Kurtosis forecasting of bearing vibration signal based on the hybrid model of empirical mode decomposition and RVM with artificial bee colony algorithm

Accepted Manuscript Kurtosis Forecasting of Bearing Vibration Signal Based on the Hybrid Model of Empirical Mode Decomposition and RVM with Artificial Bee Colony Algorithm Sheng-wei Fei PII: DOI: Reference:

S0957-4174(14)00746-5 http://dx.doi.org/10.1016/j.eswa.2014.11.047 ESWA 9702

To appear in:

Expert Systems with Applications

Please cite this article as: Fei, S-w., Kurtosis Forecasting of Bearing Vibration Signal Based on the Hybrid Model of Empirical Mode Decomposition and RVM with Artificial Bee Colony Algorithm, Expert Systems with Applications (2014), doi: http://dx.doi.org/10.1016/j.eswa.2014.11.047

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Kurtosis Forecasting of Bearing Vibration Signal Based on the Hybrid Model of Empirical Mode Decomposition and RVM with Artificial Bee Colony Algorithm

Sheng-wei Fei School of Mechanical Engineering,Donghua University,Shanghai 201620,China Email: [email protected]

Abstract Accurate prediction for kurtosis of bearing vibration signal is helpful to find out the fault of bearing as soon as possible. As it is difficult to obtain an appropriate embedding dimension in creating directly the prediction model of kurtosis of bearing vibration signal by relevance vector machine(RVM), this study is to propose a new hybrid model of empirical mode decomposition and RVM with artificial bee colony algorithm(EMD-ABCRVM) for kurtosis forecasting of bearing vibration signal. The seven decomposed signals with different frequency range can be obtained by empirical mode decomposition for kurtosis of bearing vibration signal. The prediction models of the seven decomposed signals can be established by RVM with their each appropriate embedding dimension, and artificial bee colony algorithm (ABC) is used to select the appropriate kernel parameters of their RVM models. Thus, each RVM model of the seven decomposed signals has appropriate embedding dimension and kernel parameter. In order to show the superiority of the proposed EMD-ABCRVM method, the RVM models

with several different embedding dimensions and Gaussian RBF kernel parameters are used to compare with the proposed EMD-ABCRVM method. The experimental results show that it is feasible for the proposed combination scheme to improve the prediction accuracy of RVM for kurtosis of bearing vibration signal. Keywords: kurtosis prediction of bearing vibration signal; empirical mode decomposition; embedding dimension; artificial bee colony algorithm

1 Introduction Kurtosis of bearing vibration signal is a very important indicator which can reflect the operating state of bearing(Borghesani et al., 2014, Guo et al., 2012 and Lee and Seo,2013). By the prediction for kurtosis of bearing vibration signal, the development trend of bearing can be found, and the fault of bearing can be predicted. Thus, accurate prediction for kurtosis of bearing vibration signal is helpful to find out the fault of bearing as soon as possible. Artificial neural networks and support vector machine(SVM) are two kinds of popular intelligent learning techniques, which have been widely applied to solve the nonlinear prediction problems. Practicability of artificial neural networks is limited due to the shortcomings of over-fitting and falling into local extremum easily (Jiang and Zhao,2013 and Naguib and Darwish,2012). Compared with artificial neural networks, support vector machine which is based on the structural risk minimization principle has the better generalization performance (Na’imi et al.,2014,Suganyadevi and Babulal,2014 and Yao et al.,2015 ). Relevance vector machine(RVM)

is

an

intelligent

learning

technique

based

on

sparse

Bayesian

framework(Tipping,2001 and Valente et al., 2011), which can obtain more convenience than SVM because there is no need to set the penalty parameter in RVM. In addition, the number of relevance vectors in RVM is much smaller than that of support vectors in SVM, which makes RVM have a sparser representation compared with SVM, hence, RVM has better generalization ability than SVM. As it is difficult to obtain an appropriate embedding dimension in creating directly the prediction model of kurtosis of bearing vibration signal by relevance vector machine, this study is to propose a new hybrid model of empirical mode decomposition and RVM with artificial bee colony algorithm(EMD-ABCRVM) for kurtosis forecasting of bearing vibration signal. In this study, kurtosis of bearing vibration signal can be decomposed into seven decomposed signals with different frequency range by empirical mode decomposition, the prediction models of the seven decomposed signals can be established by RVM with their each appropriate embedding dimension, and artificial bee colony algorithm (ABC) is used to select the appropriate kernel parameters of their RVM models. Thus, each RVM model of the seven decomposed signals has appropriate embedding dimension and kernel parameter. Artificial bee colony(ABC) algorithm is a relatively new swarm-based meta-heuristic method, which can excellently solve unconstrained and constrained optimization problems. In view of its attractive characteristics including lesser adjustable parameters, strong robustness, easy implementation, etc., the ABC algorithm has attracted considerable attention and has been widely applied to various research fields including clustering analysis(Karaboga and Ozturk, 2011 and Zhang et al.,2010),image processing( Draa and Bouaziz 2014 and Ma et al.,2011), economic

dispatch( Basu 2013),and so on. Recently, artificial bee colony algorithm has been extended for parameters optimization of some popular intelligent learning algorithms,such as artificial neural networks(Uzlu et al.,2014) and support vector machine (Bordoloi and Tiwari,2014). In some researches, artificial bee colony algorithm has been proved to be more effective than some other evolutionary algorithms including genetic algorithm (GA), particle swarm optimization(PSO), and so on(Karaboga and Basturk, 2008, Karaboga and Akay, 2009,Oliva et al.,2014 and Zhang et al.,2010). Hence, artificial bee colony algorithm is used to select the appropriate kernel parameter of RVM in this paper. In order to show the superiority of the proposed EMD-ABCRVM method, the RVM models with several different embedding dimensions and Gaussian RBF kernel parameters are used to compare with the proposed EMD-ABCRVM method, and mean absolute percentage prediction error is used to evaluate their prediction abilities for kurtosis of bearing vibration signal. 2 Empirical mode decomposition of kurtosis of bearing vibration signal Empirical mode decomposition (EMD) is a self-adaptive signal processing method which can solve the problem that the Hilbert transform cannot be used alone for non-stationary data(Khorramdel et al.,2014 , Lei et al.,2013 and Schlotthauer et al.,2014). By EMD,the signal can be decomposed into several simple intrinsic mode functions (IMFs) which correspond to the signal’s different frequency band ranging from high to low, and each IMF represents a kind of natural oscillatory mode embedded in the signal. The bearing vibration signal in bearings vibration data set(Loparo 2003) has been used, and a set of kurtosis data

calculated and obtained from the bearing vibration signal can be decomposed into several IMFs with different frequency range, in this study, the seven decomposed signals can be obtained and denoted as IMF1,IMF2,IMF3,IMF4,IMF5,IMF6 and IMF7 respectively, which can be shown in Fig.1. As shown in Fig.1, IMF7 is a lowest frequency signal,which reflects the variation trend of kurtosis of bearing vibration signal, and IMF1 is a highest frequency signal, which includes the detailed information of kurtosis of bearing vibration signal. As the seven decomposed signals have different characteristics, seven different prediction models must be created to fit and predict them.

Kurtosis of Bearing Vibration Signal 4

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Figure 1 Empirical mode decomposition of kurtosis of bearing vibration signal

3 Empirical mode decomposed signal prediction by ABCRVM 3.1 Kernel parameter optimization of RVM by artificial bee colony algorithm The regression model of relevance vector machine can be used to solve the nonlinear regression problems(Valente et al.,2011). Let T  x n , t n n 1 be a set of the training data,where N

xn is the input vector, and tn is the corresponding output target,the output target includes the additive noise,which can be formulated as follows:

t n  y(x n , w )  εn

(1)

where  n is assumed to be mean-zero Gaussian noise with variance  2 . The regression function of RVM consists of a linear combination of the weighted kernel functions, which can be described as follows: N

y(x, w)   wi k (x, xi )  w0

(2)

i 1

where k ( x, x i ) is the kernel function, w  [ w1 , w2 ,, wN ] is the weight vector, and w0 is the bias. Gaussian RBF kernel function has been used in this RVM, which can be expressed as follows:

 x x i j  k RBF (x i , x j )  exp   2  

2

   

(3)

where  denotes the kernel parameter of Gaussian RBF kernel function. As the selection of the parameter of the kernel function has a certain influence on the prediction results of relevance vector machine, artificial bee colony algorithm is used to select the appropriate kernel parameter  . Artificial bee colony (ABC) algorithm is an excellent

method to handle unconstrained and constrained optimization problems.The colony of artificial bees consists of three groups of bees: onlooker bees,employed bees and scout bees,among which onlooker bees and employed bees perform the exploitation process in the search space, and scout bees control the exploration process(Aydin et al.,2014 and Cura,2014). In ABC algorithm, a randomly distributed initial population of M solutions (positions) in a certain range can be created, and each solution is a D-dimensional vector which consists of the parameters to be optimized,where D is the number of the parameters to be optimized. The nectar amount of a food source is used to measure the quality (fitness value) of the associated solution(Karaboga and Basturk, 2008). The fitness value can be calculated as follows:

fitnessi 

1 1  Obj.Fun.i

(4)

where Obj.Fun.i denotes the objective function. The main steps for the ABC algorithm to select the kernel parameter of RVM are described as follows: Step 1 Randomly initialize a population of M positions in a certain range, the position of each food source can be generated by using the following equation:

xi , j  xmin, j  r1   xmax, j  xmin, j 

(5)

where xmax, j is the upper bound of the jth parameter to be optimized, xmin, j is the lower bound of the jth parameter to be optimized, j = {1, … , D}, D is the number of the parameters to be optimized,in this study, D =1; r1 is a random number between [0, 1].

Step 2

Evaluate the fitness for each food source, the fitness computing formula of

artificial bee colony algorithm is given in Eq.(4). In Eq.(4),

1 v yq  yˆ q ,i Obj.Fun.i   v q 1 yq

(6)

where yq is the actual value and yˆ q ,i is the validation value; v is the number of the training samples in training sample sets. Generate a new food source xinew for each employed bee in the neighborhood of ,j

Step 3

its present position xi , j by using the following equation:

xin, je w xi , j  r2  xi , j  xl , j 

(7)

where l  i , r2 is a random number between [−1, 1]; in this study, j=1. Step 4

Evaluate the fitness of the new food source xinew , j ,and compare it with the fitness of

new xi , j . If the fitness of xinew , j is superior to that of xi , j ,then replace xi , j with xi , j ; otherwise, xi , j is

retained. Step 5

After all employed bees complete the search process, they share the information

related to the nectar amounts and their position with the onlooker bees on the dance area, the onlooker bees evaluate the nectar amounts taken from all employed bees and choose the food sources with the probability pi calculated by using the following equation:

pi 

fitnessi M

 fitness i 1

(8)

i

where fitnessi denotes the fitness value of the solution i, and M denotes the number of food sources.

Step 6

Once the onlooker has selected its food source xi , j , it will search for a new food

source xinew in its neighborhood according to Eq.(7). Then,evaluate the fitness of new food ,j new source xinew , j and compare it with the fitness of xi , j . If the fitness of xi , j is superior to that of

xi , j ,then replace xi , j with xinew , j ; otherwise, xi , j is retained.

Step 7 If the abandoned food source exists, replace it with a new randomly produced solution xi , j for the scout by using Eq.(5). Step 8

Memorize the position(solution) of the best food source achieved so far.

Step 9

Repeat the procedure from step 3 to 8 until the maximum cycle number of the

search process is reached. 3.2 Empirical mode decomposed signal prediction by ABCRVM Empirical mode decomposed signal prediction by ABCRVM can be described in this section. As shown in Fig.1, there are data less than zero in IMF1~IMF6, in order to predict for them conveniently,IMF1~IMF6 must be offset to ensure all data in them more than zero.Define 0.5 as a offset unit in this study, it is obvious that IMF1,IMF2 need two offset units and IMF3~IMF6 need a unit offset.That is, offset values of IMF1,IMF2 are set to 1,and offset values of IMF3~IMF6 are set to 0.5. The IMFs after offset processing are defined as offset IMF signals. Fig.2 shows the offset IMF1~IMF6. Each empirical mode decomposed signal(or offset signal) data is normalized to [0,1] to improve the generalization ability of the prediction models. Assume one of the normalized empirical mode decomposed signal(or offset signal) data are a1 , a2 ,am ,, au ,, au k , among which a1 , a2 ,, am ,, au are used to establish the training sample sets, and

au 1 , , au  k are used to test the prediction model. The training sample sets can be described

by the formula:  a1  a I 2    au  m

a2 a3  au  m 1

 am   am1   a   am 1  , O   m 2            au 1   au 

(9)

where m denotes the embedding dimension, I denotes the input vectors and O denotes the corresponding outputs. Generally, in order to make the prediction model have good prediction ability, the embedding dimension is set to bigger value for low frequency signal,and is set to smaller value for high frequency signal. Offset IMF1 is the highest frequency signal among IMF7 and offset IMF1~IMF6, whose embedding dimension is set to 5; and IMF7 is the lowest frequency signal among IMF7 and offset IMF1~IMF6,whose embedding dimension is set to 10.As offset IMF2~IMF3 have higher frequency than offset IMF4~IMF6, thus, the embedding dimensions of offset IMF2~IMF3 are set to 7;and the embedding dimensions of offset IMF4~IMF6 are set to 8. Then, artificial bee colony algorithm is used to obtain the kernel parameters of the RVM prediction models of IMF7 and offset IMF1~IMF6 respectively,and the RVM prediction models of IMF7 and offset IMF1~IMF6 are established respectively. The fitting and prediction results of IMF1~IMF6 can be obtained by subtracting their the offset values. Finally, the kurtosis prediction results of bearing vibration signal can be obtained by the combination of the prediction results of the seven IMF signals.

Offset IMF1

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Figure 2 The offset processing of IMF1~IMF6 4 Experimental analysis Fig.3 gives the fitting results of the seven IMFs of kurtosis of bearing vibration signal by ABCRVM,and Fig.4 gives the prediction results of the seven IMFs of kurtosis of bearing vibration signal by ABCRVM, and the kurtosis prediction results of bearing vibration signal can be obtained by the combination of the prediction results of the seven IMFs.

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Figure 3 The fitting results of IMF1~IMF7 by ABCRVM

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In order to show the superiority of the proposed EMD-ABCRVM method, the RVM models with several different embedding dimensions and Gaussian RBF kernel parameters are used to compare with the proposed EMD-ABCRVM method.First of all,the RVM models with embedding dimension 5 including RVM5-1(embedding dimension is 5,parameter’s value of Gaussian RBF kernel is 0.5),RVM5-2(embedding dimension is 5, parameter’s value of Gaussian RBF kernel is 1) and RVM5-3(embedding dimension is 5, parameter’s value of Gaussian RBF kernel is 2) are used to compare with the proposed EMD-ABCRVM method, the comparison of kurtosis prediction results of bearing vibration signal among EMD-ABCRVM,RVM5-1,RVM5-2 and RVM5-3 is given in Fig.5. Next, the RVM models with embedding dimension 7 including RVM7-1(embedding dimension is 7,parameter’s value of Gaussian RBF kernel is 0.5),RVM7-2(embedding dimension is 7, parameter’s value of Gaussian RBF kernel is 1) and RVM7-3(embedding dimension is 7, parameter’s value of Gaussian RBF kernel is 2) are used to compare with the proposed EMD-ABCRVM method, the comparison of kurtosis prediction results of bearing vibration signal among EMD-ABCRVM,RVM7-1,RVM7-2 and RVM7-3 is given in Fig.6. Then,the RVM models with embedding dimension 8 including RVM8-1(embedding dimension is 8,parameter’s value of Gaussian RBF kernel is 0.5),RVM8-2(embedding dimension is 8, parameter’s value of Gaussian RBF kernel is 1) and RVM8-3(embedding dimension is 8, parameter’s value of Gaussian RBF kernel is 2) are used to compare with the proposed EMD-ABCRVM method, the comparison of kurtosis prediction results of bearing vibration signal among

EMD-ABCRVM,RVM8-1,RVM8-2 and RVM8-3 is given in Fig.7. Finally,the RVM models with embedding dimension 10 including RVM10-1(embedding dimension is 10,parameter’s value of Gaussian RBF kernel is 0.5), RVM10-2(embedding dimension is 10, parameter’s value of Gaussian RBF kernel is 1) and RVM10-3(embedding dimension is 10, parameter’s value of Gaussian RBF kernel is 2) are used to compare with the proposed EMD-ABCRVM method, the comparison of kurtosis prediction results of bearing vibration signal among EMD-ABCRVM,RVM10-1, RVM10-2,RVM10-3 is given in Fig.8. As shown in Fig.5~Fig.8, the prediction values of the proposed EMD-ABCRVM method are more sensitive to the future variation process of kurtosis of bearing vibration signal than those of RVM5-1,RVM5-2, RVM5-3, RVM7-1, RVM7-2, RVM7-3, RVM8-1, RVM8-2, RVM8-3, RVM10-1, RVM10-2 and RVM10-3, and the prediction curve of the proposed EMD-ABCRVM method can reflect the future variation trend of kurtosis of bearing vibration signal better than those of RVM5-1,RVM5-2, RVM5-3, RVM7-1, RVM7-2, RVM7-3, RVM8-1, RVM8-2, RVM8-3, RVM10-1, RVM10-2 and RVM10-3. Then, the superiority of the proposed EMD-ABCRVM method can be shown through quantitative analysis. Table 1 gives the comparison of mean absolute percentage prediction error for kurtosis of bearing vibration signal among EMD-ABCRVM, RVM5-1, RVM5-2 , RVM5-3, RVM7-1,RVM7-2,RVM7-3, RVM8-1,RVM8-2, RVM8-3,RVM10-1, RVM10-2 and RVM10-3. As shown in Table 1, the kurtosis prediction ability of bearing vibration signal of EMD-ABCRVM is better than those of other twelve RVMs. It is concluded that it is feasible for the proposed combination scheme to improve the prediction accuracy of RVM for kurtosis of bearing vibration signal.

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Fgure 4 The kurtosis prediction results of bearing vibration signal of EMD-ABCRVM Kurtosis of Bearing Vibration Signal 4

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Figure 5 The comparison of kurtosis prediction results of bearing vibration signal among EMD-ABCRVM,RVM5-1,RVM5-2 and RVM5-3

Kurtosis of Bearing Vibration Signal 4

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Figure 6 The comparison of kurtosis prediction results of bearing vibration signal among EMD-ABCRVM,RVM7-1,RVM7-2 and RVM7-3 Kurtosis of Bearing Vibration Signal 4

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Figure 7 The comparison of kurtosis prediction results of bearing vibration signal among EMD-ABCRVM,RVM8-1,RVM8-2 and RVM8-3

Kurtosis of Bearing Vibration Signal 4

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Figure 8 The comparison of kurtosis prediction results of bearing vibration signal among EMD-ABCRVM,RVM10-1,RVM10-2 and RVM10-3

Table 1 The comparison of mean absolute percentage prediction error for kurtosis of bearing vibration signal among EMD-ABCRVM,RVM5-1,RVM5-2,RVM5-3, RVM7-1,RVM7-2, RVM7-3,RVM8-1, RVM8-2, RVM8-3,RVM10-1,RVM10-2 and RVM10-3

Mean absolute Embedding

Parameter’s value of

dimension

Gaussian RBF kernel

5

7

8

10

Prediction model

percentage prediction error

EMD-ABCRVM

0.0793

0.5

RVM5-1

0.0997

1

RVM5-2

0.0996

2

RVM5-3

0.0904

0.5

RVM7-1

0.1191

1

RVM7-2

0.1048

2

RVM7-3

0.0912

0.5

RVM8-1

0.1350

1

RVM8-2

0.1069

2

RVM8-3

0.0918

0.5

RVM10-1

0.1049

1

RVM10-2

0.1131

2

RVM10-3

0.0935

5 Conclusions In this paper, a new hybrid model of empirical mode decomposition and artificial bee colony algorithm-based RVM is proposed for kurtosis forecasting of bearing vibration signal. The prediction models of the seven IMF signals obtained by empirical mode decomposition for kurtosis of bearing vibration signal can be established by RVM with their each appropriate embedding dimension and kernel parameter,which contributes to improve the forecasting accuracy of the entire model. The comparison results of mean absolute percentage prediction error for kurtosis of bearing vibration signal among EMD-ABCRVM and other twelve RVMs indicate that it is feasible for the proposed combination scheme to improve the prediction accuracy of RVM for kurtosis of bearing vibration signal. An effective prediction result can be obtained by the proposed method when bearing is under steady working conditions; however,it is difficult to obtain ideal prediction result when bearing is under the complex working conditions including impact load,etc. Therefore, the study on kurtosis forecasting method of bearing vibration signal under some complex working conditions is very significant in the future. In addition, it is very significant to study on fault prediction for bearing on the basis of the proposed kurtosis forecasting method of bearing vibration signal. Not only for bearing,but also for other rotating machineries, kurtosis of vibration signal is a very important indicator which can reflect their operating state.Thus, in the future, it is very significant to study on kurtosis forecasting of vibration signal for other important rotating machineries including rotor,gear,and so on.

Acknowledgements This project is supported by National Natural Science Foundation of China (Grant No. 51305076). References Aydin et al.,2014

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Highlights 1. A new forecasting model “EMD-ABCRVM” is presented in this study. 2. EMD-ABCRVM is firstly applied for kurtosis forecasting of bearing vibration signal. 3. Establish IMFs’ prediction models by RVM with each appropriate embedding dimension. 4.

Select the appropriate RVM’s kernel parameter by artificial bee colony algorithm.