial Industrial Robot Basic Components Technology Research and Development Center (Grant No. 2014H21010011).

References [1] Z. Wang, W. Du, J. Wang, J. Zhou, X. Han, Z. Zhang, L. Huang, Research and application of improved adaptive MOMEDA fault diagnosis method, Measurement 140 (2019) 63–75. doi:10.1016/j.measurement.2019.03.033. [2] K. Chi, J. Kang, R. Bajric, X. Zhang, Bearing fault diagnosis through stochastic resonance by full-wave signal construction with half-cycle delay, Measurement 148 (2019) 106893. doi:10.1016/j.measurement.2019.106893. [3] X. Yan, Y. Liu, M. Jia, Research on an enhanced scale morphological-hat product filtering in incipient fault detection of rolling element bearings, Measurement 147 (2019) 106856. doi:10.1016/j.measurement.2019. 106856. [4] K. Worden, W. J. Staszewski, J. J. Hensman, Natural computing for mechanical systems research: A tutorial overview, Mechanical Systems and Signal Processing 25 (1) (2011) 4–111. doi:10.1016/j.ymssp.2010.07.013. [5] Z. Han, B. Xu, X. Zhu, H. Jiao, Research on multi-fault diagnosis of rotor based on Approximate entropy and EEMD, China Mechanical Engineering 27 (16) (2016) 2186–2189. doi:10.3969/j.issn.1004-132X.2016.16. 010. [6] J. Cui, Q. Zheng, Y. Xin, C. Zhou, Q. Wang, N. Zhou, Feature extraction and classification method for switchgear faults based on sample entropy and cloud model, IET Generation, Transmission and Distribution 11 (11) (2017) 2938–2946. doi:10.1049/iet-gtd.2016.1459.

21

[7] X. Yan, M. Jia, Z. Zhao, A novel intelligent detection method for rolling bearing based on IVMD and instantaneous energy distribution-permutation entropy, Measurement 130 (2018) 435–447. doi:10.1016/j.measurement.2018. 08.038. [8] J. Zheng, H. Pan, J. Cheng, Rolling bearing fault detection and diagnosis based on composite multiscale fuzzy entropy and ensemble support vector machines, Mechanical Systems and Signal Processing 85 (2017) 746–759. doi:10.1016/j.ymssp.2016.09.010. [9] M. Costa, A. L. Goldberger, C. K. Peng, Multiscale entropy analysis of complex physiologic time series, Physical Review Letters 89 (6) (2002) 068102. doi:10.1103/PhysRevLett.89.068102. [10] L. Zhang, G. Xiong, H. Liu, H. Zou, W. Guo, Bearing fault diagnosis using multi-scale entropy and adaptive neurofuzzy inference, Expert Systems with Applications 37 (8) (2010) 6077–6085. doi:10.1016/j.eswa.2010.02.118. [11] Y. Li, J. Jiang, Fault diagnosis of bearing based on LMD and MSE, in: 2017 Prognostics and System Health Management Conference (PHM-Harbin), IEEE, 2017, pp. 1–4. [12] N. K. Hsieh, W. Y. Lin, H. T. Young, High-speed spindle fault diagnosis with the empirical mode decomposition and multiscale entropy method, Entropy 17 (4) (2015) 2170–2183. doi:10.3390/e17042170. [13] M. Costa, A. Goldberger, Generalized multiscale entropy analysis: application to quantifying the complex volatility of human heartbeat time series, Entropy 17 (3) (2015) 1197–1203. doi:10.3390/e17031197. [14] S. D. Wu, C. W. Wu, S. G. Lin, K. Y. Lee, C. K. Peng, Analysis of complex time series using refined composite multiscale entropy, Physics Letters A 378 (20) (2014) 1369–1374. doi:10.1016/j.physleta.2014.03.034. [15] X. Qiao, J. Bao, H. Zhang, F. Wan, D. Li, fvUnderwater sea cucumber identification based on principal component analysis and support vector machine, Measurement 133 (2019) 444–455. doi:10.1016/j.measurement.2018. 10.039. [16] N. Wang, Q. Li, A. A. El-Latif, J. Peng, X. Niu, An enhanced thermal face recognition method based on multiscale complex fusion for Gabor coefficients, Multimedia Tools and Applications 72 (3) (2014) 2339–2358. doi:10.1007/s11042-013-1551-4. [17] E. Pacola, V. Quandt, P. Liberalesso, S. Pichorim, F. Schneider, H. Gamba, A versatile EEG spike detector with multivariate matrix of features based on the linear discriminant analysis, combined wavelets, and descriptors, Pattern Recognition Letters 86 (2017) 31–37. doi:10.1016/j.patrec.2016.12.018. [18] J. Peng, Q. Li, A. A. El-Latif, X. Niu, Linear discriminant multi-set canonical correlations analysis (LDMCCA): an efficient approach for feature fusion of finger biometrics, Multimedia Tools and Applications 74 (13) (2015) 4469–4486. doi:10.1007/s11042-013-1817-x. [19] M. M. Nezhad, E. Gironacci, M. Rezania, N. Khalili, Stochastic modelling of crack propagation in materials with random properties using isometric mapping for dimensionality reduction of nonlinear data sets, International Journal for Numerical Methods in Engineering 113 (4) (2018) 656–680. doi:10.1002/nme.5630. [20] J. Ning, W. Cui, C. Chong, H. Ouyang, C. Chen, B. Zhang, Feature recognition of small amplitude hunting signals based on the MPE-LTSA in high-speed trains, Measurement 131 (2019) 452–460. doi:10.1016/j.measurement. 2018.08.035.

22

[21] J. Zhang, Z. Yin, R. Wang, Pattern classification of instantaneous cognitive task-load through GMM clustering, Laplacian Eigenmap, and ensemble SVMs, IEEE/ACM Transactions on Computational Biology and Bioinformatics 14 (4) (2017) 947–965. doi:10.1109/TCBB.2016.2561927. [22] Y. Liu, Z. Yu, M. Zeng, Y. Zhang, An improved LLE algorithm based on iterative shrinkage for machinery fault diagnosis, Measurement 77 (2016) 246–256. doi:10.1016/j.measurement.2015.09.007. [23] J. Zheng, Z. Jiang, H. Pan, Sigmoid-based refined composite multiscale fuzzy entropy and t-SNE based fault diagnosis approach for rolling bearing, Measurement 129 (2018) 332–342. doi:10.1016/j.measurement.2018. 07.045. [24] Q. Jiang, M. Jia, J. Hu, F. Xu, Machinery fault diagnosis using supervised manifold learning, Mechanical Systems and Signal Processing 23 (7) (2009) 2301–2311. doi:10.1016/j.ymssp.2009.02.006. [25] Z. Su, B. Tang, Z. Liu, Y. Qin, Multi-fault diagnosis for rotating machinery based on orthogonal supervised linear local tangent space alignment and least square support vector machine, Neurocomputing 157 (2015) 208– 222. doi:10.1016/j.neucom.2015.01.016. [26] X. Geng, D. Zhan, Z. Zhou, Supervised nonlinear dimensionality reduction for visualization and classification, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics) 35 (6) (2005) 1098–1107. doi: 10.1109/TSMCB.2005.850151. [27] M. Uyar, S. Yildirim, M. T. Gencoglu, An expert system based on S-transform and neural network for automatic classification of power quality disturbances, Expert Systems with Applications 36 (3) (2009) 5962–5975. doi: 10.1016/j.eswa.2008.07.030. [28] W. Chine, A. Mellit, V. Lughi, A. Malek, G. Sulligoi, A. M. Pavan, A novel fault diagnosis technique for photovoltaic systems based on artificial neural networks, Renewable Energy 90 (2016) 501–512.

doi:

10.1016/j.renene.2016.01.036. [29] M. Omid, Design of an expert system for sorting pistachio nuts through decision tree and fuzzy logic classifier, Expert Systems with Applications 38 (4) (2011) 4339–4347. doi:10.1016/j.eswa.2010.09.103. [30] X. Yan, M. Jia, A novel optimized SVM classification algorithm with multi-domain feature and its application to fault diagnosis of rolling bearing, Neurocomputing 331 (2018) 47–64. doi:10.1016/j.neucom.2018.05.002. [31] R. Liu, B. Yang, X. Zhang, S. Wang, X. Chen, Time-frequency atoms-driven support vector machine method for bearings incipient fault diagnosis, Mechanical Systems and Signal Processing 75 (2016) 345–370. doi:10.1016/ j.ymssp.2015.12.020. [32] S. Bansal, S. Sahoo, R. Tiwari, D. J. Bordoloi, Multiclass fault diagnosis in gears using support vector machine algorithms based on frequency domain data, Measurement 46 (9) (2013) 3469–3481. doi:10.1016/j.measurement. 2013.05.015. [33] R. Fang, L. Zheng, H. Ma, D. Huang, Fault diagnosis for rotor of induction machine based on MCSA and SVM, Chinese Journal of Scientific Instrument 28 (2) (2007) 252–257. doi:10.3321/j.issn:0254-3087.2007.02.011. [34] V. P. Kour, S. Arora, Particle swarm optimization based support vector machine (P-SVM) for the segmentation and classification of plants, IEEE Access 7 (2019) 29374–29385. doi:10.1109/ACCESS.2019.2901900.

23

[35] S. Saremi, S. Mirjalili, A. Lewis, Grasshopper optimisation algorithm: theory and application, Advances in Engineering Software 105 (2017) 30–47. doi:10.1016/j.advengsoft.2017.01.004. [36] S. Ch, S. K. Sohani, D. Kumar, A. Malik, B. R. Chahar, A. K. Nema, B. K. Panigrahi, R. C. Dhiman, A support vector machine-firefly algorithm based forecasting model to determine malaria transmission, Neurocomputing 129 (2014) 279–288. doi:10.1016/j.neucom.2013.09.030. [37] L. Li, C. Lv, M. Tseng, M. Song, Renewable energy utilization method: a novel insulated gate bipolar transistor switching losses prediction model, Journal of Cleaner Production 176 (2018) 852–863. doi:10.1016/j.jclepro. 2017.12.051. [38] X. Bian, L. Zhang, Z. Du, J. Chen, J. Zhang, Prediction of sulfur solubility in supercritical sour gases using grey wolf optimizer-based support vector machine, Journal of Molecular Liquids 261 (2018) 431–438. doi: 10.1016/j.molliq.2018.04.070. [39] H. Azami, J. Escudero, Improved multiscale permutation entropy for biomedical signal analysis: interpretation and application to electroencephalogram recordings, Biomedical Signal Processing and Control 23 (2016) 28–41. doi:10.1016/j.bspc.2015.08.004. [40] University of California, UCI repository of machine learning databases, http://archive.ics.uci.edu/ml/ machine-learning-databases/wine/.

24

Credit Author Statement Zhenya Wang: Conceptualization, Methodology, Software, Validation, Data Curation, Writing-Original Draft. Ligang Yao: Resources, Writing-Review & Editing, Project administration, Visualization, Supervision, Funding acquisition. Yongwu Cai: Formal analysis, Investigation, Writing-Review & Editing.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Declaration of interests：

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Highlights: 

GRCMSE is presented proposed for measuring the complexity and dynamic change of time series.



GOA-SVM is proposed to automatically identify the rolling bearing fault types.



A novel fault detection scheme for rolling bearing is proposed based on GRCMSE, S-Isomap and GOA-SVM.



Efficiency of the proposed method is validated on experimental data.

Credit Author Statement Zhenya Wang: Conceptualization, Methodology, Software, Validation, Data Curation, Writing-Original Draft. Ligang Yao: Resources, Writing-Review & Editing, Project administration, Visualization, Supervision, Funding acquisition. Yongwu Cai: Formal analysis, Investigation, Writing-Review & Editing.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Declaration of interests：

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Highlights: 

GRCMSE is presented proposed for measuring the complexity and dynamic change of time series.



GOA-SVM is proposed to automatically identify the rolling bearing fault types.



A novel fault detection scheme for rolling bearing is proposed based on GRCMSE, S-Isomap and GOA-SVM.



Efficiency of the proposed method is validated on experimental data.