Reliability assessment is a critical consideration in equipment engineering project. Successful reliability assessment, which is dependent on selecting features that accurately reflect performance degradation as the inputs of the assessment model, allows for the proactive maintenance of equipment. In this paper, a novel method based on kernel principal component analysis (KPCA) and Weibull proportional hazards model (WPHM) is proposed to assess the reliability of rolling bearings. A high relative feature set is constructed by selecting the effective features through extracting the time domain, frequency domain, and timefrequency domain features over the bearing’s life cycle data. The kernel principal components which can accurately reflect the performance degradation process are obtained by KPCA and then input as the covariates of WPHM to assess the reliability. An example was conducted to validate the proposed method. The differences in manufacturing, installation, and working conditions of the same type of bearings during reliability assessment are reduced after extracting relative features, which enhances the practicability and stability of the proposed method.
The rolling bearing is one of the most important components of rotating machinery [
Reliability assessment comes with two key challenges: the construction of an appropriate reliability assessment model and the selection of features which can accurately reflect performance degradation process. Reliability assessment based on realtime equipment conditions has become a popular research topic in recent years [
The WPHM is a wellestablished mathematical model. However, when it is applied in real equipment life prediction, it is problematic as far as covariates selection, setting reliability threshold, trend prediction, and other issues. In terms of covariates selection, most previous studies concern direct time domain statistical analysis where one or more time domain features are selected to build a reliability assessment model. However, one single feature or features based on one single domain cannot accurately reflect the performance degradation process and thus impact the overall accuracy of reliability assessment. Although time domain, frequency domain, and timefrequency domain features can comprehensively reflect the performance degradation process of bearings over their entire service lifetime, excessive parameters lead to data redundancy. Further, selecting more covariates of the WPHM makes the parameter estimation process more challenging. The vibration signals of faulty machinery are generally nonstationary and nonlinear under complicated operating conditions [
Kernel principal component analysis (KPCA), first proposed by Schölkopf et al. [
In order to overcome the weakness for the selection of WPHM covariates, this paper proposes a novel method for assessing the reliability of rolling bearings based on KPCA and WPHM. The novelty of this research is in improving the covariates selection method of WPHM, which has considerable value in practical application. A high relative feature set is constructed by selecting the effective features through extracting the time domain, frequency domain, and timefrequency domain features over the bearing’s life cycle data. Then the first three kernel principal components (KPCs), which can accurately reflect the performance degradation process through KPCA, are selected as WPHM covariates to assess the reliability. The feasibility and effectiveness of the method were validated using bearing’s life cycle data, and it can provide important basis for equipment proactive maintenance. The differences in manufacturing, installation, and working conditions of the same type of bearings during reliability assessment are reduced after extracting relative features, which enhances the practicability and stability of the proposed method compared to traditional assessment techniques. It enriches the theory of covariates selection and is more emphasis on application innovation.
The remainder of this paper is organized as follows. Section
KPCA essentially works by nonlinearly mapping input samples to a highly dimensional feature space
In KPCA, a set of multidimensional signals
Substituting (
The eigenvectors can be expanded as follows:
Equation (
Finally, the principal components for testing examples
The above algorithm is based on the assumption that
The cumulative contribution rate (CCR) is utilized to determine number
The cumulative contribution rate (CCR) threshold is referenced by [
There are three common types of kernel functions: polynomial kernel function, radial basis Gaussian (RBG) kernel function, and neural network kernel function. The transformation matrices of the RBG kernel function have positive definiteness and a wide convergence field. It only contains one parameter, and the calculation process is relatively simple [
The PHM builds a mathematical relationship between the feature parameters of the equipment running status and the reliability. According to the feature parameters of the realtime operation, PHM can get the device hazard rate in its current state, to assess the current reliability of the equipment. The hazard rate at time
The Weibull distribution is frequently used to model the failure time of mechanical systems. The hazard rate function of the Weibull distribution is commonly selected as the baseline hazard rate of the PHM. The hazard rate for the twoparameter Weibull distribution is written as follows:
The PHM with the Weibull baseline function is called the WPHM, the hazard function of which is defined as follows:
According to the principle of reliability analysis [
The key of using WPHM to assess the operating status of equipment is to estimate unknown parameters according to the feature data and time data of the realtime status. The maximum likelihood method is commonly applied to estimate unknown WPHM parameters. In practice, a mechanical system may be run until it fails but may be repaired prior to failure. The lifetime data usually contains failure times and suspension times to reflect this. To properly account for both types of data, the likelihood function where the covariates are timedependent is defined as follows:
In the above equations, the covariates of WPHM are timedependent. When the covariates only relate to the current time (i.e., they are nontimedependent), the reliability and the failure probability density can be, respectively, rewritten as follows:
Therefore, by substituting (
By setting the partial derivatives of (
With increase in the number of covariates, the complexity of the maximum likelihood estimation increases substantially. Therefore, the NelderMead iterative algorithm [
A flowchart of the proposed method is shown in Figure
The flowchart of the proposed method.
The reliability assessment process takes place in a stepwise manner:
Select effective features that comprehensively reflect the performance degradation process from the time domain, frequency domain, and timefrequency domain features of training bearings data to compose the feature vector.
Build a high relative feature set by extracting samples from lifetime data of training bearings.
Obtain KPCs and KPCs mapping from KPCA for the high training lifetime relative feature set, and select the first KPCs with CCR exceeding 85%.
Build the high relative feature set from lifetime data of the test bearing; obtain KPCs of the test bearing through KPC mapping of the training bearings, and then verify whether the KPCs of the test bearing reflect the performance degradation process.
Take the KPCs of the training bearings as the WPHM covariates to estimate the WPHM parameters.
Take the KPCs of the test bearing as the WPHM covariates to assess the test bearing reliability.
The rolling bearing life cycle test data used in this paper was provided by the Center for Intelligent Maintenance Systems (IMS), University of Cincinnati [
Bearing test rig and sensor placement illustration: (a) bearing test rig; (b) sensor placement illustration.
There are four test double row bearings (Rexnord ZA2115) on one shaft of the bearing test rig. The shaft is driven by an AC motor and coupled with rub belts. A radial load of 6,000 lbs was added to the shaft and bearings by a spring mechanism; the rotation speed was kept constant at 2,000 rpm during the experiment. A magnetic plug is installed in the oil feedback pipe to collect debris from the oil as evidence of bearing degradation. The test was stopped when the accumulated debris adhered to the magnetic plug exceeded a certain level, causing an electrical switch to close. Vibration data were collected every 20 minutes with a National Instruments DAQCard6062E data acquisition card (data sampling rate 20 kHz and data length 20,480 points). Data collection was conducted in the National Instruments LabVIEW program. Table
Test results.
Test order  Failure bearing  Failure modes  Censored bearing 

1  B3, B4  B3(a), B4(b&c)  B1, B2 
2  B5  B5(c)  B6, B7, B8 
Components of failure bearing: (a) inner race defect; (b) roller element defect; (c) outer race defect.
Bearing 3 (Test 1) is used as test bearing and the other bearings (Test 1 and Test 2) are used as training bearings.
More than 50 features of time domain, frequency domain, and timefrequency domain [
Time domain: root mean square (RMS), kurtosis, peakpeak value, and peak factor.
Frequency domain: spectrum mean, spectrum variance, and spectrum RMS.
Timefrequency domain: third frequency band normalized energy spectrum (E3), sample entropy (S3), seventh frequency band normalized energy spectrum (E7), and sample entropy (S7) (obtained via db10 wavelet packet decomposition at three levels).
Even the same type of bearings differ due to differences in manufacturing, installation, and working conditions; thus there are differences among them even in the same work period. Take time domain features as an example; for bearings 1–8, the time domain features for stable trend of normal work period were selected and averaged as shown in Figure
Mean time domain features in normal work period.
Time domain—RMS
Time domain—peakpeak value
Time domain—kurtosis
Time domain—peak factor
Figure
The advantages of relative features are discussed further in Section
For the seven training bearings, each had 100 samples that can reflect the process of the lifetime (at a total of 700 samples). A high training lifetime relative feature set of 700 × 11 was then composed accordingly. For the test bearing, a total of 2,152 whole life cycle samples were used to obtain the 2152 × 11 high test lifetime relative feature set shown in Figure
High test lifetime relative feature data (overall degradation process).
Time domain—RMS
Time domain—peakpeak value
Time domain—kurtosis
Time domain—peak factor
Frequency domain—spectrum mean
Frequency domain—spectrum variance
Frequency domain—spectrum RMS
Timefrequency domain—E3
Timefrequency domain—E7
Timefrequency domain—S3
Timefrequency domain—S7
The time of data collection
In order to effectively verify the following analysis, according to the features of bearing 3, the degradation process was divided into five stages as shown in Table
Point messages.
Period  Point  Date (day)  Characteristics 

Normal working stage 

0–17.69  The features are in normal range. 

17.69–28.99  

28.99–31.05  


Early failure stage 

31.05–32.53  When the surface defect just initiates, small spalling or cracks are formed. Kurtosis has great fluctuation, but RMS increases slowly [ 

32.53–33.34  


Healing stage 

33.34–33.49  The surface defect is later smoothed by the continuous rolling contact. The vibration level decreases. 


Medium wear stage 

33.49–34.11  As the damage spreads over a broader area, the vibration level rises again. 


Severe wear stage 

34.11–34.17  As the damage spreads over an enough area, the vibration level rises quickly. 
High test lifetime relative feature data (individual degradation stages).
Time domain—RMS
Time domain—peakpeak value
Time domain—kurtosis
Time domain—peak factor
Frequency domain—spectrum mean
Frequency domain—spectrum variance
Frequency domain—spectrum RMS
Timefrequency domain—E3
Timefrequency domain—E7
Timefrequency domain—S3
Timefrequency domain—S7
The time of data collection
Many previous researchers have used RMS and kurtosis as PHM covariates. Figure
High training original feature set and high training relative feature set were analyzed by PCA and KPCA (
PCA and KPCA results.
PC1  PC2  PC3  


CCR (%) 

CCR (%) 

CCR (%)  
High training original feature set  
PCA  5.4784  49.80  2.1019  68.91  1.6367  83.79 
KPCA  0.0017  49.44  0.000639  68.46  0.000502  83.40 
High training relative feature set  
PCA  7.0233  63.85  1.9339  81.43  0.9457  90.03 
KPCA  0.0021  63.24  0.000582  80.71  0.000283  89.20 
The CCR of the first three principal components of high training original feature set was lower than high training relative feature set. In other words, the dimension reduction effect of high training original feature set was lower than high training relative feature set. This fully verifies the advantages of relative features. Hence high training relative feature set was selected for subsequent analysis.
The first three principal components of high test relative feature set can be obtained by KPCA. To verify the result of KPCA, the first three KPCs were projected onto threedimensional coordinate system as shown in Figure
Kernel principal component projection: (a) the first three Kernel principal components; (b) 1st kernel principal component and 2nd Kernel principal component.
As shown in Table
By comparison, the first three principal components were projected as shown in Figure
Principal component projection: (a) the first three principal components; (b) 1st principal component and 2nd principal component.
In conclusion, the first three KPCs contain most time domain, frequency domain, and timefrequency domain information as well as nonlinear components. Between the sample points, the offset is relatively small and the trend is obvious. These first three KPCs, which fully represent the bearing performance degradation process, can be utilized as covariates to establish a WPHM model that is highly stable and reliable.
The first three KPCs were selected as the covariates of WPHM to assess the reliability. By substituting the failure and suspension data of KPCs from the high training relative feature set into (
Parameter estimation of WPHM.
Parameters 






Estimates (nontimedependent)  1.8207  110.7  4.103  −0.5075  0.3837 
Estimates (timedependent)  1.0723  36.24  7.526  −1.6423  −0.8847 
The KPCs of the high test relative feature set are plugged into the WPHM with nontimedependent covariates to calculate the reliability, as shown in Figure
Lifetime reliability (nontimedependent covariates).
The reliability of the normal working stage remained stable between 0.99 and 0.90; the reliability of the early failure stage fluctuated between 0.90 and 0.75; the reliability of the healing stage fluctuated between 0.87 and 0.84; the reliability of the medium wear state gradually fell to 0.50 from 0.75; and the reliability of the severe wear stage fell rapidly from 0.50 to 0.45. The reliability variable accurately reflects the state of the test bearing. When reliability drops to 0.90, the bearing requires attention. When reliability drops to 0.75, it urgently requires attention, and a maintenance plan should be enacted immediately. When the reliability falls to 0.50, the equipment must be stopped to avoid an accident.
By contrast, The KPCs of the high test relative feature set were plugged into the WPHM with timedependent covariates to calculate the reliability as shown in Figure
Lifetime reliability (timedependent covariates).
The reliability decreased at a generally steady rate in the normal working stage and then began to decrease at a quicker pace in the early failure stage. The reliability decline rate increased sharply in the medium wear stage and severe wear stage. Therefore, the reliability assessment of WPHM with timedependent covariates can reflect not only the degradation process of the bearing, but also the stage in its life cycle.
Considering the influence of the historical data, the reliability assessment in WPHM with timedependent covariates is highly stable and credible. Therefore, it is better suited to maintaining important equipment. However, reliability as reflected in WPHM with nontimedependent covariates is only related to the present time without considering the historical data; the complexity of calculation decreases, so it is more suitable for normal equipment. More importantly, it is more suitable for the reliability assessment of the bearings which lack historical data or have been repaired.
Timedependent covariates or nontimedependent covariates can be selected according to actual conditions. Whether the covariates are timedependent can be determined by the actual needs of the WPHM. In conclusion, the results indicate that this method can accurately assess reliability and timely provide effective maintenance decisions.
In this study, KPCs based on KPCA were successfully used as WPHM covariates to assess the reliability of rolling bearings. Based on the relative multiple features, KPCs can sufficiently describe the bearing performance degradation process. KPCs as WPHM covariates provide accurate reliability to support timely maintenance decisions. The relative features also enhance the practicability and stability of the proposed method compared to traditional assessment techniques.
The authors declare that they have no conflicts of interest.
This work was supported by the National Natural Science Foundation of China (no. 51375067) and HK Foundation of China (no. 20132163010).