Remaining life prediction of rolling bearing based on PCA and improved logistic regression model

2.1. PCA

For a given feature vector set $X_k=\left(x_{1k},x_{2k},\dots ,x_{nk}\right)$, the $n$-dimensional vector is extracted from the vibration signal of the rolling bearing. The state of machinery can be described by the vector $x_k$, the covariance matrix of which is expressed as follows:

where $N$ represents the number of training samples and $\stackrel{-}{x}$ represents average value, expressed as:

The eigenvalues ${\lambda }_i$ ($i=$1, 2,…, $N$) and eigenvectors $h_i$ are obtained from $W_k$. Then the eigenvalues are arranged in order from large to small: ${\lambda }_1>{\lambda }_2>\cdots >{\lambda }_n$ the corresponding feature vector is $h_i$ ($i=$1, 2,…, $N$). The sample $x_i$projects onto the feature vector to get the principal component in the direction [17]:

All feature vectors form an n-dimensional orthogonal space. The n-dimensional principal component could be obtained from the n-dimensional space onto which $x$ is projected. The contribution of the feature rate vector is proportional to its eigenvalues after reconfiguration. The cumulative contribution rate of the first$m$principal component of orthogonal space is expressed as follows:

$G\left(m\right)$ should be arranged in order from small to large and the contribution rate should be selected as necessary. $G\left(m\right)>$ 95 % means the first $m$ principal components consist of 95 % original data. The first $m$ principal components can thus be taken to represent the original information, which ensures it is not lost after realizing dimensional reduction.

2.2. Logistics regression model and ILRM

The bearing state is represented by a series of characteristic parameters. The logistics regression model evaluates the bearing state when the data sample is composed of characteristic parameters and the bearing state [18]. The vector $X\left(t\right)=\left\{x_1\left(t\right),x_2\left(t\right),\dots ,x_m\left(t\right)\right\}$ can represents the covariate variables which affect the state parameters of the bearing, where $m$ represents the number of covariate variables. The conditional probability of the event that does not occur ($y_t=1$) is expressed as follows:

where ${\beta }_0$, ${\beta }_1$,…, ${\beta }_m$ are the regression coefficients of the covariate variable and ${\beta }_0>0$.

The bearing state is divided into normal and failure states at $t$. The normal state is replaced by $y_t=1\text{;}$ the failure state is replaced by $y_t=0\text{.}$ The covariate variable $X\left(t\right)=\left\{x_1\left(t\right),x_1\left(t\right),\dots ,x_m\left(t\right)\right\}$ represents the characteristic signal parameters from the state of machinery at the point in question. There is a nonlinear relationship between the state of rolling bearing $y_t$ and covariate variable $X\left(t\right)$.

The covariate variable of rolling bearing is $X\left(t\right)$. The ratio of bearing reliability function $R\left(t|X\left(t\right)\right)$ to accumulation failure distribution function $F\left(t|X\left(t\right)\right)=1-R\left(t|X\left(t\right)\right)$ is expressed as follows:

The parameters can be solved by the maximum likelihood estimation method, the logarithmic form of which is expressed as follows:

The intercept ${\beta }_0$ is a constant term, ${\beta }_0$, ${\beta }_1$,…, ${\beta }_m$ are the weights of covariate variables, ${\beta }_i>0$ means indicates that the event occurrence probability increases as characteristic parameter $x_i$, increases; ${\beta }_i=0$ means the independent variable has no influence on this model. The logistics model is nonlinear, so ${\beta }_0$, ${\beta }_1$,…, ${\beta }_m$ can be estimated via maximum likelihood estimation method using $\widehat{\beta }$to denote the maximum likelihood estimate of $\beta$. The bearing reliability function is:

The logistics regression model ignores the former degradation trend and cannot adapt to fluctuation, which weakens the precision of the residual life prediction model. This paper proposes the ILRM, in which the degradation trend of the rolling bearing is considered during residual life prediction without being subject to fluctuations. The reliability function of the ILRM is expressed as follows:

where $h_m\left(t\right)$ is the function of $x_m\left(t\right)$. The function can be used to express the characteristic parameters that reflect the state of the rolling bearing.

The related functions of the ILRM are as follows:

where ${\alpha }_j$ ($j=m,m+1,\cdots ,m+n$) is related to the difference between the eigenvalues at $t$ and the eigenvalues at $t_j$ moment. $u\left(t\right)$ is the product of ${\alpha }_j$ and the corresponding covariates $x_t$. $W\left(t\right)$ is the mean values of $u\left(t\right)$ in a normal work period.

Eqs. (11-12) reflect the fact that the ILRM not only considers the influence of the bearing’s deterioration tendency and reduces the influence of random fluctuations, but also that the amplitude and characteristic trends are almost unchanged compared to the original data. Eq. (10) suggests that the ILRM is not only unaffected by the manufacturing and installation of the individual bearings, but also has a strong generality and accuracy.

The logarithmic form of the likelihood function is expressed as follows:

If covariate variable $U\left(\nu \right):t<\nu <\infty$ is predicted, the rolling bearing residual life at $t$ moment $L\left(t\right)=E\left[\left(T-t|T>t\right)\right]$ cays approximately expressed as [19]:

The logistics regression model only considers the bearing characteristic parameters and ignores the bearing degradation trend, so it is susceptible to interference. The result of residual life prediction adds error to the reliability assessment. The ILRM, conversely, considers the bearing degradation trend and decreases the effect of random signal fluctuations, so it yields more accurate prediction results.

4.1. Feature set selection and processing

Eighty-two feature parameters of time, frequency and time-frequency domain were extracted from the life cycle data of 11 groups of bearings. To reduce the amount of computation and to reduce the PCA dimensions at the same contribution rate, features that did not reflect the bearing state were discarded. Fifteen effective features were selected from the characteristic curve of 82 parameters. The selected features effectively reflect the bearing degradation process, as shown in Fig. 5.

Time domain: Kurtosis, Peak to peak, IMF1 energy spectrum, IMF2 energy spectrum, Variance, Average power, Absolute mean value (AMV), Standard deviation, Pulse factor, Clearance factor, and Kurtosis Factor.

Frequency domain: RMS, and Peak value.

Time-frequency domain: The 3rd frequency bands normalized energy spectrum (E3), and the 7th frequency bands normalized energy spectrum (E7) [21].

Errors in the installation of sensors or in the manufacturing may have caused substantial differences between the measured signals. The means of some features in time, frequency, and time-frequency domain are shown in Fig. 4 for the sake of comparison.

Fig. 4The comparison of feature parameters

a) Time-frequency domain – E3

b) Time domain – Kurtosis

c) Frequency domain – RMS

The same parameters may vastly differ among different bearings. The original data interferes with subsequent data analysis steps and RUL prediction results. In order to overcome the shortage, so relative features can be put out to mitigate this:

where $H\left(t\right)$ is the relative feature, $X\left(t\right)$ is the original feature, and $b$ is the mean value of the normal period.

A stable trend occurs in the normal period which can be selected to obtain mean values. The relative feature of the rolling bearing can then be acquired by calculating the ratio between the actual value and the mean value. The results are shown in Fig. 5.

4.2. PCA dimension reduction processing

Fig. 5 shows the feature parameters that reflect bearing degradation trends. The parameters reflected by the state of the rolling bearing are not uniform. As mentioned above, it is difficult to determine which parameters reflect the bearing degradation trend; this makes any method of extracting a useful and comprehensive bearing feature particularly important. Here, we use PCA dimension reduction processing to secure the above objectives. The results are shown in Table 2. The contribution rate of the second principal component reached 96.57 %, exceeded the target of 95 %, so the top two principal components were utilized for subsequent analysis here.

Our projection of the first two principal components obtained into two-dimensional space is shown in Fig. 6 and Table 3. Fig. 6 shows that the first principal component reflects the deterioration trend of the bearing. Normal period, early fault, medium wear stage, and severe wear stage occurred in the first principal component. The early failure stage is more prominent in the second principal component because small spalling or cracks are formed in the early failure stage. A healing stage occurs between the early failure and medium wear stage, because in this stage, the surface is smoothed by continuous rolling contact.

Fig. 5Relative characteristics of whole lifetime

a) Time-frequency domain – E3

b) Time-frequency domain – E7

c) Frequency domain – RMS

d) Frequency domain – Peak

e) Time domain – Kurtosis

f) Time domain – Peak to peak

g) Time domain – IMF1

h) Time domain – IMF2

i) Time domain – Avarage power

j) Time domain – Variance

k) Time domain – AMV

l) Time domain – Standard deviator

m) Time domain – Clearance factor

n) Time domain – Kurtosis Factor

o) Time domain – Pulse factor

p) Data acquisition time distribution

In conclusion, the top two principal components identified here do accurately express the performance degradation trend of industrial bearings. They not only contain most of the bearing degradation state information, but also greatly reduce the feature dimension. We thus selected the top two principle components as the covariates of the ILRM to predict the RUL in our subsequent assessment.

Fig. 6The top two principal components

4.3. Residual life prediction

In order to verify the effectiveness of the proposed algorithm, we used the logistics regression model to compare the results against the proposed algorithm. The maximum likelihood estimation method was used to estimate the parameters of the logistics regression model and ILRM according to Eq. (7) and Eq. (13). The results are shown in Table 4.

The first and second principal components of the large dimension were introduced into the ILRM and logistics regression model as covariates to represent the reliability of the whole life of bearings, as shown in Figs. 7-8.

Fig. 7 shows that the ILRM tends to decrease along the progression of time during the normal period, which is consistent with the actual situation. The reliability curve of the logistic regression model showed no downward trend under normal conditions (Fig. 8). Fluctuations in the ILRM curve were significantly reduced compared to the logistic regression curve during the normal period.

Fig. 7Reliability curve of improved logistics regression model

Fig. 8Reliability curve of logistics regression model

According to Williams [22], a bearing in the wake of the early fault has a healing period. In this period, bearing characteristic values decrease while reliability seems to increase, but actually decreases. This is mainly because early fault cracks are worn down. There is a rising trend in the healing stage as shown in Fig. 8, which introduces interference to the maintenance plan. The algorithm proposed in this paper mitigates this problem as it is shown in Fig. 7. This is mainly due to the logistics regression model, which does not account for the deterioration trend of the bearing prior to analysis.

Table 5 shows the RUL estimated by the ILRM and logistics regression model.

The prediction accuracy of the ILRM is much higher than that of the logistics regression model, as shown in Table 5. Take the severe wear stage for example: At 34.2 days, the actual RUL is 0.2815 days; the RUL predicted via ILRM is 0.3146 days while that predicted by the logistics regression model is 0.8371 days. The accuracies were evaluated as follows:

The accuracy of ILRM is 89.48 % while that of the logistics regression model is 33.63 %. The ILRM prediction results are consistent with the residual life of the actual bearing, but the logistics regression model prediction results are volatile and with larger margin of error. This is because the logistics regression model only considers the characteristic of the current time, ignores the degradation trend, and fails to adapt to the fluctuations; the ILRM mitigates these disadvantages.