A Roller Bearing Fault Diagnosis Method Based on LCD Energy Entropy and ACROA-SVM

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Introduction
Roller bearings are important and frequently encountered components in rotating machines, which are found in widespread industrial applications.Roller bearing fault diagnosis is therefore meaningful.Fault diagnosis includes two aspects: feature extraction and pattern recognition.When a fault occurs in a roller bearing, it is very difficult to extract the fault characteristic information from the nonstationary vibration signals [1,2].
The traditional diagnosis techniques extract the fault characteristic information from the waveforms of the vibration signals in either the time domain or the frequency domain.Then, criterion functions are constructed to identify the condition of the roller bearing.However, it is very difficult to accurately evaluate the condition of a roller bearing through an analysis in the time or frequency domain only [2,3].
The Empirical Mode Decomposition (EMD) method of feature extraction is based on the local characteristic time scale of the signal and can adaptively decompose a complicated multicomponent signal into a sum of intrinsic mode functions (IMFs) whose instantaneous frequencies have physical significance [4,5].By applying an envelope analysis to each IMF component, the characteristic information of original signal can be extracted more accurately and effectively.In addition, the frequency components involved in each IMF are related not only to the sampling frequency but to changes in the signal itself; that is, EMD can be regarded as a self-adaptive filter whose bandwidth and central frequency change with the signal itself.Therefore, EMD is a self-adaptive signal processing method that can be applied to nonlinear and nonstationary processes [6].However, amplitude and frequency information is lost because of the cubic spline and the Hilbert transform used in the EMD [7].Rilling and Flandrin asserted that two tones can be separated using EMD, and numerical experiments supported their claims [8].Wu and Huang [9] found that two components whose frequencies lie within an octave cannot be separated by EMD.However, there are currently no rules or guidelines for deciding when two separate components can be separated using EMD.Furthermore, the end effect [10], mode mixing [11], overshoot and undershoot [12], negative frequenciesinstantaneous frequency [13], and a lack of a theoretical foundation [14] are all current drawbacks of EMD.
Recently, Cheng et al. developed a new signal analysis method, the local characteristic-scale decomposition (LCD), that defines intrinsic scale components, and, with the physical instantaneous frequency, this method can decompose a complicated signal into several intrinsic scale components (ISC) [15].By analysing each resulting ISC, which involves the local characteristic of the signal, the characteristic information of the original signal can be extracted more accuracy and effectively.The LCD method is superior to the Hilbert-Huang Transform method in reducing the end effect and the iteration time and in the accuracy of the instantaneous characteristic.
Pattern recognition is the other aspect of roller bearing fault diagnosis.Traditional statistical pattern recognition methods and Artificial Neural Network (ANN) classifiers assume that sufficient samples are available, which is not always true in practice [16].The Support Vector Machine (SVM) is a powerful machine learning method, based on statistical learning theory and the structural risk minimization principle that has been successfully applied in classification and regression problems [17].SVMs not only can solve the problems of overfitting, local optimal solutions, and slow convergence rates that exist in ANNs, but they also have an excellent generalization capability in situations where there are a small number of samples.Furthermore, SVMs can solve nonlinear, high-dimensional pattern recognition problems with a limited number of samples and represent nonlinear relationships between the input and the output [18].SVMs have been found to be remarkably effective in many practical applications.This method is widely used in areas such as pattern recognition [19], time-series forecasting [20], diagnostics [21][22][23][24][25], robotics [26], signal processing [25,27], speech and word recognition [28], machine vision [29], and financial forecasting [30].In SVMs, the kernel parameters have an influence on the generalization performance, and the regularization constant  determines the trade-off between minimizing the training error and minimizing the model complexity.The parameters of the kernel function implicitly define the nonlinear mapping from the input space to the high-dimensional feature space [31].The performance of the SVM will be degraded if these parameters are not properly chosen.There are several methods for choosing the parameters of the SVM such as trial-and-error procedures [32], the grid algorithm [33], the cross-validation method [34], the generalization error estimation method [35], and the gradient descent method [36].These methods have several drawbacks; for example, both the grid method and the crossvalidation method require long and complicated calculations [33].
In recent years, heuristic algorithms have been developed and are widely used.These algorithms use simple approaches found in heuristic optimisation algorithms.Some heuristic algorithms that have been used to optimise SVM parameters are Ant Colony Optimization (ACO) [37], Particle Swarm Optimization (PSO) [38][39][40], the Genetic Algorithm (GA) [41], and the Bee Colony Algorithm (BCA) [42].The Artificial Chemical Reaction Optimisation Algorithm (ACROA), which was introduced by Bilal Alatas, is a novel computational method that was inspired by chemical reactions [43].The ACROA has been applied successfully to optimisation problems and the mining of classification rules [44].Compared with the GA, the ACROA can reach a global optimum in a very short time, and the classification result is the same.The ACROA is adapted to the solution procedure to an optimisation problem.This algorithm is based on the second law of thermodynamics; that is, a system tends toward the highest entropy and the lowest enthalpy [45].In the ACROA, the enthalpy or potential energy and the entropy can be used as the objective functions for the minimisation and maximisation problems, respectively, for the optimisation problem of interest.The ACROA is robust, and thus we chose the ACROA to solve our problem.In this study, the ACROA is applied to optimise the SVM parameters.
In this paper, LCD is applied to diagnose the roller bearing faults.First, the original acceleration vibration signal is decomposed using LCD and the ISC components are obtained, and then the concept of LCD energy entropy is introduced, which can reflect the actual condition and the fault pattern of the roller bearing.The LCD energy entropies of different vibration signals illustrate that the energy of the vibration signal in different frequency bands will change when a bearing fault occurs.To identify the condition of the roller bearing further, the ACROA-SVM serves as a classifier, and the extracted energy features of the stationary ISCs are taken as classifier input vectors so that a faulty bearing can be distinguished from a normal bearing.To verify the superiority of the LCD method, it is compared with the EMD method.As in the LCD method, the original signal is decomposed with EMD, and then the energy features are extracted.These resulting features are also used as input vectors to the ACROA-SVM to identify the condition of a roller bearing.The experimental results show that the diagnostic approach of the ACROA-SVM based on LCD energy entropy has better identification accuracy than EMD and is faster.
The remainder of this paper is organised as follows.Section 2 discusses the LCD method.In Section 3, the concept of LCD energy entropy is proposed, and the LCD energy entropies of different vibration signals are calculated to illustrate that the energy of an acceleration signal in different frequency bands changes when a roller bearing fault occurs.Section 4 explains the ACROA and the parameter optimisation of an SVM based on the ACROA.In Section 5, the fault diagnosis method based on LCD and the ACROA-SVM is given, in which the energy features extracted from a number of ISCs are used as input vectors to the ACROA-SVM.In Section 6, the fault diagnosis method is used to diagnose the condition of actual roller bearings and is compared with the EMD method.The conclusions drawn from this research are given in Section 7.

LCD Method
Figure 1 shows four types of signals having an instantaneous frequency with physical significance.In Figure 1, point A is the value of the line connecting two adjacent peaks at the time where the minimum occurs between the two maximums, and point B is the minimum point.The LCD method is developed from the simple assumptions that any complicated signal consists of several ISCs and any two ISCs are independent of each other.In this way, each signal can be decomposed into a number of ISCs, each of which must satisfy the following definition [15].
(I) In the entire data set, all the local maxima are positive, all the local minima are negative, and the signal is monotonic between any two adjacent extreme points.
(II) Among the data, let all the maximal points be denoted as (  ,   ),  = 1, 2, . . ., , where  is the number of maximal points.
The line formed by any two adjacent extreme points,   , at the  +1 as  +1 , is specified as follows: Then, the relation should be true, where Generally,  = 0.5 when  +1 = − +1 .
Based on the definition of the ISC component, a realvalued signal, () ( > 0), can be decomposed into a number of ISCs using the LCD method in the following way.
By extension, the two end extrema ( 0 ,  0 ) and ( +1 ,  +1 ) can be obtained.According to (3) and (4), we can obtain  1 and   .Otherwise, we can extend the sequence   directly from the known values.
(2) Connect all the   with a cubic spline to form the base line, denoted as  1 ().Theoretically, the difference between the original data and the base line,  1 (), is the first ISC, ℎ 1 (); that is, If ℎ 1 () meets conditions (I) and (II), then it is an ISC component, and ℎ 1 () is chosen as the first ISC.
(3) Or see the ℎ 1 () as the original data, and repeat the above step, defining If ℎ 11 () does not satisfy conditions (I) and (II), repeat this step  times until ℎ 1 () satisfies the ISC conditions; then denote ℎ 1 () as the first ISC,  1 ().
(4) Separate  1 () from the initial data, and define the residue as  1 (): (5) Next, add  1 () to the original data, and repeat steps (1)-( 4).Similarly, we obtain  2 (), . . .,   () until the residue   () is either a monotonic or a constant function.Then, () is decomposed into  ISCs and a residue   (); that is, Similar to the Cauchy convergence test, the standard deviation (SD) is defined as where  is the length of time.The sifting process is stopped when SD is less than a chosen value.Generally, a value of SD less than 0.3 is ideal for an ISC.
The decomposed results in Figure 3 show that the LCD method is superior to the EMD method with the same steps of decomposition.Intuitively, in contrast with IMFs, ISCs provide more information on the modulation characteristics.

LCD Energy Entropy
The vibration signal from a faulty roller bearing reflects the corresponding resonant frequency components, and its energy changes with the frequency distribution.Therefore, in this study, the LCD energy entropy is proposed to capture this change.
It is assumed that the vibration signal of a faulty roller bearing () has been decomposed with LCD into  ISCs and a residue   (), where the energies of the  ISCs are  1 ,  2 , . . .,   .The sum of the energies of the  ISCs should be equal to the total energy of the original signal when the residue   () is ignored.Because the ISCs  1 (),  2 (), . ..,  () include various frequency components,  = { 1 ,  2 , . . .,   } forms an energy distribution in the frequency domain of the roller bearing vibration signal.The corresponding LCD energy entropy is defined as where   =   / is the percentage of the energy of   () in the total signal energy ( = ∑  =1   ).Figures 4(a), 4(b), and 4(c) show the three cases of the roller bearing vibration signal: normal, with an outer-race fault and with an inner-race fault, respectively.Table 1 shows that the energy entropy of the vibration signal of a normal roller bearing is greater than that of the others because the energy distribution of this kind of signals in each frequency band is comparatively even and uncertain.For a roller bearing with an outer-race fault, the energy entropy is lower because the energy is distributed mainly in the resonant frequency band and the distribution uncertainty is lower.Moreover, the higher resonant frequency components are produced in the roller bearing with an inner-race fault and the impact is more severe, so the energy entropy in this case would be the least.It can be concluded from the preceding analysis that the energy entropy based on LCD can reflect the condition and the fault pattern of the roller bearing.However, for each roller bearing, the LCD energy entropy varies for the same condition.Therefore, it is not sufficient to distinguish the condition and the fault pattern only according to the LCD energy entropy; further analysis is desirable.another.Two key reactions in the ACROA are bimolecular and monomolecular reactions [43].The principle of the ACROA is presented in the flow chart in Figure 5 and consists of the following five steps [44].

Artificial Chemical Reaction Optimization
Step 1. Define the problem and the algorithm parameters.
Step 2. Initialize the reactants and evaluate them.
Step 3. Simulate the chemical reactions.
Step 5. Check the termination criterion.
The optimization problem is specified as follows: maximize  () where () is a fitness function,  = ( 1 ,  2 , . . .,   ) is the vector of decision variables,  is the number of decision variables, and   is the range of feasible values for decision variable , where   and   are the lower and upper bounds of the jth decision variable, respectively.More details about these steps can be found in [43,44].

Support Vector Machine (SVM).
The SVM is developed from the optimal separation plane under linearly separable conditions.The basic idea of the SVM is to map the training samples from the input space into a higher dimensional feature space via a mapping function  [37].Suppose there is a given training sample set  = {(  ,   ),  = 1, 2, . . ., }, where each sample   ∈   belongs to a class determined by  ∈ {+1, −1}.When the training data are not linearly separable in the feature space, the target function can be expressed as follows [17]: where  is the normal vector of the hyperplane,  is a penalty parameter,  is the bias that is a scalar,   are nonnegative slack variables, and () is a mapping function.By introducing Lagrange multipliers   ≥ 0, the optimization problem can be rewritten as follows. Maximize Subject to The decision function can be obtained as follows: The most common kernel functions used in SVMs are as follows: (i) linear kernel (ii) polynomial kernel (iii) RBF kernel where  and  are kernel parameters.In this paper, the radial basis function kernel is used because of its universal application and good performance.

Optimisation of the SVM Parameters Using the ACROA.
SVM parameters have an important effect on the classification accuracy.The parameters of the Gaussian kernel function include a penalty factor  and the standard deviation .The selection of the SVM parameters is very difficult.Generally,  and  are selected according to experience.In this paper, the ACROA is used to optimise the parameters of the SVM.These variables are  and , and the fitness function is the accuracy of the SVM.The fitness of the SVM is defined as follows: where  = (, ), and the accuracy of the SVM is defined as accuracy SVM

Number of correct classifications of test samples
Total number of samples in test set .
The flow chart of the ACROA-SVM is shown in Figure 6.

Experimental Results.
To evaluate the performance of the proposed ACROA-SVM method, we used three common benchmark data sets from the UCI benchmark, the Iris, Thyroid, and Seed data sets.The sizes of the training and test sets can be found in Table 2.
The Iris data set contains 150 instances and four attributes.In this data set, the class attribute is typed and there are three classes: Setosa, Versicolor, and Virginica.
The Thyroid data set is used for the diagnosis of hyperthyroidism or hypothyroidism.This data set contains 215 patterns and 5 attributes, and there are three classes: normal, hyper, and hypo.
The Seed data set was obtained from the high-quality visualization of the internal structure of wheat kernels.The Seed data set contains 210 instances of wheat samples with 7  inputs, and there are three classes included: Kama, Rosa, and Canadian.
Four methods, the ACROA-SVM, the GA-SVM, the PSO-SVM, and the SVM, were used to classify these data sets.For the GA, the generation and population sizes were set to 50 and 20, respectively.To make a fair comparison, the values of the ACROA were chosen to be the same, for example, iterations = 50, ReacNum = 20.For the PSO, the parameters were fixed with the values given in the literature [39,40]; that is,  = 0.75,  1 =  2 = 1.5, the numbers of particles was 20, and the iteration count was 50.In the SVM method, the values of  and  were chosen by default, and thus the computation time was not calculated.The results in Tables 3, 4, and 5 show that the values of  and  obtained by each method are different.The test error value of the ACROA-SVM method is better than those of the GA-SVM, the PSO-SVM, and the SVM method.Furthermore, the computation time of the proposed ACROA-SVM method is less than those of the GA-SVM and the PSO-SVM methods.The ACROA-SVM method was next applied to a roller bearing fault diagnosis problem.

Roller Bearing Fault Diagnosis Method
Based on LCD and ACROA-SVM It can be observed from the preceding analysis that the LCD energy entropies of the vibration signals of the roller bearings with different conditions and fault patterns are obviously different, which shows that the energy of each ISC changes when the roller bearing develops a fault.In this paper, by taking the energy feature of each ISC component as the ACROA-SVM input vector, the condition and the fault pattern of the roller bearing can be identified effectively.The flow chart of the roller bearing fault diagnosis method based on LCD and the ACROA-SVM is shown in Figure 7.
The fault diagnosis method consists of the following seven steps [47].
(1) Collect signals from the roller bearings as samples for the three conditions: normal, outer-race fault, and inner-race fault.
(2) Decompose the original vibration signals into several ISCs, and choose the first  ISCs that include the most dominant fault information to extract the feature.
(3) Calculate the total energy   of the first  ISCs from (4) Construct the feature vector  with the energy levels as its elements as follows: Considering that the energy is sometimes biggest,  is adjusted by normalizing the feature to simplify the subsequent analysis and processing.Let classifier has higher accuracy than the LCD-SVM1, the EMD-ACROA-SVM1, and the EMD-SVM1 classifier.For SVM2, the accuracy rate of the LCD-ACROA-SVM2 and the EMD-ACROA-SVM2 are the same.In summary, it can be observed from Table 7 that the ACROA-SVM method based on LCD has higher accuracy (i.e., less error) than the method based on EMD and has shorter computation times.

Conclusion
A roller bearing fault diagnosis method based on LCD energy entropy and the ACROA-SVM was investigated in this paper.First, the original vibration signals were preprocessed using the LCD method.The LCD energy entropy was used as the input to the ACROA-SVM classifier.A theoretical analysis and experimental results show that the ACROA-SVM combined with the LCD method has higher accuracy and shorter computation times than when combined with EMD.Furthermore, the analysis shows that the LCD method is a self-adaptive signal processing method and is superior to the EMD method.This signal-processing method is adaptive and suitable for nonlinear and nonstationary process.

Figure 1 :
Figure 1: Four types of typical signal owning instantaneous frequency with physical meaning.

Figure 3 :
Figure 3: The EMD (a) and LCD (b) decomposed results of the multicomponent modulated signal shown in Figure 2.

Figure 4 :
Figure 4: The vibration acceleration signal of the normal roller bearing (a), out-race fault (b), and inner-race fault (c), respectively.

Figure 6 :
Figure 6: Parameter optimization flow chart of SVM based on ACROA.

Table 1 :
The LCD energy entropies of the vibration signals of the roller bearing with different faults.

Table 2 :
Properties of the problems.

Table 3 :
The identification result of IRIS data set.

Table 4 :
The identification result of THYROID data set.

Table 5 :
The identification result of SEED data set.

Table 6 :
The identification results based on LCD and ACROA-SVM method.