Fault diagnosis of bearing based on relevance vector machine classifier with improved binary bat algorithm for feature selection and parameter optimization

Abstract

In this article, fault diagnosis of bearing based on relevance vector machine classifier with improved binary bat algorithm is proposed, and the improved binary bat algorithm is used to select the appropriate features and kernel parameter of relevance vector machine. In the improved binary bat algorithm, the new velocities updating method of the bats is presented in order to ensure the decreasing of the probabilities of changing their position vectors’ elements when the position vectors’ elements of the bats are equal to the current best location’s element, and the increasing of the probabilities of changing their position vectors’ elements when the position vectors’ elements of the bats are unequal to the current best location’s element, which are helpful to strengthen the optimization ability of binary bat algorithm. The traditional relevance vector machine trained by the training samples with the unreduced features can be used to compare with the proposed improved binary bat algorithm–relevance vector machine method. The experimental results indicate that improved binary bat algorithm–relevance vector machine has a stronger fault diagnosis ability of bearing than the traditional relevance vector machine trained by the training samples with the unreduced features, and fault diagnosis of bearing based on improved binary bat algorithm–relevance vector machine is feasible.

Introduction

It is very significant to study the reliable fault diagnosis method to prevent the bearing from malfunction. ¹ Generally, the features of bearing vibration signal have a great influence on the diagnosis results of bearing, so it is necessary to select the appropriate features to improve the diagnosis effects. In addition, it is well known that good classifier is helpful to obtain good diagnosis results. Recently, the classifiers, such as Bayesian networks,^2–4 artificial neural networks, and support vector machine (SVM), have been widely applied in fault diagnosis field, among which SVM is a machine learning method based on structure risk minimization principle, and it can solve the classification problems with small training samples, high dimensions, and nonlinearity. Until now, SVM has been applied in fault diagnosis of rolling-element bearing, ¹ fault diagnosis of turbo-pump rotor, ⁵ fault diagnosis of rotor-bearing system, ⁶ fault diagnosis of power transmission system, ⁷ and so on.

Relevance vector machine (RVM) is an intelligent learning technique based on sparse Bayesian framework. ⁸ The number of relevance vectors in RVM is much smaller than that of support vectors in SVM; thus, compared with SVM, RVM has a sparser representation. In addition, there is no need to set the penalty parameter in RVM, which makes RVM more convenient to use than SVM. Therefore, RVM has a better application prospect in fault diagnosis for bearing. As the selection of the features and the kernel parameter of RVM has a certain influence on the diagnosis results of RVM, the improved binary bat algorithm (IBBA) is used to select the appropriate features and kernel parameter of RVM. In the traditional binary bat algorithm, the change of the probabilities of changing their position vectors’ elements lacks the purpose and is in the state of disorder when the position vectors’ elements of the bats are unequal to the current best location’s element, which is unhelpful for the solution optimization. Therefore, the new velocities updating method of the bats is presented in the IBBA in order to ensure the decreasing of the probabilities of changing their position vectors’ elements when the position vectors’ elements of the bats are equal to the current best location’s element, and the increasing of the probabilities of changing their position vectors’ elements when the position vectors’ elements of the bats are unequal to the current best location’s element, which are helpful to strengthen the optimization ability of binary bat algorithm.

Therefore, fault diagnosis of bearing based on relevance vector machine classifier with improved binary bat algorithm (IBBA-RVM) is proposed in this article. Here, three IBBA-RVMs trained by the training samples with the respective appropriate features are employed to establish the diagnosis model with the form of binary tree in order to recognize the four states of bearing, including normal state, inner race fault, outer race fault, and ball fault. The traditional RVM trained by the training samples with the unreduced features can be used to compare with the proposed IBBA-RVM method. The experimental results indicate that IBBA-RVM has a stronger fault diagnosis ability of bearing than the traditional RVM trained by the training samples with the unreduced features, and fault diagnosis of bearing based on IBBA-RVM is feasible.

RVM classifier

Let $\mathbf{D}=(\mathbf{X},\mathbf{H})$ be a set of training samples, where $\mathbf{X}=\{{\mathbf{x}}_n{\}}_{n=1}^N,{\mathbf{x}}_n$ is the input vector, $H={\left\{h_n\right\}}_{n=1}^N,h_n$ is the corresponding output target, $h_n\in \{0,1\}$ , “0” and “1” denote two classes which the training samples belong to, and N is the number of training samples. The likelihood function obeys the Bernoulli distribution ⁸

$$P(\mathbf{H}|\mathbf{X})=\underset{n=1}{\overset{N}{\Pi }}\sigma {\left\{y({\mathbf{x}}_n;\mathbf{W})\right\}}^{h_n}{[1-\sigma \left\{y({\mathbf{x}}_n;\mathbf{W})\right\}]}^{1-h_n}$$

(1)

where $\mathbf{W}$ is the weight vector, and $\sigma \{y({\mathbf{x}}_n;\mathbf{W})\}$ is a predefined logistic sigmoid function, which can be expressed as follows

$$\sigma \left\{y({\mathbf{x}}_n;\mathbf{W})\right\}=\frac{1}{1+e^{-y({\mathbf{x}}_n;\mathbf{W})}}$$

(2)

The detailed description of RVM binary classifier can be shown in a study by Tipping. ⁸ Radial basis function (RBF) kernel function is employed in this RVM, which can be expressed as follows

$$K({\mathbf{x}}_n,{\mathbf{x}}_d)=\exp (-p{\Vert {\mathbf{x}}_n-{\mathbf{x}}_d\Vert }^2)$$

(3)

where $p(p>0)$ denotes the kernel parameter.

The improvement for the traditional binary bat algorithm

Binary bat algorithm is a kind of swarm-based binary intelligent optimization method inspired by the echolocation behavior of bats with the variations of pulse emission rates and loudness; ⁹ each bat represents a solution in the population, and binary coding for the solution to be optimized is performed in the binary bat algorithm. In the traditional binary bat algorithm, the velocity of each bat can be updated according to equations (4) and (5)

$$f_i=f_{min}+\beta \times (f_{max}-f_{min})$$

(4)

$${\mathbf{v}}_i(t+1)={\mathbf{v}}_i(t)+f_i\times ({\mathbf{z}}_i(t)-{\mathbf{z}}^{*})$$

(5)

where ${\mathbf{v}}_i(t+1)$ is the velocity of the ith bat at (t + 1)th time step, ${\mathbf{z}}_i(t)$ is the position vector of the ith bat at tth time step, $f_i$ is a frequency value which can be used to adjust the velocity change of the ith bat, $f_{max}$ and $f_{min}$ are maximum and minimum frequency values, respectively, $\beta$ is a randomly generated value in the range of 0–1, ${\mathbf{z}}^{*}$ is the obtained global best location (solution) among all bats so far, ${\mathbf{z}}^{*}=[\begin{array}{cccc}{z}_1^{*}\hfill & z_2^{*}\hfill & \cdots \hfill & z_L^{*}\hfill \end{array}]$ , and L denotes the length of the binary code string representing the solution.

A v-shaped transfer function and position updating rule ⁹ can be shown as follows

$$\mathrm{VT}(v_{i,j}(t+1))=\left|\frac{2}{\pi }\arctan (\frac{\pi }{2}{v}_{i,j}(t+1))\right|$$

(6)

$$z_{i,j}(t+1)=\{\begin{array}{cc}{\tilde{z}}_{i,j}(t)\hfill & \mathrm{rand}<\mathrm{VT}(v_{i,j}(t+1))\hfill \\ \multicolumn{1}{c}{z_{i,j}(t)}& \mathrm{rand}\ge \mathrm{VT}(v_{i,j}(t+1))\hfill \end{array}$$

(7)

where $v_{i,j}(t+1)$ is the velocity of the ith bat at (t + 1)th time step in jth dimension, $z_{i,j}(t+1)$ is the position of the ith bat at (t + 1)th time step in jth dimension, ${\tilde{z}}_{i,j}(t)$ is the complement of $z_{i,j}(t)$ , and rand is a randomly generated value in the range of 0–1.

The v-shaped transfer function is used to map the velocities of the bats into the probabilities of changing their position vectors’ elements. The probabilities of changing their position vectors’ elements should decrease when the position vectors’ elements of the bats are equal to the current best location’s element, and the probabilities of changing their position vectors’ elements should increase when the position vectors’ elements of the bats are unequal to the current best location’s element. However, in the traditional binary bat algorithm, the velocities of the bats keep invariability when the position vectors’ elements of the bats are equal to the current best location’s element; the velocities’ values $|v_{i,j}(t+1)|$ of the bats usually keep increasing or decreasing when the position vectors’ elements of the bats are unequal to the current best location’s element; that is, the probabilities of changing their position vectors’ elements usually keep increasing or decreasing when the position vectors’ elements of the bats are unequal to the current best location’s element, which indicates that the change of the probabilities of changing their position vectors’ elements lacks the purpose and is in the state of disorder when the position vectors’ elements of the bats are unequal to the current best location’s element, and it is unhelpful for the solution optimization.

In order to solve this problem, the new velocities updating method of the bats is presented in the IBBA, which can be shown as follows

$$\begin{array}{l}{v}_{i,j}(t+1)=v_{i,j}(t)-\omega (t)\times F(z_{i,j}(t),z_j^{\ast })\\ \times v_{i,j}(t)+f_i\times \left|z_{i,j}(t)-z_j^{\ast }\right|\times G(v_{i,j}(t))\end{array}$$

(8)

$$\omega (t)={\omega }_{\mathrm{upper}}[1-\exp (-\mu ·t)]$$

(9)

$$F(z_{i,j}(t),z_j^{*})=\{\begin{array}{cc}1\hfill & z_{i,j}(t)=z_j^{*}\hfill \\ \multicolumn{1}{c}{0}& z_{i,j}(t)\ne z_j^{*}\hfill \end{array}$$

(10)

$$G(v_{i,j}(t))=\{\begin{array}{cc}1\hfill & v_{i,j}(t)\ge 0\hfill \\ \multicolumn{1}{c}{-1}& v_{i,j}(t)<0\hfill \end{array}$$

(11)

subject to the constraints

$$\begin{array}{l}{v}_{i,j}(t+1)=v_{\max }-\omega (t)\times F(z_{i,j}(t),z_j^{\ast })\\ \times v_{\max },\mathrm{if}{v}_{i,j}(t+1)\ge v_{\max }\end{array}$$

(12)

$$\begin{array}{l}{v}_{i,j}(t+1)=-v_{\max }-\omega (t)\times F(z_{i,j}(t),z_j^{\ast })\\ \times (-v_{\max }),if{v}_{i,j}(t+1)\le -v_{\max }\end{array}$$

(13)

where $F(·)$ is the logistic function, $\omega (t)$ is the nonlinear variable weight, ${\omega }_{\mathrm{upper}}$ $(0<{\omega }_{\mathrm{upper}}<1)$ is the upper limit value of weight, and $\mu (0<\mu <1)$ is the constraint; the initial velocities of the bats are randomly generated values in the range of −0.5 to 0.5 here, and $v_{max}$ is the maximum velocity of the bats.

It can be seen that the probabilities of changing their position vectors’ elements decrease when the position vectors’ elements of the bats are equal to the current best location’s element, and the probabilities of changing their position vectors’ elements increase when the position vectors’ elements of the bats are unequal to the current best location’s element in the new velocities updating method of the bats. The nonlinear variable weight $\omega (t)$ is employed in order that the weight is smaller in the early stage and the weight keeps bigger value in the late stage; thus, the decreasing extent of the probabilities of changing their position vectors’ elements is smaller in the early stage, and the decreasing extent of the probabilities of changing their position vectors’ elements keeps bigger in the late stage when the position vectors’ elements of the bats are equal to the current best location’s element.

IBBA-based feature selection and parameter optimization of RVM classifier

As the selection of the features and the kernel parameter of RVM has a certain influence on the diagnosis results of RVM, the IBBA is used to select the appropriate features and kernel parameter p. The process of the selection of the features and the kernel parameter of RVM by using the IBBA can be described as follows

• Step 1. Code the features and kernel parameter p by binary mode. The features and kernel parameter p are been coded as $\begin{array}{cccccccc}{f}_1\hfill & f_2\hfill & \cdots \hfill & f_l\hfill & p_{l+1}\hfill & p_{l+2}\hfill & \cdots \hfill & p_L\hfill \end{array}$ . The first part of the binary code string represents the features, and l denotes the length of the binary code string representing the features; each feature in the first part of the binary code string is set to “0” or “1”, 0 denotes the unemployed feature, and 1 denotes the employed feature. The second part of the binary code string represents the kernel parameter p, and $(L-l)$ denotes the length of the binary code string representing the kernel parameter p.

• Step 2. Initialize the pulse emission rate $r_i$ and loudness $A_i$ and set the size of bat population, and so on. A population of bats are randomly generated from binary vectors with length L and number B. Check the binary code representing the features of each bat and ensure that one binary code representing the feature of each bat is “1” at least; that is, one feature is employed in each bat at least; check the binary code representing the kernel parameter and ensure that one binary code representing the kernel parameter is “1” at least.

• Step 3. Define the objective function and evaluate the performance of the location (solution) of each bat. Here, the hybrid method of 5-fold cross validation and 10-fold cross validation is used to define the objective function.

First of all, the training samples are divided equally into five subsets of samples, among which four subsets of samples are used to train the RVM model, and the remaining subset of samples are used to validate the RVM model. Each subset can be used once for validation. The total validation accuracy $A_{i,1}$ of the five subsets of samples can be obtained as follows

$$A_{i,1}=\frac{N_{i,c1}}{N_{t1}}$$

(14)

where $N_{t1}$ is the total number of the five subsets of samples, and $N_{i,c1}$ is the total number of the five subsets of samples with correct validation of the ith solution.

Then, the training samples are divided equally into ten subsets of samples, among which nine subsets of samples are used to train the RVM model, and the remaining subset of samples are used to validate the RVM model. Each subset can be used once for validation. The total validation accuracy $A_{i,2}$ of the ten subsets of samples can be obtained as follows

$$A_{i,2}=\frac{N_{i,c2}}{N_{t2}}$$

(15)

where $N_{t2}$ is the total number of the ten subsets of samples, and $N_{i,c2}$ is the total number of the ten subsets of samples with correct validation of the ith solution.

Here, the objective function can be given as follows

$$f({\mathbf{z}}_i)=1-A_i$$

(16)

where $A_i$ is the mean value of $A_{i,1}$ and $A_{i,2}$ , which can be shown as follows

$$A_i=\frac{A_{i,1}+A_{i,2}}{2}$$

(17)

The smaller the value of the objective function is, the better the solution is. Then, obtain the global best location (solution) ${\mathbf{z}}^{*}$ among all bats so far according to the objective function given by equation (16).

• Step 4. Update the velocity vectors’ elements of the bats according to equation (4) and equations (8)–(13), and update the position vectors’ elements of the bats according to the v-shaped transfer function and position updating rule given by equations (6) and (7).

• Step 5. Produce a random number rand, if $\mathrm{rand}>r_i$ ( $r_i$ is the pulse emission rate of the ith bat), then generate new solution by changing any one binary code in the binary code string of the features and the kernel parameter of global best location (solution), respectively, that is, two binary codes including any one feature’s binary code and any one binary code of the kernel parameter are changed in global best location (solution); check the binary code representing the features of each solution and ensure that one binary code representing the feature of each solution is “1” at least; that is, one feature is employed in each solution at least; and check the binary code representing the kernel parameter and ensure that one binary code representing the kernel parameter is “1” at least.

• Step 6. Evaluate the new solution according to the objective function given by equation (16).

• Step 7. Produce an another random number rand, if $\mathrm{rand}<A_i$ ( $A_i$ is the loudness of the ith bat) and $f({\mathbf{z}}_i)<f({\mathbf{z}}^{*})$ , accept the new solution, and the loudness $A_i$ and pulse emission rate $r_i$ are updated according to equations (18) and (19)

$$A_i(t+1)=\alpha A_i(t)$$

(18)

$$r_i(t+1)=r_{\mathrm{upper}}[1-\exp (-\lambda ·t)]$$

(19)

where $\alpha (0<\alpha <1)$ and $\lambda (\lambda >0)$ are the constraints, and $r_{\mathrm{upper}}$ is the upper limit value of the pulse emission rate.

• Step 8. Update the current best solution ${\mathbf{z}}^{*}$ .

• Step 9. The same procedures from Step 4 to Step 8 are repeated until the maximum iteration is reached.

• Step 10. Decode the best solution, and the optimized features and kernel parameter can be obtained.

In order to obtain excellent optimization results, the optimization process can be repeatedly carried out several or many times.

Fault diagnosis process of bearing based on RVM classifier with IBBA

Fault diagnosis process of bearing based on RVM classifier with IBBA is given in Figure 1. First, three-level decomposition for bearing vibration signal is performed based on wavelet packet transform; each wavelet packet coefficient’s reconstructed signal represents a decomposed signal based on wavelet packet transform, and eight wavelet packet coefficients’ reconstructed signals of the third level are obtained; then, the relative Shannon entropies of eight wavelet packet coefficients’ reconstructed signals of the third level are computed by equation (20), and the eight relative Shannon entropies are obtained.

$${\hat{E}}_i=\frac{\left|e_i\right|}{\sum _{i=1}^8\left|e_i\right|}$$

(20)

where $e_i$ is the Shannon entropy of the ith wavelet packet coefficient’s reconstructed signal of the third level.

Figure 1.

Fault diagnosis process of bearing based on relevance vector machine classifier with improved binary bat algorithm.

Then, the appropriate features and kernel parameter of RVM can be obtained by using the IBBA, and RVM is trained by the training samples with the appropriate features. Three IBBA-RVMs trained by the training samples with the respective appropriate features are employed to establish the diagnosis model with the form of binary tree in order to recognize the four states of bearing, including normal state, inner race fault, outer race fault, and ball fault, among which IBBA-RVM1 is used to separate normal state from fault state, IBBA-RVM2 is used to separate inner race fault from other faults, and IBBA-RVM3 is used to separate outer race fault from ball fault. Finally, the proposed IBBA-RVM model can be tested by the testing samples.

Experimental analysis

Here, the bearing vibration data are obtained from “bearings vibration data set” of Case Western Reserve University. ¹⁰ The experimental data used here are collected under the condition of motor speed: 1772 r/min, among which the fault data are collected under the condition of single-point faults with fault diameter of 0.014 inches. 720 samples composed of 180 samples representing normal state, 180 samples representing inner race fault, 180 samples representing outer race fault, and 180 samples representing ball fault are employed in this study. Lots of training samples are helpful to obtain the optimized feature set. Therefore, in this study, the first 130 samples of each state are used as the training samples, and the remaining 50 samples of each state are used as the testing data; thus, the 200 samples are used as the testing samples altogether. Here, the eight wavelet packet coefficients’ reconstructed signals of one of the samples representing normal state can be shown in Figure 2.

Figure 2.

Eight wavelet packet coefficients’ reconstructed signals of one of the samples representing normal state.

The appropriate features and kernel parameter of RVM can be obtained by using the IBBA. In the IBBA, the initial pulse emission rate is set to 0.1,the initial loudness is set to 0.8, and $r_{\mathrm{upper}}$ is set to 0.5; $f_{max}$ is set to 1, and $f_{min}$ is set to 0; $\alpha$ is set to 0.7, and $\lambda$ is set to 0.95; $v_{max}$ is set to 12, ${\omega }_{\mathrm{upper}}$ is set to 0.7, and $\mu$ is set to 0.25. As the optimized features are obtained in the eight relative Shannon entropies, the length of the binary code string representing the features is 8. The value range of the kernel parameter p is [1, 2¹⁰−1], and the intervals of the adjacent values are 1; thus, the length of the binary code string representing the kernel parameter p is 10.

The traditional RVM trained by the training samples with the eight relative Shannon entropies can be used to compare with the proposed IBBA-RVM method. Here, grid method is used to select the kernel parameter of the traditional RVM; the value range of the kernel parameter is [1, 2¹⁰−1], and the intervals of the adjacent values are 1. Three traditional RVMs trained by the training samples with the eight relative Shannon entropies are employed to establish the diagnosis model with the form of binary tree, among which RVM1 is used to separate normal state from fault state, RVM2 is used to separate inner race fault from other faults, and RVM3 is used to separate outer race fault from ball fault. The kernel parameters of RVM1, RVM2, and RVM3 are respectively selected by grid method; that is, RVM1, RVM2, and RVM3 have respective kernel parameters.

Two kinds of IBBA-RVMs with different diagnosis results of bearing can be obtained by a series of repeated experiments, which can be defined as IBBA-RVM (the first situation) and IBBA-RVM (the second situation), respectively. Table 1 gives the diagnosis results of bearing by using IBBA-RVM (the first situation). As shown in Table 1, all the testing samples are correctly classified by using IBBA-RVM (the first situation). Thus, the diagnosis accuracy of IBBA-RVM (the first situation) for bearing is 100%. Table 2 gives the diagnosis results of bearing by using IBBA-RVM (the second situation). As shown in Table 2, only one sample representing inner race fault is misclassified by using IBBA-RVM (the second situation). Thus, 199 testing samples are correctly diagnosed in the 200 testing samples by using IBBA-RVM (the second situation), and the diagnosis accuracy of IBBA-RVM (the second situation) for bearing is 99.5%. It can be seen that IBBA-RVM can excellently recognize the four states of bearing. That is because three IBBA-RVMs including IBBA-RVM1, IBBA-RVM2, and IBBA-RVM3 have their respective appropriate features and kernel parameters, which make them have a strong classification ability.

Table 1.

The diagnosis results of bearing by using IBBA-RVM (the first situation).

	The number of testing samples	IBBA-RVM (the first situation)
		Normal state	Inner race fault	Outer race fault	Ball fault
Normal state	50	50	0	0	0
Inner race fault	50	0	50	0	0
Outer race fault	50	0	0	50	0
Ball fault	50	0	0	0	50

IBBA-RVM: improved binary bat algorithm–relevance vector machine.

Table 2.

The diagnosis results of bearing by using IBBA-RVM (the second situation).

	The number of testing samples	IBBA-RVM (the second situation)
		Normal state	Inner race fault	Outer race fault	Ball fault
Normal state	50	50	0	0	0
Inner race fault	50	0	49	0	1
Outer race fault	50	0	0	50	0
Ball fault	50	0	0	0	50

IBBA-RVM: improved binary bat algorithm–relevance vector machine.

Table 3 gives the diagnosis results of bearing by using the traditional RVM. As shown in Table 3, the traditional RVM with the eight relative Shannon entropies can excellently recognize normal state and outer race fault; however, the fault type of 12 samples representing inner race fault is misclassified as ball fault, and the fault type of 8 samples representing ball fault is misclassified as inner race fault. It can be seen that the traditional RVM with the eight relative Shannon entropies cannot excellently recognize inner race fault and ball fault. Thus, 180 testing samples are correctly diagnosed in the 200 testing samples by using the traditional RVM with the eight relative Shannon entropies, and the diagnosis accuracy of the traditional RVM with the eight relative Shannon entropies is 90%.

Table 3.

The diagnosis results of bearing by using the traditional RVM.

	The number of testing samples	RVM
		Normal state	Inner race fault	Outer race fault	Ball fault
Normal state	50	50	0	0	0
Inner race fault	50	0	38	0	12
Outer race fault	50	0	0	50	0
Ball fault	50	0	8	0	42

RVM: relevance vector machine.

The experimental results indicate that IBBA-RVM has a stronger fault diagnosis ability of bearing than the traditional RVM trained by the training samples with the unreduced features, and fault diagnosis of bearing based on IBBA-RVM is feasible.

Conclusion

In this article, fault diagnosis of bearing based on RVM classifier with IBBA is proposed, and the IBBA is used to select the appropriate features and kernel parameter of RVM. In the IBBA, the new velocities updating method of the bats is presented to ensure the decreasing of the probabilities of changing their position vectors’ elements when the position vectors’ elements of the bats are equal to the current best location’s element, and the increasing of the probabilities of changing their position vectors’ elements when the position vectors’ elements of the bats are unequal to the current best location’s element, which are helpful to strengthen the optimization ability of binary bat algorithm. Three IBBA-RVMs trained by the training samples with the respective appropriate features are employed to establish the diagnosis model with the form of binary tree in order to recognize the four states of bearing, including normal state, inner race fault, outer race fault, and ball fault. The traditional RVM trained by the training samples with the unreduced features can be used to compare with the proposed IBBA-RVM method. The experimental results indicate that IBBA-RVM has a stronger fault diagnosis ability of bearing than the traditional RVM trained by the training samples with the unreduced features, and fault diagnosis of bearing based on IBBA-RVM is feasible.

The bearing vibration data used here are acquired under the condition of steady load; however, it is very significant to study on fault diagnosis of bearing under varying load conditions; thus, the feasibility of application of IBBA-RVM in fault diagnosis of bearing under varying load conditions will be studied in the future.