Comparison of Support Vector Machine-Based Techniques for Detection of Bearing Faults

-is paper presents a method that combines Shuffled Frog Leaping Algorithm (SFLA) with Support Vector Machine (SVM) method in order to identify the fault types of rolling bearing in the gearbox. -e proposed method improves the accuracy of fault diagnosis identification after processing the collected vibration signals through wavelet threshold denoising. -e global optimization and high computational efficiency of SFLA are applied to the SVMmodel. Simulation results show that the SFLASVM algorithm is effective in fault diagnosis. Compared with SVM and Particle Swarm Optimization SVM (PSO-SVM) algorithms, it is demonstrated that the SFLA-SVM algorithm has the advantages of better global optimization, higher accuracy, and better reliability of diagnosis. Its accuracy is further improved through the integration of the wavelet threshold denoising method.


Introduction
Rolling bearing is an important rotating mechanical part widely used in the fields of aerospace and metallurgy and in automotive, manufacturing, and chemical industry.Its working condition affects the entire equipment as well as the whole production.Its failure can cause economic losses and possible personal injury [1].For example, a major derailment accident of Lanzhou Railway Branch 1479 train happened on November 30, 1991, due to poor-quality bearing and cage broken [2].In June 1992, a 600 MW supercritical forming unit from Japan Kansai Electric Power Company Hainan Power Plant in the speeding test caused a strong unit vibration due to the unit bearing failure and the critical speed drop.It not only damaged the aircraft but also resulted in an economic loss of up to 5 billion yen [3]. erefore, it is very important to detect and diagnose the rolling running bearings.
Around 60% of the mechanical equipment failures are caused by the gearbox, of which the failure caused by the bearings accounts for about 19% [4].Many methods such as PSO, Genetic Algorithm, and Ant Colony Algorithm were developed for the bearings fault analysis.In this paper, a combined SFLA-SVM method is proposed to diagnose the fault of the gearbox through the determination of the type of failure of the rolling bearing in the gearbox for the first time.
e performance of the new method is then compared with the SVM method and PSO-SVM method.

Shuffled Frog Leaping Algorithm (SFLA). SFLA is a swarm intelligence optimization algorithm proposed by
Eusuff and Lasey in 2001, which was refined in 2003 and 2006 [5].e algorithm is inspired by the biological foraging behavior in nature, in which the methods of local search and information sharing in the population are combined to carry out the computation of the global optimization of randomness and certainty.SFLA has the advantages of Mimetic Algorithm (MA) and PSO with local search capacities.By using the components-mixing-division of each meme form, it can achieve global information sharing and find the global optimal solution rather than the local optimum.e advantageous characteristics of SFLA include ease of setting up, high precision, fast convergence, and global optimization [6].e SFLA model is as follows: (1) Initialization of the SFLA parameters, including deciding the total number of frogs N , analyzing the experimental data, proposing frog heron number m, and setting the maximum distance D max that frog individuals can move to.Mimetic group evolution algebra is L. Algorithm mixed sort iteration number is G . e frog individual biggest search range is r max .(2) Calculation of the fitness value: Assume the first frog population is where ] is calculated first, and then the data are stored according to the size of the value of the sort, which is recorded as (3) Division of the population: e frog population is divided into m memes M 1 (t), . . ., M j (t), . . ., M m (t) j � 1, 2, . . ., m, using the following equation, and the best frog individual X j B (t) and the worst frog individual X j w (t) in each group are recorded: (4) Local search: According to the rules of e frog jumping step is decided by Equation ( 4) is used to update the value and calculate its fitness value.If the updated frog is better than the original frog, it will replace the original frog.Otherwise, replace X j B (t) with X g (t).Equations ( 3) and ( 4) are used to iterate each mimetic group.If the optimization fails, a new frog individual will be randomly formed to replace the original X j w (t).According to this process, there are L times of mimetic group to gain a new mimetic group (5) Mixing of populations: e evolved frog populations are mixed again to form U(t + 1), and the global best frog X g (t + 1) � U 1 (t + 1) is updated and recorded.en, the frogs in U(t + 1) are grouped once again.(6) If the number of iterations of the algorithm satisfies the condition t < G, go back to step (4); otherwise, output the best frog individual.

Support Vector Machine.
In 1995, Vapnik proposed a machine learning method, Support Vector Machine (SVM).SVM is a learning method based on statistical theory and risk minimization [7].Its core idea is to transform the problem of linear inseparability through the kernel formula, and then find the best classification surface in the space and the solution to the problem using convex quadratic programming [8].It successfully solved over learning and the local minimum problem.It also has the better generalization ability.e SVM algorithm model is as follows: (1) Give sample x i (i � 1, 2, . . ., l) and the matching Although there is no unified method to decide the best features C and σ, the methods of network search and cross-validation are normally selected [10].In this paper, SFLA-SVM model is proposed, whose process is outlined below: (  y i a * k(x i , x) + b * } and then decide the classification vector x.

Wavelet reshold Denoising.
A significant problem in wavelet threshold denoising is how to choose the threshold.If the chosen threshold is too small, the noise will largely remain in the signal.However, if the threshold is too large, it will remove useful and important characteristic information from the signal resulting in deviation.
erefore, the threshold will directly affect the denoising effect [11].
Another problem in wavelet threshold denoising is how to choose the threshold formula.Wavelet threshold denoising includes hard threshold denoising, soft threshold denoising, and default threshold denoising [12].Hard threshold formula and soft threshold formula are the two most commonly used threshold formulas.
e expression for a hard threshold formula is e expression of a soft threshold formula is In Equations ( 5) and ( 6), ω j,k is a wavelet coefficient,  ω j,k is the denoised wavelet coefficient, and T is a threshold value, whose formula is where σ is the standard deviation of the noise and N is the strength of the signal.After breaking up the noisy signal by wavelet, the signal has a larger amplitude than that the noise does.erefore, choosing the wavelet coefficients is achieved by setting the threshold.
e basic steps of the wavelet threshold denoising using (6) are as follows: (1) Use wavelet transform to break up the noisy signal f(k) and to obtain a set of wavelet coefficients ω j,k .(2) reshold the wavelet coefficient ω j,k to decide the estimated value of the wavelet coefficient  ω j,k , so that ‖ ω j,k − ω j,k ‖ is minimum.
(3) Use wavelet inverse transform to remake  ω j,k to gain the estimated signal  f(k), which is the signal after denoising.
ere is a difference between the wavelet coefficient gained by the soft threshold formula and that of the original signal.e hard threshold formula is not continuous at the threshold point.ese defects affect the effect of denoising.
erefore, in order to overcome the shortcomings of the traditional wavelet threshold, soft threshold and hard threshold, it is necessary to improve the selection of the threshold.e improved threshold is where j is the resolution scale.As the scale j increases, the threshold T decreases.Compared with the original method, the new threshold is more adaptive to separate noise at all levels.e improved threshold formula is where ω j,k is the wavelet coefficient,  ω j,k is the denoised wavelet coefficient, T is the threshold, and n is the adjustment parameter.e improved threshold formula has the advantages of both the soft and hard threshold formulas.

Application of SFLA-SVM Algorithm in
Gearbox Bearing Fault Diagnosis e table shows that some parameters are not independent.For example, the variance indicator, skewness indicator, and kurtosis indicator are related.
e margin indicator and square root amplitude are related as well.Due to the nonlinear and nonstationary nature of the vibration signal, the correlation of these features does not exhibit a linear relationship and there is no collinear relationship between these data.
e dimensionless eigenvalue indicators are calculated from the collected data, and they are then normalized to form a uni ed basis for the determination of the fault type.Table 2 shows the results of the eigenvalue indicators calculated using the training data while Table 3 is with the test data.e normalization formula used here is as follows [13]: where x ig is the normalized eigenvalue, x k is the kth eigenvalue, and x max and x min are the maximum and the minimum of x k , respectively.

Signal Denoising.
In order to reduce the in uence of noise on the vibration signal, the vibration signal under di erent conditions is denoised by the wavelet threshold and then identi ed by fault diagnosis.At rst, the simulation signal x(t) sin(2πt) and the noise signal n(t) rand(1, N), where t 0 : 0.01 : 4π and N length(t), are constructed and synthesized.e synthesized signal is then denoised by the wavelet adaptive threshold and the given threshold, respectively.e results are shown in Figure 7, from which it can be observed that with the given threshold denoising process, noise part of the signal is removed   and the remaining signal is evenly distributed.e e ect of adaptive threshold denoising is not ideal, and the noise is unevenly included in the original signal.erefore, the acquired signal should be denoised by adjusting the given threshold.

Shock and Vibration
ere are three ways to adjust the threshold: hard threshold, soft threshold, and default threshold denoising.e denoising results for di erent conditions are shown in Figures 8-11.Shock and Vibration e comparisons among the three denoising methods indicate that the characteristic information of the vibration signal obtained using the given soft-threshold denoising is better preserved without causing loss of signal characteristic information.erefore, we choose the signal with the given soft threshold to denoise under di erent working conditions and to identify fault by the SFLA-SVM algorithm.Shock and Vibration

Results and Discussion
With the denoised data signals under four working conditions, the fault diagnosis is simulated by SFLA-SVM algorithm, as outlined below.Firstly, the SFLA parameters are set as follows: (1) N is the total frog population, which is the primary parameter of the algorithm.It is related to the complexity and dimension of the problem to be solved.Considering the experiments undertaken previously in this, N is set to be 100.
(2) m is the number of subpopulations.If m is too small, the optimal information in the subpopulation cannot be completely shared in the local range.If m is too large, the evolution process becomes complicated and is easy to fall into local optimum.e number of subpopulations follows the relationship N m * n, where n represents the number of frogs contained in each subpopulation.In this experiment, m 10 and n 10. (3) G is the number of subpopulation iteration.A large G value will cause the "frog" to change its position frequently and ignore the information exchange

Shock and Vibration
between individuals, while a small G value may cause the meme group to fail to find an optimal solution and fall into a local optimum.G is set to be 10 in this experiment.(4) L is the total number of iterations.If L is too small, it will cause the "frog" to ignore the information exchange between individuals.If L is too large, it will increase computational workload and lead to local optimum.L is set to be 20 in this experiment.A more comprehensive comparison of SVM, PSO-SVM, and SFLA-SVM diagnostic results and running time are shown in Table 4.
Figures [12][13][14] indicate that the SFLA-SVM algorithm has the highest accuracy, and only 3 of its results are misclassi ed.In comparison, SVM and PSO-SVM algorithms have 13 and 9 misclassi cations, respectively.Meanwhile, the comprehensive comparative analysis of Table 4 indicates that optimizing

Shock and Vibration
SVM by the PSO and SFLA algorithms can improve the accuracy of the diagnosis results although the running time increases by 27.46 s and 55.30 s, respectively.is demonstrates that SFLA-SVM algorithm can signi cantly improve the accuracy of SVM recognition and does not cause much diagnostic e ciency loss, but with a marginally higher running time compared to the other two algorithms.
e vibration signal under di erent working conditions and the corresponding improved threshold denoised signal are analyzed in the frequency domain.
eir Fouriertransform diagram and power spectral density diagram are shown in Figures 15-18.
e comparative analysis of these gures indicates that the denoising with the improved threshold preserves the most characteristic information of the signal and generates very little distortion.e fault identi cation results are shown in Figure 19.
As shown in Figure 19, the SFLA-SVM algorithm produces only two misclassi cations after the wavelet  Shock and Vibration threshold denoising, which means that its accuracy has been improved to 97.5%.
Secondly, the SVM parameters are set as follows: ere are two main parameters of the SVM, namely, the penalty parameter C and the kernel function coefficient σ.In the Gaussian kernel function, k(x, z) � exp(−(‖x − z‖/2σ 2 )) has the kernel function coefficient σ. e SVM initialization parameters are C � 100, σ � 20. e parameters that have been optimized by SFLA are C � 94.3910, σ � 11.3558.e simulation results of the fault diagnosis by the SFLA-SVM algorithm are shown in Figure 12.

Figure 16 :Figure 15
Figure 16: (a) Rolling wear signal, rolling wear Fourier transform, and power spectral density plot; (b) denoising signal, Fourier transform, and power spectral density plot after improving threshold.

Figure 17 Figure 18 :
Figure 17: (a) Inner wear signal, inner wear Fourier transform, and power spectral density plot; (b) denoising signal, Fourier transform, and power spectral density plot after improving threshold.

Figure 19
Figure 19: SFLA-SVM predicted results from improving the threshold denoising.
In this paper, the experiment uses 6 vibration channels and 1 speed channel to collect signals simultaneously.With the time-domain signal gaining module, the maximum sampling points is 32768 and the maximum analysis frequency is 50 kHz.Once the signal is Selection of the Characteristic Value of the Fault.e eigenvalues of bearing faults are closely correlated to the accuracy of diagnosis.Various methods are proposed to extract the eigenvalue information.Eigenvalue indicators can be classi ed into the dimensional indicators, such as square root amplitude and variance, and the dimensionless indicators, such as waveform indicator, margin indicator, kurtosis indicator, and skewness indicator.eformulasfor calculating of the eigenvalue indicators are shown in Table1.

Table 3 :
Gearbox rolling bearing part of the test data.

Table 1 :
Calculation formulas of eigenvalue indicators.

Table 2 :
Gearbox rolling bearing part of the training data.