# Research on Fault Diagnosis Method of Rolling Bearing Based on Feature Optimization and Self-Adaptive SVM

Authors: Min Mao; Chengjiang Zhou; Jingzong Yang; Bin Fang; Fang Liu; Xiaoping Liu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Aiming at the problems of the difficulty of extracting the fault sensitive features in the mixed domain of bearings under complex working conditions and the lack of self-adaptability of support vector machine (SVM) parameters, to solve these problems, we investigate a diagnosis method combining optimal feature selection and self-adaptive SVM. Firstly, the optimal feature space through multicluster feature selection (MCFS) is constructed. Then, the differential evolution (DE) strategy is used to improve the search performance of gray wolf optimizer (GWO) to improve the diagnosis accuracy of SVM and make it strong self-adaptability. The experimental results show that, with a small number of sensitive features containing significant category differences, the proposed method not only guarantees the construction of the optimal feature space under the minimum feature dimension, but also greatly improves the accuracy of fault recognition. Simultaneously, compared with the traditional feature reduction method (PCA), the proposed MCFS combined diagnosis model improves the accuracy from 98.5% to 100% with 1.5%. Concurrently, compared with MPA-SVM, GWO-SVM, and PSO-SVM, the convergence performance is improved by 61.54%, 78.26%, and 92.64%, respectively. It can seek the best classification performance in the shortest time. The effectiveness and superiority of the proposed method are fully verified.


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## Body

## 1. Introduction

Rolling bearing are used to reduce the loss caused by friction between the shaft and the shaft during operation, and its health directly affects the working performance of the mechanical equipment. Rolling bearing damage may lead to immeasurable economic losses and even more serious casualties. So, it is particularly important to diagnose its faults. In recent years, many scholars use single features such as time-frequency domain, time domain, or frequency domain to represent fault information and diagnose faults through intelligent methods. The classical time-frequency domain signal processing methods include wavelet transform (WT), empirical mode decomposition (EMD), variational mode decomposition (VMD), and so on. Among them, WT was used to get the time-frequency diagram of the harmonic removal signal, which can provide relatively clear information on the health status of rolling bearing [1]. In particularly, wavelet packet decomposition was recognized as the most effective method. EMD was combined with the chaos synchronisation detection method to draw the voltage and current discharge signal captured by the high-frequency oscilloscope into a 3D chaotic error scatter diagram as the fault diagnosis characteristic image [2]. VMD was used to decompose the water pipe acoustic vibration signal, and the IMF component was selected to extract the kurtosis vector of VMD, the sample entropy vector of VMD, the center frequency vector of VMD [3], etc. However, due to insufficient fault features, the classification effect needs to be improved [4]. For this reason, a feature extraction method based on the combination of the time-frequency domain and time domain was proposed. Because the features of fault characterization are not comprehensive, although the effect can be improved, it is still not ideal [5]. To fully extract the time, frequency, and time-frequency domain features, a construction method based on mixed domain features is proposed, and the diagnosis effect is further improved. However, the problem of high dimensionality and complicated calculation of the feature set of the mixed domain will follow. Also, redundant features often reduce the reliability of diagnosis [6]. Because of the abovementioned problems of mixed domain features, the following three feature dimensionality reduction methods are often used: (1) Sun et al. used the locally linear embedding (LLE) algorithm to find the low-dimensional manifold structure hidden by high-dimensional features, so that the reduced features maintain the original topology. However, the overall distribution of data will be deformed in the low-dimensional space after dimensionality reduction [7]. (2) Ye et al. used linear local Tangent space alignment (LLTSA) to achieve dimensionality reduction, but it cannot fuse the category information into the dimensionality reduction process to improve the readability of low-dimensional features [8]. (3) Allegretta et al. used PCA to extract the covariance matrix of features to reduce dimensions, but PCA only considered the global reconstruction information and did not use the correlation within the data and between samples [9]. The above dimension reduction method based on feature fusion can make up for the deficiency of mixed domain features, but it is not the optimal method. With the continuous deepening of research, some scholars put forward the following three methods based on feature ranking and selection [10]: (1) Xiaoan and Minping proposed the feature selection method of max-relevance min-redundancy (MRMR), which can automatically select useful features from candidate multiscale features without any prior knowledge. However, the only local optimum is considered to some extent [11]. (2) Si et al. used a feature selection method based on Laplacian score (LS), which uses a filtering feature selection method to screen the original feature set to get the top n-dimensional feature subset [12]. Since the rigidity between each feature is not weighed, this method may select redundant features. (3) Yan et al. used MCFS for extraction [13, 14]. This method combines sparse subspace learning based on spectral regression with unsupervised feature selection, which can better compensate for the problems of other methods [15]. The above methods based on feature ranking and optimization, especially MCFS, can accurately select the features reflecting the operating state of the equipment and greatly improve the efficiency of fault diagnosis. Keerthana and Shyla used MCFS for unsupervised feature selection and wild animal detection; this technique provides better results in terms of recall, positive rate accuracy, and F-measure [16]. An optimal feature set with high accuracy was selected by using the supervised MCFS algorithm, which can effectively help neuroscience researchers and sleep physicians by more accurately and robustly promoting the visual sleep scoring process [17].In the construction of fault diagnosis model after fault feature extraction, support vector machine (SVM) was widely used because of its good classification effect [18]. However, traditional SVM does not have the parameter self-adaptability, the classification accuracy changes with the change of parameters, and the generalization ability of the model needs to be improved [19, 20]. At present, models based on heuristic optimization algorithm to optimize SVM parameters are not uncommon. Recently, some new metaheuristic algorithms have been proposed. (1) Monarch butterfly optimization (MBO) algorithm is a kind of swarm intelligence metaheuristic algorithm inspired by the migration behavior of monarch butterflies. Update the individual in MBO through migration and butterfly adjustment. In solving global numerical optimization and engineering problems, MBO can surpass many of the most advanced optimization technologies [21]. (2) Slime mould algorithm (SMA) has several new characteristics and unique mathematical model. Adaptive weights were used to simulate the process of positive and negative feedback generated by slime mould propagation wave based on biological oscillator, form the optimal path, and connect food with excellent exploration ability and development tendency [22]. (3) Hunger games search (HGS) has a simple structure, special stability characteristics, and highly competitive performance, which can more effectively solve constrained and unconstrained problems [23]. In addition, there are moth search algorithm (MSA), Runge Kutta method (RUN), colony predation algorithm (CPA), Harris hawks optimization (HHO), etc. [24–27], which can be used to optimize the two parameters of SVM. In the last few years, with the continuous development of deep learning technology, it was also widely used in the field of fault diagnosis. The methods based on intelligent algorithms mainly include convolutional neural networks (CNN), deep belief networks (DBN), recurrent neural networks (RNN), generative adversarial networks (GAN), and stacked autoencoders, etc. In particular CNN is a supervised deep learning method. Compared with the traditional fault diagnosis methods based on signal analysis, the method does not rely on the ability of artificial feature extraction and automatically extracts the deep features of signals. However, most intelligent algorithms still exist in the fault diagnosis of rolling bearing. The network model is not deep enough to fully extract features, resulting in the diagnosis accuracy that cannot reach the ideal accuracy, and there are still some errors [1]. The above methods are significant for fault diagnosis methods, no matter the manual designed feature extraction methods or deep learning methods. The change of working condition will lead to the performance drop of the trained classification model. Han et al. presented a novel diagnosis framework, which combines the spatiotemporal pattern network (STPN) method with CNN to construct a hybrid ST-CNN scheme. It was tested on two data sets to diagnose invisible operating conditions and fault severity, respectively, to evaluate its generalization ability [28]. And Han et al. proposed a hybrid diagnosis network based on domain generalization for deploying to invisible working conditions [29].In the selection of SVM parameters, some intelligent optimization algorithm has already proved to be effective: (1) particle swarm optimization (PSO) seeks the optimal solution through the collaboration and information sharing among individuals in a population. Xiaokai et al. used the PSO algorithm to optimize the penalty coefficient and kernel function radius of SVM, to improve the accuracy of classification [30]. (2) Genetic algorithm (GA) is a method to find the optimal solution by simulating the process of natural evolution. Tao et al. obtained the optimal value of the key parameters of SVM through the GA parameter optimization method [31]. (3) Dragonfly algorithm (DA) is a new swarm intelligent algorithm based on the theory of dragonfly miracle tiger foraging and avoiding natural enemies. Zhang et al. combined the parameters of SVM as the solution position of dragonflies and its prediction accuracy as the current fitness value of dragonflies. The best position of the dragonfly is the best parameter of SVM [32]. Although the above swarm intelligent algorithms and the latest marine predators algorithm (MPA) [33] have optimized the parameters of the SVM model, they all may have the problems of low solution accuracy and very easy precocious stagnation. To solve the above problems, a new swarm intelligence optimization algorithm gray wolf optimizer (GWO) was proposed. GWO algorithm was realized by simulating the social class and hunting behavior of gray wolves [34]. In recent years, this algorithm has been widely used in the field of model optimization: (1) Liao et al. used the GWO algorithm to optimize the internal parameters of SVM, established a prediction model, and completed online real-time monitoring. However, for high-dimensional nonlinear features, the evaluation performance of GWO-SVM is not high [35]. (2) Bian et al. proposed a new metaheuristic GWO-SVM model. This model can accurately predict the solubility of sulfur in supercritical acid gas, but its effectiveness required more reliable experimental data [36]. (3) Bian et al. combined GWO with SVM to propose an effective wax disappearance temperature (WDT) estimation framework, but there is still a problem of poor GWO searchability [37]. However, the initial population of GWO algorithm was randomly generated, which lead to the lack of diversity of gray wolf individuals in the search space and was easy to fall into local optimization. In response to this phenomenon, Islam and Islam proposed a simulated annealing (SA) algorithm, which introduced the idea of annealing into the field of combinatorial optimization, effectively solving the problem of local optimal solution [38]. But its parameters are difficult to control, and there is no guarantee that it will converge to the optimal value once. Furthermore, Gupta and Deep proposed a random walk (RW) which is not easy to fall into the local minimum [39]. Although the overall optimal point can be better achieved, it depends on the selection of the initial point. Storn and Price proposed the differential evolution (DE) algorithm, which is a more efficient global optimization algorithm [40]. It is worth noting that Elazia et al. used the DE algorithm to generate a more suitable initial population position for the GWO algorithm, which can effectively avoid the GWO algorithm from falling into local optimal solution [41, 42]. More worthy of attention, to improve the performances of DE, many research teams developed some improved differential evolution algorithms. They all have good optimization ability, scalability, efficiency, and stability, which can provide some new methods for solving large-scale optimization problem [43–46]. Therefore, there is still a large space for optimization of SVM model parameters; an improved swarm intelligence optimization algorithm is considered in this paper. And it should be robust to the change of working condition or unseen condition during the training of model.In summary, to improve the existing problems of feature selection and diagnosis model, a fault diagnosis method of rolling bearing based on feature optimization and self-adaptive SVM is proposed. The main innovations and contributions of this article are summarized as follows: (1) MCFS is introduced to select the most favorable features for diagnosis. It is of great significance to select a small number of sensitive features containing significant category difference information from the mixed domain set of rolling bearing fault. It can not only reduce the dimension of feature vector, but also improves the reliability and timeliness of diagnosis. (2) The local optimization and slow convergence of GWO are improved by DE strategy, and the convergence efficiency and accuracy are improved. The combination of optimal feature and optimal DEGWO-SVM model has strong self-adaptability, higher optimization accuracy, and faster convergence speed and can seek optimality in a shortest time. The process is as follows: Firstly, the time domain, frequency domain, and time-frequency domain features of the vibration signals of rolling bearings under various working conditions are extracted to obtain high-dimensional features of the mixed domain. Secondly, MCFS is used to sort and optimize the features to get the optimal feature space. Then the DE strategy is used to improve the local optimal problem when GWO optimizes SVM, and a self-adaptive SVM diagnostic model is constructed. Finally, the optimal features are input into the self-adaptive SVM to diagnose rolling bearing faults. Through experiments and comparative analysis, the applicability and superiority are verified.The general structure of the entire manuscript: Theories of mixed domain and MCFS are introduced briefly in Section2. With theories of GWO, DE, and DEGWO, the process of using DEGWO to optimize SVM parameters is introduced in Section 3. In Section 4, the validity and reliability of the approach are validated by CWRU datasets. The experimental process, algorithm, and model comparative analysis and preliminary conclusions are discussed. The conclusions of the experiment are summarized in Section 5.


## 2. Construction of Optimal Feature Space Based on MCFS

The characteristic parameters of rolling bearing vibration signals usually only reflect the single equipment fault information (a fault location or fault state), and a single characteristic parameter cannot fully describe the fault state of the rolling bearing [47]. To characterize the fault characteristics more comprehensively and accurately, this paper introduces mixed domain feature parameters to describe the state of rolling bearings. In [35], 9-time domain features (including mean, root mean square value, peak value, etc.), 3-frequency domain features (including the center of gravity frequency, mean square frequency, frequency variance, etc.), and wavelet packet decomposition were used to realize feature extraction, comprehensive tool wears information could be obtained, and the classification and prediction effect was better.Therefore, in this paper, the time domain features are obtained through statistical probability analysis of the original vibration signals, and 16-time domain feature parameters including dimensional and dimensionless are extracted. At the same time, when a rolling bearing fails, frequency domain analysis is often used to describe information that cannot be obtained in the time domain, including frequency features that reflect the spectrum energy and the degree of spectrum dispersion (or concentration) [48]. To obtain complete fault information as possible, 13-frequency domain characteristic parameters need to be introduced. For each signal, the detailed calculation formulas of time domain and frequency domain features can be found in [35, 47, 48]. The above specific features are shown in Tables 1 and 2. The constructed vector by fusion is(1)Xt&f=Xtime,Xfrequency.Table 1
Time domain feature set.
NumberTime domain features1TF1=1/N∑n=1Nxn (mean value)2TF2=1/N∑n=1Nxn2 (square root amplitude)3TF3=1/N∑n=1Nxn2 (root mean square value)4TF4=1/N∑n=1Nxn (absolute mean value)5TF5=1/N∑n=1Nxn3 (skewness)6TF6=1/N∑n=1Nxn4 (kurtosis)7TF7=maxxn (peak-to-peak value)8TF8=maxxn (maximum value)9TF9=minxn (minimum value)10TF10=1/N∑n=1Nxn2 (variance)11TF11=TF3/TF4 (waveform index)12TF12=TF8/TF3 (peak index)13TF13=TF8/TF4 (Pulse index)14TF14=TF8/TF2 (margin index)15TF15=TF5/TF103 (skewness)16TF16=TF6/TF104 (kurtosis index)Table 2
Frequency domain feature set.
NumberFrequency domain features1FF1=1/K∑k=1KSk2FF2=1/K−1∑k=1KSk−FF123FF3=1/K∗FF23∑k=1KSk−FF134FF4=1/K∗FF22∑k=1KSk−FF145FF5=∑k=1KfkSk/∑k=1KSk6FF6=∑k=1Kfk−FF52Sk/K7FF7=∑k=1Kfk2Sk/∑k=1KSk8FF8=∑k=1Kfk4Sk/∑k=1Kfk2Sk9FF9=∑k=1Kfk2Sk/∑k=1KSk∑k=1Kfk4Sk10FF10=FF6/FF511FF11=∑k=1Kfk−FF53Sk/K⋅FF6312FF12=∑k=1Kfk−FF54Sk/K⋅FF6413FF9=∑k=1Kfk−FF5Sk/FF6KHowever, the rolling bearing vibration signal under the background of strong noise has obvious nonstationary and nonlinear characteristics, and only extracting the time domain and frequency domain features is not enough to fully characterize the state characteristics. Many studies have shown that the joint time-frequency domain space contains abundant fault feature information of vibration signals [48, 49]. The calculation method of wavelet packet is(2)di,j,2m=∑hk−2idk,j+1,mdi,j,2m+1=∑gk−2idk,j+1,m.The reconstruction of the wavelet packet is calculated as follows:(3)di,j+1,m=∑khi−2kdi,j,2m+∑kgi−2kdi,j,2m+1.The energy of the signal in each frequency range of wavelet decomposition resultdi,j,k is(4)Ei,j=∑k=1Ndi,kk2j=1,2,…,2i.To facilitate subsequent processing, the energy values in each frequency band of the signal are normalized to form the wavelet packet energy spectrum:(5)E⌢=Ei,1,Ei,2,…Ei,j,…Ei,2i∑n=12iEi,n.In this paper, a three-layer wavelet packet decomposition method is used to decompose the signal into 23 frequency bands, which are, respectively, recorded as E1,E2,…,E8. Finally, a set of vectors containing 23 components is obtained as the time-frequency domain characteristics of the signal:(6)Xtf=E1,E2,…,E8.Combining formulas (1) and (6), we can get(7)XMixed=Xt&f,XtfN∗M.In this paper,XMixed contains 37 feature components. However, the calculation of the above extraction process is complex, and the features are inevitably redundant and affect the calculation speed. Considering the MCFS is used for unsupervised feature selection, it only involves the sparse eigenproblem and ℓ1 regularization least squares problem. In the process of dimensionality reduction, MCFS can best retain the multicluster structure of data and provide the most effective feature samples of the signals. To fully improve the classification accuracy of rolling bearing fault diagnosis and save the time of subsequent evaluation, MCFS is introduced for feature dimensionality reduction and optimization. The algorithm specifically includes three steps [50]:(1)
Spectral Embedding for Cluster Analysis. The process first constructs a nearest neighbor graph, and the sample setX=E1,E2,…,EN,Ei∈RM is given. Set an edge between the sample points with the nearest neighbor relationship, and the corresponding weight Wij can be defined in the following ways:(i)
0-1 weight: IfEi is the nearest neighbor of Ej (or Ej is Ei), then Wij=1; otherwise Wij=0.(ii)
Thermonuclear weight: If there is a neighbor relationship betweenEi and Ej, the weight between the two nodes is(8)Wij=e−Ei−Ej2/σ.(iii)
Dot product weight: If there is a neighbor relationship betweenEi and Ej, the weight between the two nodes is(9)Wij=EiTEj.According to the adjacent weight matrix, the Laplacian matrixL=D−W can be obtained, where D is a diagonal matrix and satisfies Dii=∑jWij. Spectral clustering solves(10)Ly=λDy.In the above formula, the feature vector corresponding to the minimum eigenvalue constitutesY=y1,y2,…,yk, i.e., clustering result, k is usually the number of clusters, and yk reflects the data distribution of each cluster.(2)
Learn the sparse coefficient vector. After getting the flat embeddingy of the data points, the minimum fitting error is used to measure the importance of each feature yk relative to the 8-dimension, and the sparse coefficient vector is learned. Its optimization function is(11)minakyk−ETak2+βak,whereak is an M-dimensional vector (E is an N ∗ M-dimensional matrix) and contains coefficients for each feature when approximating yk. The above formula can be solved by the Least Angle Regression algorithm.(3)
Feature extraction. Using the original data to sparsely reconstruct the clustering results, define the MCFS score of each feature, as follows:(12)MCFSj=maxkak,j,whereak,j is the jth element in ak. According to the MCFS score, all features are sorted in descending order, and the first d features are selected.Finally, through the above analysis and optimization process, the optimal feature vector is obtained:(13)XMCFS=E1,E2,…,EN,Ei∈Rd.


## 3. Construction of an Optimal Fault Diagnosis Model Based on Self-Adaptive SVM

### 3.1. The Improved Algorithm of GWO

Traditional GWO algorithm optimizes employing wolves tracking, encircling, chasing, attacking, and so on. It has the characteristics of a few parameters, easy implementation, and strong convergence performance [51]. The mathematical model of surrounding behavior is as follows:(14)D=CXpt−Xt,Xt+1=Xpt−AD.Among them,D is the distance between the gray wolf and its prey, t is the current iteration number, Xp is the prey position vector, X is the gray wolf position vector, and A, C are the parameter vectors (C=2⋅r1, A=2a⋅r2−a, a, linearly decrease from 2 to 0 during the iteration, r1 and r2 are random vectors between [0, 1]). Therefore, the gray wolf group updates its position according to the position of α, β, δ wolves to simulate the hunting behavior, which is described as follows:(15)Dα=C1Xαt−Xt,(16)Dβ=C2Xβt−Xt,(17)Dδ=C3Xδ−Xt,(18)X1t+1=Xβt−A2Dα,(19)X2t+1=Xβt−A2Dβ,(20)X3t+1=Xδt−A3Dδ,(21)Xt+1=X1+X2+X33.From equations (15)–(20), we can calculate the distance between other gray wolf individuals and α,β,δ and the location of the updated gray wolf. From equation (21) we can determine the location of the prey.However, GWO has the phenomenon of the disparity decrease when the population iterates to a certain region [52, 53]. To this end, the DE strategy is introduced to improve the global search ability, effectively avoiding the problems of precocious stagnation and falling into local optima.The optimization function of the DE algorithm includes three operations: mutation, crossover, and selection. Defined in the D-dimensional search space, the population size isN, Xit is the ith individual of the tth generation (XiLt≤Xit≤XiUt), and then(22)Xit=Xi1t,Xi2t,…,Xint,i=1,2,…,N,t=1,2,…,tmax.In the equation,XiUt is the upper bound, XiLt is the lower bound of the individual population, tmax is the maximum number of iterations, and the calculation steps are as follows: N initial population individuals are randomly generated in the whole search space.(23)Xij0=XijL+rand0,1XijU−XijL.In the equation,rand0,1 is a random number obeying uniform distribution on [0, 1].(1)
Mutation operation. The individuals are generated by the following equation:(24)Dijt+1=Xr1t+FrXr2t−Xr3t,wheret is the current number of iterations, XX∈0,2 is the scaling factor, r1, r2, and r3 are random integers that are not equal to i, and they are not equal to each other r1,r2,r3∈1,N.(2)
Cross operation. Complete the exchange of some elements between the old and new individuals to increase the diversity of the population. The operation is as follows:(25)Uijt+1=Dijt+1,rand0,1<CRorj=rand1,nXijt,rand0,1≥CRorj≠rand1,n,whereCR is the crossover probability, and rand0,1 is a random number in the interval [0, 1].(3)
Select operation. Using the greedy algorithm, the vectorUit+1 and Xit are compared by the evaluation function:(26)Xit+1=Uit+1,fUit+1≤fXitXit,fUit+1>fXit.In summary, the idea of the DEGWO algorithm is as follows: Firstly, the mutation and selection operation of the DE algorithm is used to maintain the diversity of the population [43]. Secondly, it is regarded as the initial population of the GWO algorithm. Thirdly, calculate the target function value of the gray wolf individual, select the optimal Xα, Xβ, Xδ, and update other gray wolf individual positions accordingly. Then use crossover and selection operations to update the individual positions of the gray wolf, and repeat iterative updating until the optimal target function value is selected and output.


### 3.2. The Self-Adaptive Model of SVM

Firstly, considering that SVM is a generalization of a simple and intuitive Maximal Margin Classifier, it does not have self-adaptability. The theory of SVM can be consulted in [20, 54]. By introducing the algorithm of DEGWO, an adaptive model can be built, which is to realize the self-optimization of the penalty coefficient c and the radius σ in the kernel function κ. in the SVM [55]. Finally, the specific implementation steps of the rolling bearing fault classification model (DEGWO-SVM) are as follows:(1)
Initialization of GWO parameters. According to the parameters setting of the optimization algorithm in [34], group is 20, iteration is 200, and the crossover probability is 0.1. For details, please refer to [34, 35]. To verify the superiority of the proposed method in this paper, the specific initialization parameters of GWO are set, the population size is 10, the maximum number of iterations is 100, the crossover probability is 0.2, the search dimension is 2, and the search scope and scaling factor range is [0.2, 0.8].(2)
Initialization of the gray wolf population. Firstly, the mutation operation of the DE algorithm is carried out for the individuals of the population according to equation (24), and intermediates are generated. Then, the selection operation of the DE algorithm according to equation (26) is carried out to generate the individuals of the initial population, and the number of iterations t=1 is set.(3)
Evaluation of individual fitness. For any individual in the population, the SVM model is used to calculate the objective function of every individual of gray wolf in the population:(27)fitness=100−accuracyt100,wheret is the number of iterations and accuracyt is the classification accuracy of the tth iterations. The smaller the objective function value, the more accurate the SVM model classification.(4)
Determine whether the termination criterion is met (ortmax is reached). If not, perform Step 5∼Step 6. Otherwise, execute Step 7.(5)
Using the algorithm of DE to optimize GWO parameters [40, 56–58]. Firstly, the crossover operation of the DE algorithm is performed on the individual position of the population to retain the more excellent components according to equation (25). Then, select and generate new individuals of gray wolf according to equation (26), calculate the objective function values of all individuals, and t=t+1 is executed.(6)
The evolution of the gray wolf population. Sort according to the value of fitness function, and select the best three individualsXα, Xβ, and Xδ. Then, calculate the distance between other gray wolf individuals and the optimal Xα, Xβ, and Xδ according to the principle equation of the GWO algorithm and update the current positions of each gray wolf individual. Finally, the next generation of the gray wolf population is generated, and jump to Step 3.(7)
Training and prediction of SVM model. At the end of the execution of DEGWO algorithm,Bestc and Bestσ are obtained. Then, give SVM; use bearing rolling samples to train the model of SVM. Finally, the self-adaptive model of SVM is constructed, and the test samples are input to the model of SVM for prediction and the results are output.The execution flow of the DEGWO-SVM model is shown in Figure1.Figure 1
Execution process based on DEGWO-SVM model.


## 3.1. The Improved Algorithm of GWO

Traditional GWO algorithm optimizes employing wolves tracking, encircling, chasing, attacking, and so on. It has the characteristics of a few parameters, easy implementation, and strong convergence performance [51]. The mathematical model of surrounding behavior is as follows:(14)D=CXpt−Xt,Xt+1=Xpt−AD.Among them,D is the distance between the gray wolf and its prey, t is the current iteration number, Xp is the prey position vector, X is the gray wolf position vector, and A, C are the parameter vectors (C=2⋅r1, A=2a⋅r2−a, a, linearly decrease from 2 to 0 during the iteration, r1 and r2 are random vectors between [0, 1]). Therefore, the gray wolf group updates its position according to the position of α, β, δ wolves to simulate the hunting behavior, which is described as follows:(15)Dα=C1Xαt−Xt,(16)Dβ=C2Xβt−Xt,(17)Dδ=C3Xδ−Xt,(18)X1t+1=Xβt−A2Dα,(19)X2t+1=Xβt−A2Dβ,(20)X3t+1=Xδt−A3Dδ,(21)Xt+1=X1+X2+X33.From equations (15)–(20), we can calculate the distance between other gray wolf individuals and α,β,δ and the location of the updated gray wolf. From equation (21) we can determine the location of the prey.However, GWO has the phenomenon of the disparity decrease when the population iterates to a certain region [52, 53]. To this end, the DE strategy is introduced to improve the global search ability, effectively avoiding the problems of precocious stagnation and falling into local optima.The optimization function of the DE algorithm includes three operations: mutation, crossover, and selection. Defined in the D-dimensional search space, the population size isN, Xit is the ith individual of the tth generation (XiLt≤Xit≤XiUt), and then(22)Xit=Xi1t,Xi2t,…,Xint,i=1,2,…,N,t=1,2,…,tmax.In the equation,XiUt is the upper bound, XiLt is the lower bound of the individual population, tmax is the maximum number of iterations, and the calculation steps are as follows: N initial population individuals are randomly generated in the whole search space.(23)Xij0=XijL+rand0,1XijU−XijL.In the equation,rand0,1 is a random number obeying uniform distribution on [0, 1].(1)
Mutation operation. The individuals are generated by the following equation:(24)Dijt+1=Xr1t+FrXr2t−Xr3t,wheret is the current number of iterations, XX∈0,2 is the scaling factor, r1, r2, and r3 are random integers that are not equal to i, and they are not equal to each other r1,r2,r3∈1,N.(2)
Cross operation. Complete the exchange of some elements between the old and new individuals to increase the diversity of the population. The operation is as follows:(25)Uijt+1=Dijt+1,rand0,1<CRorj=rand1,nXijt,rand0,1≥CRorj≠rand1,n,whereCR is the crossover probability, and rand0,1 is a random number in the interval [0, 1].(3)
Select operation. Using the greedy algorithm, the vectorUit+1 and Xit are compared by the evaluation function:(26)Xit+1=Uit+1,fUit+1≤fXitXit,fUit+1>fXit.In summary, the idea of the DEGWO algorithm is as follows: Firstly, the mutation and selection operation of the DE algorithm is used to maintain the diversity of the population [43]. Secondly, it is regarded as the initial population of the GWO algorithm. Thirdly, calculate the target function value of the gray wolf individual, select the optimal Xα, Xβ, Xδ, and update other gray wolf individual positions accordingly. Then use crossover and selection operations to update the individual positions of the gray wolf, and repeat iterative updating until the optimal target function value is selected and output.


## 3.2. The Self-Adaptive Model of SVM

Firstly, considering that SVM is a generalization of a simple and intuitive Maximal Margin Classifier, it does not have self-adaptability. The theory of SVM can be consulted in [20, 54]. By introducing the algorithm of DEGWO, an adaptive model can be built, which is to realize the self-optimization of the penalty coefficient c and the radius σ in the kernel function κ. in the SVM [55]. Finally, the specific implementation steps of the rolling bearing fault classification model (DEGWO-SVM) are as follows:(1)
Initialization of GWO parameters. According to the parameters setting of the optimization algorithm in [34], group is 20, iteration is 200, and the crossover probability is 0.1. For details, please refer to [34, 35]. To verify the superiority of the proposed method in this paper, the specific initialization parameters of GWO are set, the population size is 10, the maximum number of iterations is 100, the crossover probability is 0.2, the search dimension is 2, and the search scope and scaling factor range is [0.2, 0.8].(2)
Initialization of the gray wolf population. Firstly, the mutation operation of the DE algorithm is carried out for the individuals of the population according to equation (24), and intermediates are generated. Then, the selection operation of the DE algorithm according to equation (26) is carried out to generate the individuals of the initial population, and the number of iterations t=1 is set.(3)
Evaluation of individual fitness. For any individual in the population, the SVM model is used to calculate the objective function of every individual of gray wolf in the population:(27)fitness=100−accuracyt100,wheret is the number of iterations and accuracyt is the classification accuracy of the tth iterations. The smaller the objective function value, the more accurate the SVM model classification.(4)
Determine whether the termination criterion is met (ortmax is reached). If not, perform Step 5∼Step 6. Otherwise, execute Step 7.(5)
Using the algorithm of DE to optimize GWO parameters [40, 56–58]. Firstly, the crossover operation of the DE algorithm is performed on the individual position of the population to retain the more excellent components according to equation (25). Then, select and generate new individuals of gray wolf according to equation (26), calculate the objective function values of all individuals, and t=t+1 is executed.(6)
The evolution of the gray wolf population. Sort according to the value of fitness function, and select the best three individualsXα, Xβ, and Xδ. Then, calculate the distance between other gray wolf individuals and the optimal Xα, Xβ, and Xδ according to the principle equation of the GWO algorithm and update the current positions of each gray wolf individual. Finally, the next generation of the gray wolf population is generated, and jump to Step 3.(7)
Training and prediction of SVM model. At the end of the execution of DEGWO algorithm,Bestc and Bestσ are obtained. Then, give SVM; use bearing rolling samples to train the model of SVM. Finally, the self-adaptive model of SVM is constructed, and the test samples are input to the model of SVM for prediction and the results are output.The execution flow of the DEGWO-SVM model is shown in Figure1.Figure 1
Execution process based on DEGWO-SVM model.


## 4. Analysis of Rolling Bearings Fault Diagnosis Experiment

To verify the effectiveness of the proposed method, the vibration signals of rolling bearing under different operating conditions are used. By calculating the time domain, frequency domain, and wavelet packet energy spectrum of each signal, a mixed domain high-dimensional feature set is obtained. Then the MCFS algorithm is used for feature optimization, and the optimal feature vector is input to the DEGWO-SVM model for classification training. Finally, verify the accuracy of classification; the specific implementation steps are as follows:(1)
Using the original vibration signals of rolling bearings, 16 time domain features, 13 frequency domain features, and 8 wavelet packet energy spectra are obtained, and then fusion is carried out with the 37-dimensional mixed domain high-dimensional feature setXMixed. It is directly input into the SVM model for training and testing, and the effectiveness of XMixed is preliminarily analyzed.(2)
According to the idea of the MCFS algorithm,X′Mixed is analyzed by spectral embedding clustering, and then the sparse coefficient vector is learned to obtain the first N-dimensional optimal feature vector X′MCFS.(3)
According to the implementation process of the optimization algorithm based on DEGWO, the optimal fault diagnosis model of self-adaptive rolling bearing is obtained.(4)
The firstnn=1,…,d,…,N-dimension XMCFS is input into the DEGWO-SVM model in turn, and then the classification training and testing are carried out, respectively. According to the classification results, the interference components introduced in the mixed domain feature set are eliminated, and the first D-dimensional feature which is most convenient for recognition is selected as the optimal feature vector XMCFS′.(5)
The optimal feature vectorXMCFS′ is input into the optimal self-adaptive rolling bearing fault diagnosis model to analyze the optimal fitness curve and classification recognition accuracy.The overall flowchart of the scheme is shown in Figure2.Figure 2
Test flowchart.


### 4.1. Construction and Optimization of Feature Space

The experimental data in this article was obtained by the Electrical Engineering Laboratory of Case Western Reserve University (CWRU) [59]. The experimental platform consists of a motor, torque sensor, power meter, and electronic control equipment. The following uses the drive end vibration signal collected at a speed of 1750 r/min and a sampling frequency of 12 kHz as data samples. The three fault diameters of Inner Race, Outer Race, and Ball are 0.007 inches, 0.014 inches, and 0.021 inches, respectively, and the corresponding fault labels are B07, B14, B21, IR07, IR14, IR21, Normal, OR07, OR14, OR21, respectively. The schematic diagram and illustration of the experiment test platform of the rolling bearing are shown in Figure 3. Each kind of fault signal is divided into 60 groups, including 40 groups of training samples and 20 groups of testing samples. The signals of four kinds of bearing fault states are shown in Figure 4(a), in which the periodic shock of the Outer Race and Inner Race signals are more obvious. However, it is difficult to accurately diagnose 10 fault states only by time domain waveform, so we first extract 37-dimensional fault features of each fault state signal by the proposed method, as shown in Figures 4(b)–4(d). Meanwhile, all the experiments were completed by using MATLAB R2018b 64 bit under the environment of Intel(R) Core (TM) i5-7300HQ CPU @ 2.50 GHz 2.50 GHz, 4GRAM.Figure 3
Schematic diagram and illustration of the bearing experiment test platform.
(a)(b)Figure 4
Vibration signal sampling and mixed domain feature map.
(a)(b)(c)(d)If the wavelet packet energy is only used as the fault feature, we get more than 90% diagnosis accuracy. That is because, in the time-frequency domain, the wavelet packet can effectively separate both high-frequency and low-frequency components, and wavelet packet energy features can effectively distinguish fault states. For the 37-dimensional high-dimensional features we extracted, when the SVM parameters arec=1 and σ=1, respectively, the SVM diagnosis model gets 92.5% diagnosis accuracy. As shown in Figure 5(a), seven samples of B7 were misdiagnosed as IR21 and eight samples of B14 as IR21. When c=0.1 and σ=1, the diagnostic accuracy of the SVM model is 97%. As shown in Figure 5(b), six samples of B7 were misdiagnosed as B14. The results show that the diagnostic accuracy of SVM needs to be improved under the condition of random parameters, so we use the proposed DEGWO to self-adaptively determine the optimal parameters of the SVM mode.Figure 5
Test and prediction classification chart based on SVM model. (a) Test set classification chart. (b) Test set classification chart.
(a)(b)On the other hand, not all features can effectively distinguish different fault states. If all the features are used in the training model, the calculation amount is large and the time cost is high. Besides, interference components that are not conducive to diagnosis may be introduced. Therefore, MCFS is used to sort and optimize the mixed domain features. To verify the effectiveness of this method, we reduce the high-dimensional fault features through LS and MCFS, respectively, and the first 20 optimal features are shown in Tables3 and 4.Table 3
The no. 1–no. 10 features based on feature optimization.
Serial number12345678910MCFSE22E11E18E6E2E24E27E16E20E5LSE36E27E19E25E33E17E10E14E13E23Table 4
The no. 11–no. 20 features based on feature optimization.
Serial number11121314151617181920MCFSE8E23E19E26E25E29E9E28E30E32LSE21E15E11E8E26E9E34E37E12E7The new feature setXNew=E1,E2,…EN is constructed by the above method, and the first n (n=1,…,d,…,N) items are selected in turn as fault features for classification training and testing. According to the classification effect, XOptimal=E1,E1,…,Ed is selected as the optimal feature vector to further improve the efficiency of rolling bearing fault identification.


### 4.2. Rolling Bearing Fault Diagnosis Based on DEGWO-SVM

To solve the problem that the penalty coefficientc and kernel parameter σ of SVM are not self-adaptive, we, respectively, use GWO and the proposed DEGWO to optimize the parameters of SVM. Besides, the first n (n=1,…,d,…,N) optimal features obtained from LS, MCFS, and PCA are taken as training and testing sets of the self-adaptive SVM model, respectively, and the change of diagnostic accuracy with n is shown in Figure 6.Figure 6
Decomposition results evaluation of simulated signalx(t). (a) Recognition rate based on GWO-SVM classifier. (b) Recognition rate based on DEGWO-SVM classifier.
(a)(b)As shown in Figure6(a), when the GWO-SVM model is used to identify the first n-dimensional features obtained by LS, PCA, and MCFS, respectively, PCA can get the best accuracy of 98.5% at E19 (i.e., the first 13-dimensional best features), and LS is 99.5% at E27 (n=8). It is worth noting that MCFS can get the best precision of 100% at E2 (n=5). The results show that the optimal features of MCFS can more accurately represent the fault state and improve diagnostic accuracy. In addition, the optimal feature dimension obtained by MCFS is the least, which greatly reduces the calculation amount of the diagnosis model and improves the diagnosis efficiency.As shown in Figure6(b), when using the DEGWO-SVM model to identify the first n-dimensional features obtained by LS, PCA, and MCFS, respectively, PCA can get the best accuracy of 98.5% at E5 (i.e., the first 10-dimensional best features), and LS is 99.5% at E16 (n=8). It is worth noting that MCFS is 100% at E2 (n=5), and the results also show that the optimal features of MCFS can more accurately represent the fault state and improve the diagnostic accuracy. In addition, compared with the GWO-SVM model, the DEGWO-SVM model can get the best accuracy of 98.5% at E5 (i.e., the first 10-dimensional best features) when the PCA feature is input, and the GWO-SVM is 98.5% at E19 (n=13). When LS features are input, the best precision of DEGWO-SVM is 100% at E16 (i.e., the first 8-dimensional best features), while GWO-SVM is 99.5% at E27 (n=8). When MCFS features are input, the best precision of DEGWO-SVM is 100% at E2 (i.e., the first 5-dimensional best features), and the GWO-SVM is 100% at E2 (n=5). The results show that the self-adaptability of DEGWO-SVM to fault features is stronger, and it only needs low-dimensional fault features to get better diagnosis results. In addition, the diagnostic accuracy of the DEGWO-SVM model is higher than that of GWO-SVM under the condition of the same features, and the convergence speed and accuracy of the model based on DEGWO are higher.Whenn∈1,10, due to the small amount of information that can represent the fault features, the recognition accuracy of the two models shows a rapid upward trend with the increase of n (especially when n=10, the classification accuracy reaches more than 99%, which can be used as the optimal feature vector). And MCFS > LS > PCA, so MCFS becomes the first choice to reduce the dimension of rolling bearing fault features. For the DEGWO-SVM model, the features E2, E24, E27, E16, E20, E5, and E8 are the most favorable for classification and identification, which need to be introduced. In the GWO-SVM model, only features E2 and E24 are the most advantageous. When two models identify the first n-dimensional features obtained by MCFS with n∈11,20, their accuracy decreases with increasing n. For the DEGWO-SVM model, the features E26, E25, E29, E9, and E28 are not conducive to recognition, so the interference components should be removed. GWO-SVM model also includes feature E19. When n∈21,37, because there are many features that can characterize the fault, although there are interference components, the recognition accuracy of the two models is over 99.5%. This shows that there is data redundancy in the high-dimensional features of a mixed domain. However, as shown in Table 5, when iterating to 100th times, the time consumption gap between the optimal feature vector and the mixed domain is significant. For DEGWO-SVM model, it takes 23.34 seconds to realize the fault diagnosis of the rolling bearing by using the features of the mixed domain. But the optimal feature vector obtained by MCFS only takes 9.34 seconds, and the efficiency improvement is particularly obvious. For GWO-SVM model, the time consumption is almost reduced by half. This is due to the high dimension of the feature set in mixed domain; the correlation between the features is very serious. The abovementioned problems will not only cause data redundancy, but also increase the computational pressure of the fault diagnosis model. Therefore, it is of great significance to select a small number of sensitive features containing significant category difference information from the mixed domain set of rolling bearing fault.Table 5
Comparison table of time consumption between optimal feature vector and mixed domain.
Optimization modelDEGWO-SVM (s)GWO-SVM (s)Optimal feature vectorBased on MCFS9.344.42Based on PCA15.585.85Based on LS14.153.79Mixed domain features23.348.01Based on the above analysis, the optimal feature vectorXOptimal=E1,E1,…,Ed constructed by MCFS, PCA, and LS is, respectively, used to train and test the models of DEGWO-SVM and GWO-SVM. In addition, the models of MPA-SVM and PSO-SVM are specially introduced for comparison. The parameters initialization settings of each model remain consistent. The comparison of the optimal fitness curves of the four self-adaptive models is shown in Figure 7.Figure 7
Comparison chart of the best fitness curve. (a) The best fitness curve based on DEGWO (MCFS). (b) The best fitness curve based on MPA (MCFS). (c) The best fitness curve based on GWO (MCFS). (d) The best fitness curve based on PSO (MCFS). (e) The best fitness curve based on DEGWO (PCA). (f) The best fitness curve based on MPA (PCA). (g) The best fitness curve based on GWO (PCA). (h) The best fitness curve based on PSO (PCA). (i) The best fitness curve based on DEGWO (LS). (j) The best fitness curve based on MPA (LS). (k) The best fitness curve based on GWO (LS). (l) The best fitness curve based on PSO (LS).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)As shown in Figures7(a), 7(e), and 7(i), the optimal feature vectors obtained by MCFS, PCA, and LS are, respectively, identified by using the DEGWO-SVM model. Most notably, (1) the fitness of MCFS has as low as 0.06 initially, the best fitness is 0, and the number of iterations to find the optimal solution is 5 (the time consumption is 0.47s). The best recognition rate is 100%. What’s more, the best fitness tends to be more stable after the 27th iterations. When iterating to 100th iterations, it only takes 9.34 seconds. (2) The initial fitness of PCA is 0.225, the best fitness is 0.015, and the number of iterations to find the optimal solution is 4 (the time consumption is 0.62 s). The best recognition rate is 98.5%. The best fitness tends to be more stable after the 70th iterations. Compared with the traditional feature reduction method (PCA), the proposed MCFS combined diagnosis model improves the accuracy from 98.5% to 100% with 1.5%, the initial fitness of MCFS is reduced by 0.165, and the time consumption is reduced by 66.8%. (3) The initial fitness of LS is 0.155, the best fitness of LS is 0, and the number of iterations to find the optimal solution is 2 (the time consumption is 0.28 s). The best recognition rate is 100%. However, compared with LS, the initial fitness of MCFS is reduced by 0.095. The initial fitness of LS is not as good as MCFS, and the time consumption is also increased by 51.5%. In summary, when using the optimal features of MCFS, the DEGWO-SVM model can obtain the highest classification accuracy and the fastest convergence speed in the shortest time. The performance of MCFS is remarkable, better than others.As shown in Figures7(a)–7(d), the DEGWO-SVM, MPA-SVM, GWO-SVM, and PSO-SVM models are, respectively, used to identify the optimal feature vectors obtained by MCFS. For the model of DEGWO-SVM, the number of iterations to find the optimal solution is as low as 5, while MPA-SVM is 13 (the time consumption is 7.79 s). Compared with MPA-SVM, the convergence performance of DEGWO-SVM is improved by 61.54%. Also compared with GWO-SVM and PSO-SVM, it is improved by 78.26% and 92.64%, respectively. The model of PSO-SVM even needs 68 (the time consumption is 6.46 s); the best fitness tends to stable. But the best fitness level of GWO-SVM will not be relatively stable until the 87th iterations. In addition, the initial fitness of the MPA-SVM, GWO-SVM, and PSO-SVM models is 0.205, 0.675, and 0.225. The effect of the above models is not as good as DEGWO-SVM. Experiments have proved that the above swarm intelligent algorithms and the latest MPA may have the problems of low solution accuracy and very easy precocious stagnation in the process of SVM parameter optimization. The model of DEGWO-SVM constructed by this method has strong self-adaptability and can obtain higher optimization accuracy and faster convergence speed and seek optimal performance in a shorter time. This also fully verifies the applicability and superiority of the method based on MCFS and DEGWO-SVM.Furthermore, the indexes ofBestc, Bestσ, accuracy, iteration, and time consumption are obtained by using the above scheme, and the specific detailed comparison is shown in Tables 6–9.Table 6
Comparison table of the DEGWO-SVM classification model.
Optimization modelDEGWO-SVMIndexBestcBestσAccuracyIterationsTime consumptionMCFS100.0059.52100%50.47 sPCA98.5075.4698.5%40.62 sLS100.0019.17100%20.28 sTable 7
Comparison table of the MPA-SVM classification model.
Optimization modelMPA-SVMIndexBestcBestσAccuracyIterationsTime consumptionMCFS61.300.12100%137.79 sPCA95.371.0198.50%1513.10 sLS89.160.09100%53.49 sTable 8
Comparison table of the GWO-SVM classification model.
Optimization modelGWO-SVMIndexBestcBestσAccuracyIterationsTime consumptionMCFS41.450.20100%231.02 sPCA60.901.0298.5%402.34 sLS78.130.06100%60.23 sTable 9
Comparison table of the PSO-SVM classification model.
Optimization modelPSO-SVMIndexBestcBestσAccuracyIterationsTime consumptionMCFS3.402.06100%686.46 sPCA49.131.0697.50%495.23 sLS43.750.63100%131.22 s


## 4.1. Construction and Optimization of Feature Space

The experimental data in this article was obtained by the Electrical Engineering Laboratory of Case Western Reserve University (CWRU) [59]. The experimental platform consists of a motor, torque sensor, power meter, and electronic control equipment. The following uses the drive end vibration signal collected at a speed of 1750 r/min and a sampling frequency of 12 kHz as data samples. The three fault diameters of Inner Race, Outer Race, and Ball are 0.007 inches, 0.014 inches, and 0.021 inches, respectively, and the corresponding fault labels are B07, B14, B21, IR07, IR14, IR21, Normal, OR07, OR14, OR21, respectively. The schematic diagram and illustration of the experiment test platform of the rolling bearing are shown in Figure 3. Each kind of fault signal is divided into 60 groups, including 40 groups of training samples and 20 groups of testing samples. The signals of four kinds of bearing fault states are shown in Figure 4(a), in which the periodic shock of the Outer Race and Inner Race signals are more obvious. However, it is difficult to accurately diagnose 10 fault states only by time domain waveform, so we first extract 37-dimensional fault features of each fault state signal by the proposed method, as shown in Figures 4(b)–4(d). Meanwhile, all the experiments were completed by using MATLAB R2018b 64 bit under the environment of Intel(R) Core (TM) i5-7300HQ CPU @ 2.50 GHz 2.50 GHz, 4GRAM.Figure 3
Schematic diagram and illustration of the bearing experiment test platform.
(a)(b)Figure 4
Vibration signal sampling and mixed domain feature map.
(a)(b)(c)(d)If the wavelet packet energy is only used as the fault feature, we get more than 90% diagnosis accuracy. That is because, in the time-frequency domain, the wavelet packet can effectively separate both high-frequency and low-frequency components, and wavelet packet energy features can effectively distinguish fault states. For the 37-dimensional high-dimensional features we extracted, when the SVM parameters arec=1 and σ=1, respectively, the SVM diagnosis model gets 92.5% diagnosis accuracy. As shown in Figure 5(a), seven samples of B7 were misdiagnosed as IR21 and eight samples of B14 as IR21. When c=0.1 and σ=1, the diagnostic accuracy of the SVM model is 97%. As shown in Figure 5(b), six samples of B7 were misdiagnosed as B14. The results show that the diagnostic accuracy of SVM needs to be improved under the condition of random parameters, so we use the proposed DEGWO to self-adaptively determine the optimal parameters of the SVM mode.Figure 5
Test and prediction classification chart based on SVM model. (a) Test set classification chart. (b) Test set classification chart.
(a)(b)On the other hand, not all features can effectively distinguish different fault states. If all the features are used in the training model, the calculation amount is large and the time cost is high. Besides, interference components that are not conducive to diagnosis may be introduced. Therefore, MCFS is used to sort and optimize the mixed domain features. To verify the effectiveness of this method, we reduce the high-dimensional fault features through LS and MCFS, respectively, and the first 20 optimal features are shown in Tables3 and 4.Table 3
The no. 1–no. 10 features based on feature optimization.
Serial number12345678910MCFSE22E11E18E6E2E24E27E16E20E5LSE36E27E19E25E33E17E10E14E13E23Table 4
The no. 11–no. 20 features based on feature optimization.
Serial number11121314151617181920MCFSE8E23E19E26E25E29E9E28E30E32LSE21E15E11E8E26E9E34E37E12E7The new feature setXNew=E1,E2,…EN is constructed by the above method, and the first n (n=1,…,d,…,N) items are selected in turn as fault features for classification training and testing. According to the classification effect, XOptimal=E1,E1,…,Ed is selected as the optimal feature vector to further improve the efficiency of rolling bearing fault identification.


## 4.2. Rolling Bearing Fault Diagnosis Based on DEGWO-SVM

To solve the problem that the penalty coefficientc and kernel parameter σ of SVM are not self-adaptive, we, respectively, use GWO and the proposed DEGWO to optimize the parameters of SVM. Besides, the first n (n=1,…,d,…,N) optimal features obtained from LS, MCFS, and PCA are taken as training and testing sets of the self-adaptive SVM model, respectively, and the change of diagnostic accuracy with n is shown in Figure 6.Figure 6
Decomposition results evaluation of simulated signalx(t). (a) Recognition rate based on GWO-SVM classifier. (b) Recognition rate based on DEGWO-SVM classifier.
(a)(b)As shown in Figure6(a), when the GWO-SVM model is used to identify the first n-dimensional features obtained by LS, PCA, and MCFS, respectively, PCA can get the best accuracy of 98.5% at E19 (i.e., the first 13-dimensional best features), and LS is 99.5% at E27 (n=8). It is worth noting that MCFS can get the best precision of 100% at E2 (n=5). The results show that the optimal features of MCFS can more accurately represent the fault state and improve diagnostic accuracy. In addition, the optimal feature dimension obtained by MCFS is the least, which greatly reduces the calculation amount of the diagnosis model and improves the diagnosis efficiency.As shown in Figure6(b), when using the DEGWO-SVM model to identify the first n-dimensional features obtained by LS, PCA, and MCFS, respectively, PCA can get the best accuracy of 98.5% at E5 (i.e., the first 10-dimensional best features), and LS is 99.5% at E16 (n=8). It is worth noting that MCFS is 100% at E2 (n=5), and the results also show that the optimal features of MCFS can more accurately represent the fault state and improve the diagnostic accuracy. In addition, compared with the GWO-SVM model, the DEGWO-SVM model can get the best accuracy of 98.5% at E5 (i.e., the first 10-dimensional best features) when the PCA feature is input, and the GWO-SVM is 98.5% at E19 (n=13). When LS features are input, the best precision of DEGWO-SVM is 100% at E16 (i.e., the first 8-dimensional best features), while GWO-SVM is 99.5% at E27 (n=8). When MCFS features are input, the best precision of DEGWO-SVM is 100% at E2 (i.e., the first 5-dimensional best features), and the GWO-SVM is 100% at E2 (n=5). The results show that the self-adaptability of DEGWO-SVM to fault features is stronger, and it only needs low-dimensional fault features to get better diagnosis results. In addition, the diagnostic accuracy of the DEGWO-SVM model is higher than that of GWO-SVM under the condition of the same features, and the convergence speed and accuracy of the model based on DEGWO are higher.Whenn∈1,10, due to the small amount of information that can represent the fault features, the recognition accuracy of the two models shows a rapid upward trend with the increase of n (especially when n=10, the classification accuracy reaches more than 99%, which can be used as the optimal feature vector). And MCFS > LS > PCA, so MCFS becomes the first choice to reduce the dimension of rolling bearing fault features. For the DEGWO-SVM model, the features E2, E24, E27, E16, E20, E5, and E8 are the most favorable for classification and identification, which need to be introduced. In the GWO-SVM model, only features E2 and E24 are the most advantageous. When two models identify the first n-dimensional features obtained by MCFS with n∈11,20, their accuracy decreases with increasing n. For the DEGWO-SVM model, the features E26, E25, E29, E9, and E28 are not conducive to recognition, so the interference components should be removed. GWO-SVM model also includes feature E19. When n∈21,37, because there are many features that can characterize the fault, although there are interference components, the recognition accuracy of the two models is over 99.5%. This shows that there is data redundancy in the high-dimensional features of a mixed domain. However, as shown in Table 5, when iterating to 100th times, the time consumption gap between the optimal feature vector and the mixed domain is significant. For DEGWO-SVM model, it takes 23.34 seconds to realize the fault diagnosis of the rolling bearing by using the features of the mixed domain. But the optimal feature vector obtained by MCFS only takes 9.34 seconds, and the efficiency improvement is particularly obvious. For GWO-SVM model, the time consumption is almost reduced by half. This is due to the high dimension of the feature set in mixed domain; the correlation between the features is very serious. The abovementioned problems will not only cause data redundancy, but also increase the computational pressure of the fault diagnosis model. Therefore, it is of great significance to select a small number of sensitive features containing significant category difference information from the mixed domain set of rolling bearing fault.Table 5
Comparison table of time consumption between optimal feature vector and mixed domain.
Optimization modelDEGWO-SVM (s)GWO-SVM (s)Optimal feature vectorBased on MCFS9.344.42Based on PCA15.585.85Based on LS14.153.79Mixed domain features23.348.01Based on the above analysis, the optimal feature vectorXOptimal=E1,E1,…,Ed constructed by MCFS, PCA, and LS is, respectively, used to train and test the models of DEGWO-SVM and GWO-SVM. In addition, the models of MPA-SVM and PSO-SVM are specially introduced for comparison. The parameters initialization settings of each model remain consistent. The comparison of the optimal fitness curves of the four self-adaptive models is shown in Figure 7.Figure 7
Comparison chart of the best fitness curve. (a) The best fitness curve based on DEGWO (MCFS). (b) The best fitness curve based on MPA (MCFS). (c) The best fitness curve based on GWO (MCFS). (d) The best fitness curve based on PSO (MCFS). (e) The best fitness curve based on DEGWO (PCA). (f) The best fitness curve based on MPA (PCA). (g) The best fitness curve based on GWO (PCA). (h) The best fitness curve based on PSO (PCA). (i) The best fitness curve based on DEGWO (LS). (j) The best fitness curve based on MPA (LS). (k) The best fitness curve based on GWO (LS). (l) The best fitness curve based on PSO (LS).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)As shown in Figures7(a), 7(e), and 7(i), the optimal feature vectors obtained by MCFS, PCA, and LS are, respectively, identified by using the DEGWO-SVM model. Most notably, (1) the fitness of MCFS has as low as 0.06 initially, the best fitness is 0, and the number of iterations to find the optimal solution is 5 (the time consumption is 0.47s). The best recognition rate is 100%. What’s more, the best fitness tends to be more stable after the 27th iterations. When iterating to 100th iterations, it only takes 9.34 seconds. (2) The initial fitness of PCA is 0.225, the best fitness is 0.015, and the number of iterations to find the optimal solution is 4 (the time consumption is 0.62 s). The best recognition rate is 98.5%. The best fitness tends to be more stable after the 70th iterations. Compared with the traditional feature reduction method (PCA), the proposed MCFS combined diagnosis model improves the accuracy from 98.5% to 100% with 1.5%, the initial fitness of MCFS is reduced by 0.165, and the time consumption is reduced by 66.8%. (3) The initial fitness of LS is 0.155, the best fitness of LS is 0, and the number of iterations to find the optimal solution is 2 (the time consumption is 0.28 s). The best recognition rate is 100%. However, compared with LS, the initial fitness of MCFS is reduced by 0.095. The initial fitness of LS is not as good as MCFS, and the time consumption is also increased by 51.5%. In summary, when using the optimal features of MCFS, the DEGWO-SVM model can obtain the highest classification accuracy and the fastest convergence speed in the shortest time. The performance of MCFS is remarkable, better than others.As shown in Figures7(a)–7(d), the DEGWO-SVM, MPA-SVM, GWO-SVM, and PSO-SVM models are, respectively, used to identify the optimal feature vectors obtained by MCFS. For the model of DEGWO-SVM, the number of iterations to find the optimal solution is as low as 5, while MPA-SVM is 13 (the time consumption is 7.79 s). Compared with MPA-SVM, the convergence performance of DEGWO-SVM is improved by 61.54%. Also compared with GWO-SVM and PSO-SVM, it is improved by 78.26% and 92.64%, respectively. The model of PSO-SVM even needs 68 (the time consumption is 6.46 s); the best fitness tends to stable. But the best fitness level of GWO-SVM will not be relatively stable until the 87th iterations. In addition, the initial fitness of the MPA-SVM, GWO-SVM, and PSO-SVM models is 0.205, 0.675, and 0.225. The effect of the above models is not as good as DEGWO-SVM. Experiments have proved that the above swarm intelligent algorithms and the latest MPA may have the problems of low solution accuracy and very easy precocious stagnation in the process of SVM parameter optimization. The model of DEGWO-SVM constructed by this method has strong self-adaptability and can obtain higher optimization accuracy and faster convergence speed and seek optimal performance in a shorter time. This also fully verifies the applicability and superiority of the method based on MCFS and DEGWO-SVM.Furthermore, the indexes ofBestc, Bestσ, accuracy, iteration, and time consumption are obtained by using the above scheme, and the specific detailed comparison is shown in Tables 6–9.Table 6
Comparison table of the DEGWO-SVM classification model.
Optimization modelDEGWO-SVMIndexBestcBestσAccuracyIterationsTime consumptionMCFS100.0059.52100%50.47 sPCA98.5075.4698.5%40.62 sLS100.0019.17100%20.28 sTable 7
Comparison table of the MPA-SVM classification model.
Optimization modelMPA-SVMIndexBestcBestσAccuracyIterationsTime consumptionMCFS61.300.12100%137.79 sPCA95.371.0198.50%1513.10 sLS89.160.09100%53.49 sTable 8
Comparison table of the GWO-SVM classification model.
Optimization modelGWO-SVMIndexBestcBestσAccuracyIterationsTime consumptionMCFS41.450.20100%231.02 sPCA60.901.0298.5%402.34 sLS78.130.06100%60.23 sTable 9
Comparison table of the PSO-SVM classification model.
Optimization modelPSO-SVMIndexBestcBestσAccuracyIterationsTime consumptionMCFS3.402.06100%686.46 sPCA49.131.0697.50%495.23 sLS43.750.63100%131.22 s


## 5. Conclusions

A fault diagnosis method for rolling bearings based on feature optimization and self-adaptive SVM is proposed in this paper. Compared with those of the traditional methods of feature reduction and diagnostics, the effectiveness and superiority of the proposed method above are fully verified. The main contributions of this method are as follows:(a)
For the high-dimensional features of mixed domain of the rolling bearing vibration signal, the dimension is reduced by MCFS algorithm. It can ensure the best recognition accuracy and realize the construction of the best feature space under the selection of the minimal feature dimension. Compared with the methods based on PCA and LS, the initial fitness and recognition accuracy are all significantly improved. Moreover, compared with the method based on LS and DEGWO-SVM, the method in this paper can enter the state of higher recognition accuracy more quickly. All these fully prove that, after using MCFS algorithm to sort the mixed domain features, a small number of sensitive features containing significant category difference information can be directly selected.(b)
The optimization algorithm based on the mixture of DE and GWO can improve the global search ability and effectively avoid the problems of precocious stagnation and falling into local optimality, etc. The model of DEGWO-SVM constructed by this method has strong self-adaptability and can obtain higher optimization accuracy and faster convergence speed and seek optimal performance in a shorter time. Compared with the methods based on MPA-SVM, GWO-SVM, and PSO-SVM and traditional SVM, the number of iterations to find the optimal solution is reduced obviously. The applicability and superiority are fully verified. This manuscript has fully demonstrated the strong self-adaptability of this method through large number of experiments and comparisons, and it can seek the best performance in the shortest time.Therefore, this method is expected to be used in the field of composite fault diagnosis and provide new reference for fault diagnosis technology.However, as new algorithms for feature extraction, feature selection, metaheuristics, machine learning, and deep learning are constantly being proposed, the authors will continue to make up for the shortcomings of this manuscript. For example, the metaheuristic algorithm is not novel enough, the generalization ability and robustness of the model need to be further improved, etc. Through the improvement of the above algorithms, the innovation of research work will be continuously improved. In addition, multiple test functions and data sets will be used to verify the methods and provide stronger evidence. Finally, some new methods are applied to actual engineering.



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