Performance Prediction of Rolling Bearing Using EEMD and WCDPSO-KELM Methods

Abstract

Research on bearings performance degradation trend is significant, and can greatly reduce the loss caused by potential faults in the whole life-cycle of rolling bearings. It is also a very important part of Prognostic and Health Management (PHM). This paper proposed a new performance degradation prediction method based on optimized kernel extreme learning machine (KELM), improved particle swarm optimization (PSO) and Ensemble Empirical Mode Decomposition (EEMD). Firstly, the particle swarm optimization algorithm was improved by adjusting the inertia weight and linear learning factor and introducing a disturbance term, namely WCDPSO. Then, the penalty coefficient and kernel parameters of KELM were optimized by the WCDPSO, and the WCDPSO-KELM model was obtained. Subsequently, the EEMD method was used to extract original features from sample data, and a performance degradation index is selected from the EEMD feature space, which was input into the WCDPSO-KELM model in order to build a bearing performance degradation prediction trend model. Finally, the proposed method was verified by datasets of rolling bearings that were provided by the PRONOSTIA platform. Experimental results confirmed that the proposed method can efficiently predict the performance degradation trend of rolling bearings.

1. Introduction

Rolling bearings are widely used in industrial installations and are one of the most important parts of rotating machinery. The running state of rolling bearings is vital for reliable operation of the whole mechanical equipment, as bearing faults represent 41% of all mechanical fault occurrences [1,2,3], Mostly equipment faults are caused by rolling bearing faults [4]. Bearing faults account for about 20% among all kinds of gear-box faults [5]. Bearing faults are responsible for 40 to 90% of unexpected induction motor shutdowns [6]. Monitoring and diagnosis bearings faults can reduce the accident percentage by about 75%, according to statistical data, and the corresponding maintenance cost is cut down by 25–50% [7].

Predicting the residual life and performance degradation trend of rolling bearings have become a research hotspot. It is also a very important part of Prognostics and Health Management (PHM) [8,9]. Much of the literature has employed a variety of techniques for predicting bearings’ residual life. There are many data-driven technologies; traditional prediction methods include regression-based models [10] such as the Wiener process [11], Gamma process [12], random filtering model [13], Hidden Markov model and semi-hidden Markov model [14,15]. For nonlinear systems, there is pattern recognition and machine-learning technology. In recent years, artificial neural networks and machine learning have been applied to the rolling bearings research field. In [6], Artificial Neural Networks models were proposed to estimate the remaining useful life of a defective bearing. In References [16,17], the authors proposed variation mode decomposition and a deep belief network to diagnosis rolling bearing faults. In [18], the authors proposed using high-order differential mathematical morphology gradient spectrum entropy and an extreme learning machine to predict bearing performance. Reference [19] used artificial intelligence and machine-learning approaches to design a classifier. The wavelet decomposition and support vector data description (SVDD) were introduced to train and predict health prediction index of rolling bearings in the whole life-cycle in [20]. In [21], the authors proposed the prediction residual life model of rolling bearings based on an optimized support vector machine (SVM), and parameters of the SVM model were optimized using phase-space reconstruction and the nearest point of neighbor domain method. Reference [22] put forward Principal Component Analysis (PCA) and Least Squares Support Vector Machines (LS-SVM); the LS-SVM model was built and trained for bearing degradation process prediction, and the multi-features fusion technique PCA was used to merge the original features and reduce the dimension. In [23], proportional risk model and logarithmic regression model were introduced to predict bearing residual life. The residual life of rolling bearings in helicopters was predicted using mainstream approaches and their basic framework in [24]. Reference [25] used feature spectrum and SVM classifier to predict the residual life of rolling bearings; the whole life-cycle of the monitored bearing was divided into different stages by a SVM classifier. The degradation state of the rolling bearing can be assessed by comparing distance between the newly measured data and the trained dataset. Edwin et al. [26] proposed using SVR based on a soft computing model to predict the residual life of rolling bearings.

In short, research on the residual life of bearings has become a hotspot, and many related research literatures have appeared in this field.

This paper presents a new method to predict the residual life of rolling bearings, based on the WCDPSO-KELM and Ensemble Empirical Mode Decomposition (EEMD) method. The penalty coefficient and kernel parameters of KELM are optimized by improved particle swarm optimization algorithm, then a different performance index is selected from feature space of the time–frequency domain, wavelet packet and EEMD, which are then divided into a training dataset and test dataset. This training dataset is input to train the WCDPSO-KELM model, and the test dataset is input to test performance of the proposed method. Finally, the experimental results verify the validity of the proposed method.

This paper is organized as follows: Section 2 presents the proposed methodology for building a prediction model and comparison with other models. Section 3 presents signal acquisition and processing. Section 4 presents experimental data results and analyzes the performance of different feature spaces under different load levels, while the conclusion is shown in Section 5.

2. Methodology

In this section, we first present an overview of the improving details of WCDPSO in Section 2.1. Subsequently, Section 2.2 presents the optimization workflow of the KELM. Finally, Section 2.3 assesses the performance of the WCDPSO-KELM Model by comparing it with other models.

2.1. The WCDPSO Algorithm

Fundamental particle swarm optimization (PSO) has become a research hotspot. It has simple structure and powerful searching capability [27,28,29]. However, improper PSO parameters can reduce the searching speed and accuracy of the algorithm, which causes particles to easily fall into local optimal value and poor late convergence. In order to compensate for these deficiencies, Shi et al. [30] modified the particle swarm optimization algorithm by introducing inertia weight, then renamed it WPSO.

The particle swarm optimization algorithm is improved by adjusting inertia weight and linear learning factor, and adding a disturbance term, then the WCD-PSO algorithm in the paper.

2.1.1. Improving Inertia Weight

The inertia weight $w$ linearly descends according to research results in [30], which gives particles strong searching capability at the early iteration stage and more large solution space. The convergence speed is faster at the late iteration stage. However, if the optimal particles are not obtained at the initial iteration stage, although the local search capability of particles is increased, it is easy to fall into the local optimal value. Reference [31] proposed the differential descending method to compensate for the deficiency of the PSO algorithm to some extent; as a result, the performance of the PSO algorithm could be improved.

Reference [32] proposed a nonlinear descending method by introducing exponential factor $\lambda$ and iterative threshold value $\max ge{n}_o$ to prevent particles falling into the local optimal value. The specific formulas are described as follows.

$$w\left(i\right)=w_{\max }-{\left(\frac{i-1}{\max ge{n}_0-1}\right)}^{\lambda }\left(w_{\max }-w_{\min }\right)$$

(1)

where, $w_{\max }$ and $w_{\min }$ are the maximum inertia weight and the minimum inertia weight, $i$ is the current iteration number, $\lambda$ is the exponential factor, and $\max ge{n}_0$ is the iteration threshold. The nonlinear decreasing method effectively prevents particles from falling into the local optimal value and speeds up its convergence.

The adjusting inertia weight method is improved according to the above-described methods; the inertia weight $w$ decreases with the increase in iteration number $i$, so it can improve the convergence speed of particles and the accuracy of the PSO algorithm. Thus, the differential descending method is proposed in the paper, and the specific formulas are described as follows.

$$w(i)=\left(w_{\max }-w_{\min }\right){\left(\frac{i}{\max gen}-1\right)}^2+0.4$$

(2)

where $\max gen$ is the maximum iteration number and the most proper offset is thought to be 0.4 after multiple simulations. The PSO algorithm is improved by adjusting inertia weight $w$, then it is renamed the WPSO algorithm.

2.1.2. Linear Adjustment Learning Factor

The linear adjusting learning factors method is proposed in order to search the large solution space and avoid falling into the local optimal value. The specific formulas are described as follows in Equations (3) and (4). The larger $c_1$ is, the more powerful the local searching capability of particles is; the larger $c_2$ is, the more powerful the global searching capability of particles is.

$$c_1=c_{1\max }+\left(c_{1\min }-c_{1\max }\right)\left(\frac{i}{\max gen}\right)$$

(3)

$$c_2=c_{2\max }+\left(c_{2\min }-c_{2\max }\right)\left(\frac{i}{\max gen}\right)$$

(4)

where $i$ is the current iteration number; $\max gen$ is the maximum iteration number; $c_1{}_{\max }$ and $c_2{}_{\max }$ are the initial values of $c_1$ and $c_2$, respectively; and $c_1{}_{\max }$ and $c_2{}_{\max }$ are the final values of $c_1$ and $c_2$, respectively. The performance of the PSO algorithm is improved by linearly adjusting the learning factor, then it is renamed as the CPSO algorithm.

2.1.3. Adding Disturbance Term

The global and local optimal solutions are adjusted by adding a disturbance term because there may be better particles in the vicinity of the global or local optimal particles. Thus, adding disturbance term can prevent particles from falling into the local optimal solution. Now, the velocity-updating formula of each particle is described as follows:

$$V_{id}(t+1)=w{V}_{id}(t)+c_1{r}_1(P_{id}(t)\ast (1\pm \delta \ast rand)-X_{id}(t))+c_2{r}_2(P_{gd}(t)\ast (1\pm \delta \ast rand)-X_{id}(t))$$

(5)

$1\pm \delta \ast rand$ is the added disturbance term and $\delta$ is a constant. The PSO algorithm is improved by adding a disturbance term, then it is renamed as the DPSO algorithm. The $1\pm \delta \ast rand$ is so small that its influence on the searching capability may be ignored at the early iteration stage, but the particle convergence speed becomes smaller and smaller at the late iteration stage. Now, the disturbance term $1\pm \delta \ast rand$ can prevent the particle searching speed from dropping to zero and falling into the local optimal.

2.2. Optimizing Parameters of KELM Model

Extreme Learning Machine (ELM) is a single hidden layer feedforward neural network [33]. Its learning speed is very fast, and the weights in the hidden layer and offset values can be randomly specified. The structure of ELM is shown in Figure 1.

Figure 1. The structure of the ELM.

N samples are $(x_i,t_i)\in R^{n\times m}$, the equations of ELM are the following:

$${\displaystyle \sum _{i=1}^L{\beta }_{i}g(W_i\ast x_j+b_i)}=o_j, j=1,2,\cdots ,N$$

(6)

where L is the number of hidden layer nodes, $g(\cdot )$ is the activation function, $W_i$ is the input weight vector, ${\beta }_i$ is the output weight vector, and $b_i$ is the offset of the $i^{th}$ hidden node. The learning purpose is to minimize the training errors. That is, ${\displaystyle {\sum }_{j=1}^L\left|\left|o_j-t_j\right|\right|}=0$ is the expression of ${\beta }_i$, $b_i$ and $W_i$:

$${\displaystyle \sum _{i=1}^L{\beta }_{i}g(W_i\ast x_j+b_i)}=t_j, j=1,2,\cdots ,N$$

(7)

The expression can be described.

$$H\beta =T$$

(8)

$$H=\left[\begin{array}{c}h(x_1)\\ ⋮\\ h(x_n)\end{array}\right]=\left[\begin{array}{c}g(W_1\ast x_1+b_1)\cdots g(W_L\ast x_1+b_L)\\ ⋮\\ g(W_1\ast x_n+b_1)\cdots g(W_L\ast x_n+b_L)\end{array}\right]$$

(9)

$$\beta ={\left[{\beta }_1^T,{\beta }_2^T,\cdots ,{\beta }_L^T\right]}_{L\times m}$$

(10)

$$T={\left[t_1^T,t_2^T,\cdots ,t_n^T\right]}_{n\times m}$$

(11)

Therefore, it is simple to solve the least-squares solution of linear systems. The output weight $\beta$ is described.

$$\beta =H^{+}T$$

(12)

where $H^{+}={(H^{T}H)}^{-1}{H}^T$ is pseudo inverse matrix of $H$. After the $\beta$ is calculated, the new expression is obtained.

$$f(x_t)=\beta h(x_t)$$

(13)

For the KELM, the kernel function needs to be introduced into the ELM. That is, the kernel matrix is $\Omega =H{H}^T$ and its element is $\Omega (i,j)=h(x_i)\ast h(x_j)=K(x_i,x_j)$. The paper selects the Gaussian Kernel Function:

$$K\left(x_i,x_j\right)=\exp \left(-\frac{{\left|x_i-x_j\right|}^2}{{\sigma }^2}\right)$$

(14)

The Kernel parameter $\sigma$ reflects the mapping relationship between the input space and the feature space. The mapping function determines the magnitude of the sample space, and affects the prediction accuracy of the KELM model. The optimizing parameters of the Kernel extreme learning machine are usually transformed into parameter estimation of multiple linear regression function [34]. The penalty coefficient and Kernel parameter $\sigma$ play an important role in KELM model, whether the penalty coefficient is too big or too small will affect the prediction results of KELM model. The WCDPSO algorithm is used to optimize the penalty coefficient and Kernel parameters $\sigma$ in the paper.

The WCDPSO algorithm is used to solve the parameter combinatorial optimization of the KELM model; the flowchart is shown in Figure 2.

Figure 2. The flowchart of parameter combinatorial optimization based on WCDPSO.

The specific steps of the WCDPSO-KELM model are as follows.

Step 1: Set the size of the particle swarm, the random speed and the position of the initial particle swarm.

Step 2: Select the fitness function of the WCDPSO algorithm. Here, the Root Mean Squared Error (RMSE) between the output of the trained dataset and the actual output is used as the fitness function, and fitness values of all particles are calculated; the formula is described as follows.

$$f\left(i\right)=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(y\left(i\right)-y^{\ast }\left(i\right)\right)}^2}}$$

(15)

where $f(i)$ is the fitness of the particle $i$, $N$ is the number of samples, $y(i)$ is the actual output of the particle $i$ and $y^{\ast }(i)$ is the predicted output of the particle $i$.

Step 3: Calculate the fitness of each particle according to Equation (15) and compare the current location of the particle with its historical optimal location and the optimal location of the population, then select the smaller one as the new optimal location of the individual and the new optimal location of the population.

Step 4: Update the velocity and position of each particle based on two extreme values which were obtained in Step 3.

Step 5: the iteration is yes or not; if not, return to Step 2. If yes, the iteration end and the optimal parameters are obtained.

Step 6: The WCDPSO-KELM prediction model is built.

2.3. Assessming the WCDPSO-KELM Model

The Kernel parameters and penalty coefficient of KELM are optimized by the WCDPSO algorithms, and the WCDPSO-KELM model is built to predict the residual life of rolling bearings.

The specific assessing steps of the WCDPSO-KELM model are described as follows.

Step 1: Acquiring whole life-cycle data of rolling bearings.

Step 2: The sample data are processed, different features are extracted using different methods, and the performance degradation index is selected from different features.

Step 3: The performance degradation index of each feature space is divided into the training dataset and the test dataset; the training dataset is input to train the WCDPSO-KELM model, then obtain the performance degradation prediction model.

Step 4: The testing dataset is input to the trained model to test its performance.

A flowchart of the performance degradation prediction model based on WCDPSO-KELM is shown in Figure 3.

Figure 3. Flowchart of prediction model based on WCDPSO-KELM method.

Mean Absolute Percentage Error (MAPE) is used as the assessment index to analyze the prediction performance of the WCDPSO-KELM model; output error is used to assess the validity of the prediction model. The specific formula is described as follows.

$$X_{MAPE}=\frac{1}{N}{\displaystyle \sum _{i=1}^N\left|\frac{{\widehat{y}}_i-y_i}{y_i}\right|}\times 100\%$$

(16)

where $N$ is the number of testing datasets, $y_i$ is the actual output, and ${\widehat{y}}_i$ is the prediction output.

The performance of the WCDPSO-KELM model is compared with that of other models such as the PSO-KELM model, the WPSO-KELM model, the CPSO-KELM model and the DPSO-KELM model. In order to reduce errors and avoid randomness, each prediction model runs 20 times to calculate the average value and standard deviation. The MAPE values of various models are listed in Table 1 and shown in Figure 4.

Table 1. MAPE value of different models.

Model	MAPE										Average	Standard Deviation
PSO-KELM	0.0738	0.0707	0.0729	0.0741	0.0688	0.0699	0.0705	0.0681	0.0732	0.0728	0.0717	0.0020
	0.0719	0.0726	0.0722	0.0701	0.0708	0.0733	0.0764	0.0727	0.0687	0.0713
WPSO-KELM	0.0686	0.0670	0.0727	0.0723	0.0738	0.0682	0.0715	0.0684	0.0714	0.0717	0.0704	0.0020
	0.0738	0.0703	0.0692	0.0671	0.0718	0.0700	0.0720	0.0687	0.0698	0.0696
CPSO-KELM	0.0693	0.0724	0.0692	0.0703	0.0689	0.0732	0.0728	0.0701	0.0714	0.0706	0.0715	0.0018
	0.0751	0.0725	0.0744	0.0727	0.0709	0.0690	0.0723	0.0704	0.0712	0.0741
DPSO-KELM	0.0723	0.0679	0.0664	0.0706	0.0706	0.0726	0.0664	0.0694	0.0685	0.0677	0.0699	0.0021
	0.0700	0.0688	0.0730	0.0729	0.0698	0.0707	0.0696	0.0678	0.0705	0.0738
WCDPSO-KELM	0.0687	0.0688	0.0671	0.0692	0.0675	0.0680	0.0682	0.0673	0.0660	0.0689	0.0686	0.0013
	0.0698	0.0704	0.0691	0.0701	0.0658	0.0697	0.0708	0.0693	0.0685	0.0689

Figure 4. The MAPE value of different models.

The WCDPSO-KELM model has a smaller average MAPE value and smaller standard deviation than other models. The experimental results show that the WCDPSO-KELM model has the smallest error, so it has higher prediction accuracy.

3. Data Acquisition and Processing

In this section, we first present the experiment platform in Section 3.1. Subsequently, Section 3.2 presents data processing and selection of the performance degradation index.

3.1. Pronostia Experiment Platform

This paper used rolling bearing data from the PRONOSTIA experimental platform, which is related to bearings’ degradation tests [35]. Figure 5a shows the platform dedicated to testing and validating bearings fault detection, as well as diagnostic and prognostic approaches. The platform was designed and realized at the AS2M 1 department of the FEMTO-ST 2 institute. The main objective of PRONOSTIA is to provide real experimental data that characterize the degradation of ball bearings along their whole operational life (until their total failure).

Figure 5. Test Rig. (a) Overview of PRONOSTIA; (b) normal and degraded bearing.

The degradation may be different for the same bearing if load and speed are distinct. The data are collected via the sensors located around the bearings, and data-driven techniques has to be applied to predict the degradation trend (balls, inner or outer races, cage, etc.), Figure 5b depicts an example of what one can observe on the ball bearing components before and after an experiment. According to the bearing and to the way the degradation evolves, the degradation “patterns” may have particular characteristics, as illustrated in the following. In this paper, we considered several experiments for which different features (health index) were extracted. The choice of features is typical of bearings diagnostics and prognostics [35,36].

Experimental dataset datasets are selected from different working conditions. The motor speed is 1800 rpm and the load is 4000 N in the first working condition; the motor speed is 1650 rpm and the load is 4200 N in the second working condition; the motor speed is 1500 rpm and the load is 5000 N in the third working condition. The specific information is shown in Table 2.

Table 2. The specific information of the sample data.

Working Condition	Speed (Rpm)	Load (N)	Sample Data
1	1800	4000	bearing1_k, k = 1,2,…,7
2	1650	4200	bearing2_k, k = 1,2,…,7
3	1500	5000	bearing3_k, k = 1,2,3

In order to test the effectiveness of different feature spaces, one-dimensional time-domain vibration signals are transformed into three different feature spaces by the time–frequency domain algorithm, the EEMD algorithm and the wavelet packet algorithm, then performance degradation indexes are selected from these features for construct training matrix and the testing matrix, which are the input of the performance degradation prediction model.

3.2. Selecting Performance Degradation Index

3.2.1. Feature of Time–Frequency Domain

Because the statistical index of the vibration signal contains rich information regarding rolling bearings’ running state, it can be used to assess performance degradation trends of rolling bearings or diagnose bearing faults. The running state of rolling bearings is effectively described by multiple time–frequency domain indexes of the whole life-cycle, then PCA is used to extract the first principal component from features of the time–frequency domain. Figure 6a shows performance degradation indexes from time–frequency domain features in three working conditions.

Figure 6. Different performance degradation index. (a) Time–frequency domain; (b) wavelet packet space; (c) EEMD space.

3.2.2. Features of Wavelet Packet Space

Vibration signals are processed using a wavelet packet method to extract features of wavelet packet space. The signal frequency band is divided into multiple sub-bands by wavelet packet decomposition; thus, the feature information of the original signal is also divided into each sub-band. The total energy of a signal in the time domain is equal to its total energy in the frequency domain according to Parseval′s theorem in [37]. The total energy of original signal is also divided into each sub-band.

When a signal $x(t)$ is decomposed to the $j$-layer by wavelet packet, the energy of each sub-band can be calculated according to Equation (17).

$$E_{ij}={\displaystyle \int {\left|S_{i,j}(t)\right|}^2}dt={\displaystyle \sum _{k=1}^n{\left|x_{i,j}(k)\right|}^2}$$

(17)

The total energy can be calculated as follows:

$$E={\displaystyle \sum _{j=0}^{2^i-1}{E}_{ij}}$$

(18)

The percentage of all sub-bands’ energy can be as the feature vectors of the vibration signal, then the expression of the feature vector as shown as follows in Equation (19):

$$T=[t_1,t_2,\cdots ,t_{2^i}]=\left[\frac{E_{i,0}}{E},\frac{E_{i,1}}{E},\cdots ,\frac{E_{i,(2^i-1)}}{E}\right]$$

(19)

The first principal component is extracted by PCA from the energy feature of vibration signals in different working conditions, which are shown in Figure 6b.

3.2.3. Features of EEMD Space

The Ensemble Empirical Mode Decomposition (EEMD) method [38] is used to extract features; vibration signal $x(t)$ is decomposed into multiple Intrinsic Mode Function (IMF) by the EEMD method. In order to judge which $IMF$ is the real component of the signal or meaningless components, the correlation coefficient between the IMF and $x(t)$ is as an index. All $IMFs$ are normalized, and the normalized correlation coefficient between $IMFs$ and $x(t)$ is described as follows,

$$r_j=\frac{{\displaystyle \sum _{t=1}^N(x(t)-\overline{x})(c_j(t)-{\overline{c}}_j)}}{\sqrt{{\displaystyle \sum _{t=1}^N{(x(t)-\overline{x})}^2\ast {\displaystyle \sum _{t=1}^N{(c_j(t)-{\overline{c}}_j)}^2}}}}$$

(20)

where, $x(t)$ is the original signal, $c_j(t)$ is the $j^{th}$ $IMF$, $r_j$ is the correlation coefficient of the $j^{th}$ $IMF$, and $t=1,2,\cdots ,N$ is the sampling points of the signal.

The threshold value $TH$ is set, which is the standard deviation of the correlation coefficient $TH=std(r_j)$.

$$TH=\sqrt{\frac{1}{n-1}{\displaystyle \sum _{j=1}^n{(r_j-\overline{r})}^2}}$$

(21)

If $r_j>TH$, the $j^{th}$$IMF$ is retained; otherwise, it is removed.

The major performance degradation index of EEMD space is extracted by SVD in different working conditions, which are shown in Figure 6c.

4. Experimental Results

In this section, the performance of the proposed method is tested in Section 4.1. Subsequently, Section 4.2 presents analysis of the experimental results.

4.1. Testing the Performance of the WCDPSO-KELM Model

In this section, the performance of the proposed method is tested by performance degradation indexes from different feature spaces in different working conditions. Experimental data are selected in different working conditions, and the specific information of sample datasets is listed in Table 3.

Table 3. Selected experimental datasets in different working conditions.

Working Condition	Numbered Sample Data	The Data Dimension
1	bearing1_5	2463 × 2560
2	bearing2_5	2304 × 2560
3	bearing3_2	1637 × 2560

The sample data numbered bearing1_5 is processed according to Figure 3 to obtain performance degradation indexes of time–frequency domain, wavelet packet space and EEMD space. From a total of 2442 datasets, the first 2000 datasets are taken as training datasets, and the remaining 442 as the testing dataset. A different performance degradation index is input to the WCDPSO-KELM model. The outputs of the WCDPSO-KELM model and actual outputs are shown in Figure 7.

Figure 7. Actual outputs and prediction outputs in working condition 1. (a) phase space; (b) wavelet packet space; (c) EEMD space.

The sample data numbered bearing2_5 is processed as described above. From a total of 2304 datasets, the first 1900 datasets are taken as training datasets, and the remaining 404 as testing datasets. The outputs of the WCDPSO-KELM model and actual outputs are shown in Figure 8.

Figure 8. Actual outputs and prediction outputs in working condition 2. (a) Phase space; (b) wavelet packet space; (c) EEMD space.

The sample data numbered bearing3_2 is processed as described above. From a total of 1625 datasets, the first 1200 datasets are taken as training datasets, and the remaining 425 as testing datasets. The outputs of the WCDPSO-KELM model and actual outputs are shown in Figure 9.

Figure 9. Actual outputs and prediction outputs in working condition 3. (a) Phase space; (b) wavelet packet space; (c) EEMD space.

4.2. Result Analysis

In order to compare the performance of different features from the time–frequency domain, wavelet packet space and EEMD space, the performance degradation index of rolling bearings is selected from different feature spaces, which then divided into the training data and the testing data. The experimental results of the WCDPSO-KELM model are shown in Figure 7, Figure 8 and Figure 9, When rolling bearings run in different working condition, the performance index distribution does not exactly satisfy the same distribution condition, as can be seen in Figure 6.

The performance degradation index of different feature spaces is input to the WCDPSO-KELM model to calculate the MAPE value. The MAPE values of the WCDPSO-KELM model are listed in Table 4 and shown in Figure 10.

Table 4. MAPE value of different feature spaces.

Working Condition	MAPE
	phase	EEMD	wavelet
1	0.0629877	0.005971	0.141041
2	0.1859479	0.076398	0.182735
3	0.1917164	0.003721	0.007825

Figure 10. MAPE value in different feature space.

The experimental result shows the MAPE value of EEMD space is smaller than that of the time–frequency domain, wavelet packet space in same working conditions. Therefore, compared with the feature extraction method of wavelet packet or time–frequency algorithms, EEMD can effectively extract the features of the vibration signal of rolling bearings, and takes on higher efficiency.

5. Conclusions

This paper proposed the WCDPSO-KELM model to predict the performance degradation trend of rolling bearings. Firstly, the particle swarm optimization algorithm was improved by adjusting the inertia weight and linear learning factor, adding the disturbance term, which has more accuracy than the PSO algorithm. Secondly, the penalty coefficient and kernel parameters of KELM model were optimized by an improved particle swarm optimization algorithm (WCDPSO). Subsequently, a training dataset was input to train the WCDPSO-KELM model, the performance degradation trend prediction model was built, then its performance was compared with that of other models. Thirdly, features of the whole life cycle were extracted from the time–frequency domain, wavelet packet space and EEMD space, and major performance degradation indexes were selected from these features spaces, then input to the trained WCDPSO-KELM model to calculate MAPE values in different feature spaces. The experimental result shows the MAPE value in the EEMD space is smaller than in the other two feature spaces. Therefore, compared with other prediction models, the WCDPSO-KELM model has higher accuracy for predicting the performance degradation trend. Compared with the feature extraction methods of wavelet packet, time–frequency algorithms, EEMD can effectively extract the features of vibration signal of rolling bearings. The proposed method based on WCDPSO-KELM and EEMD is an efficient and accurate performance degradation prediction method.