# Prediction on the Remaining Useful Life of Rolling Bearings Using Ensemble DLSTM

Authors: Miao Jiang; Yang Xiang
Journal: Shock and Vibration
 (2023)

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## Abstract

As a significant component of rotation machinery, bearing plays a role in supporting and transmitting power. However, bearings are subject to complex operating conditions and are prone to failure. To avoid ineffectiveness and improve the reliability of bearings, a data-driven method is used to predict the remaining useful life (RUL). However, this method is less stable and can only forecast the RUL of bearings under training sample conditions. An ensemble deep, long-term, and short-term memory (EDLSTM) method is proposed to solve this problem. First, the feature of the forecast-bearing RUL was extracted including time-domain features, frequency-domain energy features, and Shannon entropy. Then, a deep long- and short-term memory network prediction model of the bearing RUL was constructed. To resolve the instability of DLSTM predictions, multiple DLSTMs were ensembled using the maximum information component (MIC) criterion. The model i trained using bearing data with different failure modes under difficult operating conditions to improve the predictive stability of the model. Finally, an EDLSTM was constructed to achieve the bearing RUL prediction. In the prediction result of the training set, the cumulative relative accuracy (CRA) was above 0.9 for most of the bearings. According to the experimental results in the test set, the mean CRA was over 0.80. For some of the bearing’s RUL, the CRA was more than 0.90. The above results show that the proposed approach can effectively predict the RUL of a bearing and has a more stable prediction ability than the bagging integration method.


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## Body

## 1. Introduction

Rotating machinery plays a significant role in modern society, generally being used in aerospace, transport, and industrial production, and therefore must ensure that the machinery has a high degree of safety and reliability. Rolling bearings are one of the critical components of rotating machinery, and their operating condition will directly affect the working condition of rotating machinery [1]. Researchers have proposed predictive and healthy management (PHM) to maintain rotating machinery and promote its reliability and safety to guarantee the healthy operation of rotating machinery over a long period. According to Reference [2], researchers are concerned with the implementation of mechanical PHM into the following three categories: model-based approaches, data-driven approaches, and hybrid approaches. In practice, it is difficult to achieve complete mathematical modeling of mechanical systems due to the external conditions of the working environment and the intrinsic factors such as the complexity of the machinery system. With unprecedented advances in science and technology for signal collection sensors, communication technology, and data storage technology, data-driven methods are becoming increasingly widespread [3].To improve the reliability of bearings, Zheng et al. [1] used K-ELM to identify faults in the bearings. Zhang et al. [4] proposed a ResNet-STAC-Tanh fault diagnosis method for the bearing. To achieve accurate diagnosis of small sample data, Bai et al. [5] proposed a multichannel-based convolutional neural network combining multiscale clipping fusion (MSCF) bearing fault diagnosis method. However, as these methods are based on the state in which the bearing failure has already occurred, they are unable to provide trend prediction in advance to provide decision support for the bearing maintenance strategy. In response to this issue, remaining life prediction was proposed to prevent the spread of failures that can lead to the failure collapse of the entire mechanical system, avoiding catastrophic accidents and casualties [6, 7]. Chen et al. [8] proposed a life prognosis framework for engines based on similarity theory and support vector machines, which used full life-cycle data and a failure-free performance deterioration index. Wu et al. [9] proposed a random forest (RF)-based tool wear prediction method, which was validated using milling test data. Berghout et al. [10] proposed an online sequential extreme learning machine (OS-ELM) method for RUL prognosis and validated it using the C-MAPSS data set. However, they used traditional machine methods. One advantage of deep learning methods is that they can handle high-dimensional big data, allowing for the deep extraction of features and the mining of deep mechanical state information. Babu et al. [11] proposed a deep convolutional neural network-based regression method to predict RUL, which was the first time that CNN had been used to predict RUL. Ding et al. [12] proposed a deep convolutional neural network (DCNN) architecture to predict RUL by improving CNN for prognosis model accuracy and stability. By addressing the limitations of traditional data-driven prediction, Deutsch et al. [13] proposed a method combining deep belief networks (DBNs) with particle filtering to achieve RUL prediction of ceramic bearings. By filtering the original vibration signals and extracting the spectral energy values of these five different bands, Chen et al. [14] obtained the vibration signals of five frequency bands. Finally, a recurrent neural network (RNN) based on an encoding-decoding framework and an attention mechanism was used to predict the health values of the bearing RUL.Nevertheless, after extensive research, researchers discovered that RNN in the processing of time series will lose the previous status information due to the overlarge of the previous state information time interval. RNN suffers not only from this problem but also from gradient disappearance and explosion. Long- and short-term memory was proposed by researchers to solve a series of problems [15]. To date, the LSTM algorithm has been successfully applied in various fields. Huang et al. [16] selected the PRONOSTIA dataset as the database for RUL research and calculated the root mean square (RMS) of the full life-cycle bearing data as a degradation index. Meanwhile, the LSTM’s attention mechanism was used to optimize the prognosis model. Finally, the RMS was applied to forecast the bearing RUL. Wu et al. [17] proposed a prediction model for multisensor time series singles with a depth long short-term memory (DLSTM) network and adaptive matrix estimation to improve accuracy and stability. Hu and Zheng [18] proposed a multilevel attention network to improve the accuracy of a deep learning model for time series data prediction. Qin et al. [19] proposed an LSTM with a macroscopic attention mechanism to improve the accuracy of gear RUL prediction. Tian and Wang [20] proposed a bearing RUL prediction method based on the bearing degradation index in response to the problem of low generalization and poor prediction ability of a single prognosis model. First, the Fast Correlation-Based Filter with Approximate Markov Blankets (FCBF-AMB) with the maximal information coefficient (MIC) was used to obtain a robust degradation index, and then the AdaBoost algorithm was used to optimize the RUL prognosis model. Finally, the bearing RUL was predicted using the degradation index.To summarize, the current studies on bearing RUL prediction used data from a single operating situation for training and validation. These studies cannot demonstrate the stability of the method because they rarely use one operating condition data to train the model and then another to validate it. In this paper, a data-driven approach is used to both accurately predict the bearing RUL and improve the generalization capacity of the model. To begin, an LSTM network is chosen to construct the bearing RUL prediction model for the accuracy of the rolling bearing. To address the issue of the poor generalization capability of a single LSTM prediction model, a multifeatured EDLSTM method is proposed to predict rolling bearing under various operating conditions.The main contributions of this study are as follows:(1)
To achieve the RUL prediction of the bearing degradation stage, the bearing’s initial degradation point was divided using a standard root mean square. (2) A multifeature extraction method was used to improve the accuracy of the bearing RUL prediction. The LSTM was then chosen to construct the prediction model because it is more advantageous in dealing with time series. (3) To address the issue of the prognosis model, it is generally unstable. This study proposed using the EDLSTM to improve the prediction model’s learning abilities and stability.(2)
The following are the main contents of this study: First, the feature extraction methods and LSTM theory are introduced before demonstrating the detailed process of EDLSTM. Next, the experimental data from Xi’an Jiaotong University’s XJTU-SY rolling bearing are used to validate the proposed method. Finally, the study is summarised and the next steps in this research are described.


## 2. Methodology and Theory

Because the bearing is a slowly changing process from normal to degradation, predicting the normal stage RUL would result in a waste of resources. Therefore, this study chooses to perform life prediction for the deterioration stage of the bearing. The main work was as follows: using SRMS to classify the different bearing stages to realize the bearing degradation stage RUL prediction. The theoretical basis of LSTM has been introduced, and the prognosis model is optimised using EDLSTM.


### 2.1. Initial Degradation Point Assessment

The root mean square (RMS) of vibration can be used to assess the mechanical health state [21, 22]. When the machine moves from normal to abnormal, the RMS value rises at a set rate. As a result, it is reasonable to use RMS to determine the degradation point of a bearing. The collected data in the paper include horizontal and vertical directions. First, equation (1) is used to compute the RMS for both directions. Then, equation (2) is used to compute the root mean square (RMS). Equation (3) is used for standardization to avoid differences in the magnitude of the RMS.(1)Xrms=1N∑i=1Nxi2,(2)Xs=Xrms−h2+Xrms−v2,(3)Xsrms=Xs−minXsmaxXs−minXs.In the above equations,xi is the collected signals collected at different periods and N is the length of the signals at selected periods. Xrms−h is the RMS of the horizontal sensor collection collected signals, and Xrms−v is the RMS of the vertical sensor collection collected signals.In this paper, to identify the RUL initial prediction point, assume that the current bearing collection signals isn. The mean RMS of the previous n−1 signals is multiplied by a threshold range to get ∆Xsrms, when the current period RMS of n is over than ∆Xsrms, that is the initial point of deterioration. The degradation point determination logic is shown in Algorithm 1.Algorithm 1: Degradation point determination logic.
(1)
Compute the average RMS of the previousn − 1 series to get ∆Xsrms.(2)
Compute the current periodn root mean square Xrmsn.(3)
Set the thresholdkk∈2.5–3.(4)
Ifk∗∆Xsrms≤Xrmsn(5)
N is the initial degradation point.(6)
Else(7)
N is the normal stage, n = n + 1, repeat the above process.(8)
endThe initial degradation point of different bearings is obtained according to the judgment condition, and Figure1 shows the initial degradation points of various bearings.Figure 1
The evaluation of various bearing initial degradation points. (a) Bearing 1-2. (b) Bearing 2-1.
(a)(b)Figure1 of the different bearings shows that the signal fluctuations change significantly as the operation time increases, and the initial degradation point cannot be determined accurately. On the other hand, the initial degradation point can be determined visually using SRMS, and a comparison of the time-domain plots with the SRMS graphs clearly demonstrate that the degradation stages classified using SRMS are compatible with reality.


### 2.2. Multifeature Extraction Methods

#### 2.2.1. Spectral Energy Characteristics

The use of spectrum energy to reflect the health state of the bearing at a specific moment was proposed in Reference [14]. The spectrum energy was calculated in this study using fast Fourier transform (FFT) of the time step signal. The specific equation for integrating the time step frequency to obtain a characteristic value that can comprehensively reflect the current state of the bearing is as follows:(4)Et,i=14500∫i×ni+1×nxtfdf,where Et,i is the ith energy feature value at time step t and xtf is the vibration frequency of the vibration signal at time step t. The vibration signal in this study includes both vertical and horizontal directions. Figure 2 shows that two spectral energy features obtained in the end.Figure 2
DELSTM training set testing results. (a) Bearing 1-1. (b) Bearing 1-3. (c) Bearing 1-5. (d) Bearing 2-2. (e) Bearing 2-3. (f) Bearing 2-4. (g) Bearing 3-1. (h) Bearing 3-2. (i) Bearing 3-3. (j) Bearing 3-4. (k) Bearing 3-5.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)


#### 2.2.2. Shannon Entropy

The Shannon entropy, also known as information entropy, is defined as the discrete sequence of signals being measured as when each component isa state and then X contains N status. p is the probability component corresponding to the N states, which are defined as follows:(5)∑i=0Npi=0≤pi≤1.It shows the amount of uncertain information in each of these state quantitiesxi in the following equation:(6)Ixi=−logapii=1,2,3,…,N.Different values ofα in the above equation will give distinct units of Shannon entropy, generally making α equal to 2.Since all the uncertain information quantities inX are random variables that do not accurately express the measured discrete sequence signal information, define the Shannon entropy HX as shown in the following formula:(7)HX=∑i=1NpiIxi=−∑i=1Npilogapi.The inconsistent magnitude of the feature reduces the predictive power of the model, and normalizing the feature magnitudes can eliminate this effect. As a consequent, when consolidated into the prediction model, all the predictive features in this paper are normalized by the following equation:(8)yi=xi∑i=1nxi.


### 2.3. LSTM Theory

The RNN is exceptionally adept at time series processing. However, the network loses information about previously separated states during backpropagation, as well as the disappearance or explosion of gradients, rendering the RNN unsuitable for addressing long-term dependence problems [23]. The LSTM was proposed to solve the RNN problems [24]. The LSTM repeating cell network layer differs from the RNN’s single network layer in which it consists of four network layers that adjust the current cell state by deleting and adding information via three logical gate structures (forget gate, input gate, and output gate).A typical LSTM cell structure is shown in Figure3. The forgetting gate ft is used to decide which information ct−1 has been kept or deleted; the input gate it determines which information in input xt has been added to the cell state and updates candidate cell information ct˜, and then the current state cell state ct is obtained by integrating over the previous cell state ct−1 as well as ct˜. The output gate ot determines the output characteristics, and the final output ht is determined by the output gate and the current cell state ct. The expressions for the forget gate, input gate, and output gate, are shown as follows [25, 26]:(9)ft=σwfhht−1+wfxxt+bf,it=σwihht−1+wixxt+bi,ct˜=tanhwchht−1+wcxxt+bc,ct=ft⊗Ct−1+it⊗ct˜,ot=σwohht−1+woxxt+bo,ht=ot⊗tanhct.Figure 3
LSTM cell frame.In the above equation,wfh, wih, wch,andwoh are the weighting factors of output ht. wfx,wix,wcx,andwox are the weighting factors of input xt. σandtanh are the activation functions where σ is the sigmoid function. ft,it,andot are the forget gate, input gate, and output gate outputs, and the symbol ⊗ denotes the multiplication of the corresponding elements of the two vectors.


### 2.4. EDLSTM Theory

The operating conditions of bearings are complex and variable in real-world applications. Through training on one-bearing failure mode data, it is challenging for a single prediction model to learn other failure modes. To improve the RUL prediction model with various failure modes and to increase stability, an RUL prognosis method for EDLSTM has been proposed in this study.The framework of the EDLSTM prognosis model constructed in this study is shown in Figure4. It can be seen that the EDLSTM is composed of several individual DLSTM, and that the DLSTM has the same structure. During the process of offline training, one failure mode corresponds to one DLSTM network architecture. To enhance the computational efficiency of the prediction model and decrease the running time, all DLSTM in the model is run in parallel. The RUL estimates Rti for each DLSTM were obtained through some calculations where i=1,2,3,…,n and n is the total number of DLSTM. The n should be set reasonably according to the total number of failure modes and the training time. Finally, the obtained series of predicted RUL values are brought into the MIC-based multimodel ensemble method to estimate the optimal RUL.Figure 4
A framework for the EDLSTM predictive model.


#### 2.4.1. MIC Ensemble

MIC is a way to measure the degree of correlation between two variables [27]. When calculating the degree of correlation, this method is independent of the data distribution, is not restricted to a specific form of the correlation function, and is fairer and more extensive than other methods. This process of computing MIC for two-state variables X1,X2 in a data set M is as follows.To compute the mutual information (MI) of the features, assume that the two-state feature variablesX1=x11,…,xn1andX2=x12,…,xn2, then the MI of the feature state variables is as follows [27]:(10)MIX1,X2=∑xi2∈X2∑xj1∈X1pxj1,xi2logpxj1,xi2pxj1pxi2.In the above equation,pxj1,xi2 is the joint probability density function between the state feature variables X1,X2, and pxj1 and pxi2 are the marginal probability density functions of X1andX2.The state featuresX1andX2 are divided into a network of r rows and s columns, denoted as G=r,s, and it will obtain a variety of ways G division schemes when given r and s. According to the mutual information formula (10), the MI of different divisions G is computed. Finally, the maximum MI value of a division scheme is selected. Meanwhile, the MI value is normalized to obtain the maximum mutual information value.(11)MDGX1,X2=MIDG∗X1,X2log2minr,s,where MIDG∗X1,X2 represents the maximum MI of the state feature variables X1andX2 for the dataset D under the network G.The maximum MI under differentr, s values is then calculated based on the above finally using the following equation to obtain the MIC under diverse network divisions:(12)MICD=minr∗s<n0.6MDGX1,X2,where n is the number of data samples.In this study, each prediction of the bearing’s remaining useful lifeRi corresponds to a set of state feature variables Xi, while different DLSTM training corresponds to various failure mode data. To determine the optimum predicted RUL value of the EDLSTM, the MIC of the currently predicted bearing and each independent DLSTM training sample is calculated as a weighted value w to achieve the optimum RUL estimation. The details have been shown.Step 1.
The MIC values of the current feature variable and the different training sample feature variables are computed. The current predicted bearing RUL feature variable isXo, the training sample feature variable is Ytk, t=1,...,n,k=1,...K, n is the length of training model features, and K is the number of training models. The MIC values for X1 and Ytk are calculated.(13)wtk=MICXo,Ytk,t=1,…,n.k=1,…,K,where t is the length of each training sample and K is the total number of different failure mode training samples for the EDLSTM.Step 2.
The mean MIC value of the current state variableXo and the training sample feature variable Ytk is computed.(14)wtok¯=1n∑t=1nMICXo,Ytk=1n∑t=1nwtk.Step 3.
Summingwtok¯ to get wall.(15)wall=1K∑k=1Kwtok¯.
The above equationK is the number of DLSTMs constituted in the EDLSTM.Step 4.
To accurately calculate the EDLSTM optimal RUL, a weight correlation factoru is proposed in this study to adjust the weighted value of the independent DLSTM predicted RUL.(16)Yk¯=1n∑t=1nYtk,u=MICXo,Yk¯,where Yk¯ is the mean of the state features for the different training sets.Step 5.
The weights obtained from the different DLSTM prediction state variables are multiplied by the corresponding RUL. Finally, obtained the optimal EDLSTM prediction RUL.(17)Rt¯=u1K∑k=1Kwtok¯wall∗Rtk,where Rt¯ is the optimal value of the predicted bearing RUL of the EDLSTM at time t and Rtk is the kth DLSTM predicted bearings RUL at time t.


## 2.1. Initial Degradation Point Assessment

The root mean square (RMS) of vibration can be used to assess the mechanical health state [21, 22]. When the machine moves from normal to abnormal, the RMS value rises at a set rate. As a result, it is reasonable to use RMS to determine the degradation point of a bearing. The collected data in the paper include horizontal and vertical directions. First, equation (1) is used to compute the RMS for both directions. Then, equation (2) is used to compute the root mean square (RMS). Equation (3) is used for standardization to avoid differences in the magnitude of the RMS.(1)Xrms=1N∑i=1Nxi2,(2)Xs=Xrms−h2+Xrms−v2,(3)Xsrms=Xs−minXsmaxXs−minXs.In the above equations,xi is the collected signals collected at different periods and N is the length of the signals at selected periods. Xrms−h is the RMS of the horizontal sensor collection collected signals, and Xrms−v is the RMS of the vertical sensor collection collected signals.In this paper, to identify the RUL initial prediction point, assume that the current bearing collection signals isn. The mean RMS of the previous n−1 signals is multiplied by a threshold range to get ∆Xsrms, when the current period RMS of n is over than ∆Xsrms, that is the initial point of deterioration. The degradation point determination logic is shown in Algorithm 1.Algorithm 1: Degradation point determination logic.
(1)
Compute the average RMS of the previousn − 1 series to get ∆Xsrms.(2)
Compute the current periodn root mean square Xrmsn.(3)
Set the thresholdkk∈2.5–3.(4)
Ifk∗∆Xsrms≤Xrmsn(5)
N is the initial degradation point.(6)
Else(7)
N is the normal stage, n = n + 1, repeat the above process.(8)
endThe initial degradation point of different bearings is obtained according to the judgment condition, and Figure1 shows the initial degradation points of various bearings.Figure 1
The evaluation of various bearing initial degradation points. (a) Bearing 1-2. (b) Bearing 2-1.
(a)(b)Figure1 of the different bearings shows that the signal fluctuations change significantly as the operation time increases, and the initial degradation point cannot be determined accurately. On the other hand, the initial degradation point can be determined visually using SRMS, and a comparison of the time-domain plots with the SRMS graphs clearly demonstrate that the degradation stages classified using SRMS are compatible with reality.


## 2.2. Multifeature Extraction Methods

### 2.2.1. Spectral Energy Characteristics

The use of spectrum energy to reflect the health state of the bearing at a specific moment was proposed in Reference [14]. The spectrum energy was calculated in this study using fast Fourier transform (FFT) of the time step signal. The specific equation for integrating the time step frequency to obtain a characteristic value that can comprehensively reflect the current state of the bearing is as follows:(4)Et,i=14500∫i×ni+1×nxtfdf,where Et,i is the ith energy feature value at time step t and xtf is the vibration frequency of the vibration signal at time step t. The vibration signal in this study includes both vertical and horizontal directions. Figure 2 shows that two spectral energy features obtained in the end.Figure 2
DELSTM training set testing results. (a) Bearing 1-1. (b) Bearing 1-3. (c) Bearing 1-5. (d) Bearing 2-2. (e) Bearing 2-3. (f) Bearing 2-4. (g) Bearing 3-1. (h) Bearing 3-2. (i) Bearing 3-3. (j) Bearing 3-4. (k) Bearing 3-5.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)


### 2.2.2. Shannon Entropy

The Shannon entropy, also known as information entropy, is defined as the discrete sequence of signals being measured as when each component isa state and then X contains N status. p is the probability component corresponding to the N states, which are defined as follows:(5)∑i=0Npi=0≤pi≤1.It shows the amount of uncertain information in each of these state quantitiesxi in the following equation:(6)Ixi=−logapii=1,2,3,…,N.Different values ofα in the above equation will give distinct units of Shannon entropy, generally making α equal to 2.Since all the uncertain information quantities inX are random variables that do not accurately express the measured discrete sequence signal information, define the Shannon entropy HX as shown in the following formula:(7)HX=∑i=1NpiIxi=−∑i=1Npilogapi.The inconsistent magnitude of the feature reduces the predictive power of the model, and normalizing the feature magnitudes can eliminate this effect. As a consequent, when consolidated into the prediction model, all the predictive features in this paper are normalized by the following equation:(8)yi=xi∑i=1nxi.


## 2.2.1. Spectral Energy Characteristics

The use of spectrum energy to reflect the health state of the bearing at a specific moment was proposed in Reference [14]. The spectrum energy was calculated in this study using fast Fourier transform (FFT) of the time step signal. The specific equation for integrating the time step frequency to obtain a characteristic value that can comprehensively reflect the current state of the bearing is as follows:(4)Et,i=14500∫i×ni+1×nxtfdf,where Et,i is the ith energy feature value at time step t and xtf is the vibration frequency of the vibration signal at time step t. The vibration signal in this study includes both vertical and horizontal directions. Figure 2 shows that two spectral energy features obtained in the end.Figure 2
DELSTM training set testing results. (a) Bearing 1-1. (b) Bearing 1-3. (c) Bearing 1-5. (d) Bearing 2-2. (e) Bearing 2-3. (f) Bearing 2-4. (g) Bearing 3-1. (h) Bearing 3-2. (i) Bearing 3-3. (j) Bearing 3-4. (k) Bearing 3-5.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)


## 2.2.2. Shannon Entropy

The Shannon entropy, also known as information entropy, is defined as the discrete sequence of signals being measured as when each component isa state and then X contains N status. p is the probability component corresponding to the N states, which are defined as follows:(5)∑i=0Npi=0≤pi≤1.It shows the amount of uncertain information in each of these state quantitiesxi in the following equation:(6)Ixi=−logapii=1,2,3,…,N.Different values ofα in the above equation will give distinct units of Shannon entropy, generally making α equal to 2.Since all the uncertain information quantities inX are random variables that do not accurately express the measured discrete sequence signal information, define the Shannon entropy HX as shown in the following formula:(7)HX=∑i=1NpiIxi=−∑i=1Npilogapi.The inconsistent magnitude of the feature reduces the predictive power of the model, and normalizing the feature magnitudes can eliminate this effect. As a consequent, when consolidated into the prediction model, all the predictive features in this paper are normalized by the following equation:(8)yi=xi∑i=1nxi.


## 2.3. LSTM Theory

The RNN is exceptionally adept at time series processing. However, the network loses information about previously separated states during backpropagation, as well as the disappearance or explosion of gradients, rendering the RNN unsuitable for addressing long-term dependence problems [23]. The LSTM was proposed to solve the RNN problems [24]. The LSTM repeating cell network layer differs from the RNN’s single network layer in which it consists of four network layers that adjust the current cell state by deleting and adding information via three logical gate structures (forget gate, input gate, and output gate).A typical LSTM cell structure is shown in Figure3. The forgetting gate ft is used to decide which information ct−1 has been kept or deleted; the input gate it determines which information in input xt has been added to the cell state and updates candidate cell information ct˜, and then the current state cell state ct is obtained by integrating over the previous cell state ct−1 as well as ct˜. The output gate ot determines the output characteristics, and the final output ht is determined by the output gate and the current cell state ct. The expressions for the forget gate, input gate, and output gate, are shown as follows [25, 26]:(9)ft=σwfhht−1+wfxxt+bf,it=σwihht−1+wixxt+bi,ct˜=tanhwchht−1+wcxxt+bc,ct=ft⊗Ct−1+it⊗ct˜,ot=σwohht−1+woxxt+bo,ht=ot⊗tanhct.Figure 3
LSTM cell frame.In the above equation,wfh, wih, wch,andwoh are the weighting factors of output ht. wfx,wix,wcx,andwox are the weighting factors of input xt. σandtanh are the activation functions where σ is the sigmoid function. ft,it,andot are the forget gate, input gate, and output gate outputs, and the symbol ⊗ denotes the multiplication of the corresponding elements of the two vectors.


## 2.4. EDLSTM Theory

The operating conditions of bearings are complex and variable in real-world applications. Through training on one-bearing failure mode data, it is challenging for a single prediction model to learn other failure modes. To improve the RUL prediction model with various failure modes and to increase stability, an RUL prognosis method for EDLSTM has been proposed in this study.The framework of the EDLSTM prognosis model constructed in this study is shown in Figure4. It can be seen that the EDLSTM is composed of several individual DLSTM, and that the DLSTM has the same structure. During the process of offline training, one failure mode corresponds to one DLSTM network architecture. To enhance the computational efficiency of the prediction model and decrease the running time, all DLSTM in the model is run in parallel. The RUL estimates Rti for each DLSTM were obtained through some calculations where i=1,2,3,…,n and n is the total number of DLSTM. The n should be set reasonably according to the total number of failure modes and the training time. Finally, the obtained series of predicted RUL values are brought into the MIC-based multimodel ensemble method to estimate the optimal RUL.Figure 4
A framework for the EDLSTM predictive model.


### 2.4.1. MIC Ensemble

MIC is a way to measure the degree of correlation between two variables [27]. When calculating the degree of correlation, this method is independent of the data distribution, is not restricted to a specific form of the correlation function, and is fairer and more extensive than other methods. This process of computing MIC for two-state variables X1,X2 in a data set M is as follows.To compute the mutual information (MI) of the features, assume that the two-state feature variablesX1=x11,…,xn1andX2=x12,…,xn2, then the MI of the feature state variables is as follows [27]:(10)MIX1,X2=∑xi2∈X2∑xj1∈X1pxj1,xi2logpxj1,xi2pxj1pxi2.In the above equation,pxj1,xi2 is the joint probability density function between the state feature variables X1,X2, and pxj1 and pxi2 are the marginal probability density functions of X1andX2.The state featuresX1andX2 are divided into a network of r rows and s columns, denoted as G=r,s, and it will obtain a variety of ways G division schemes when given r and s. According to the mutual information formula (10), the MI of different divisions G is computed. Finally, the maximum MI value of a division scheme is selected. Meanwhile, the MI value is normalized to obtain the maximum mutual information value.(11)MDGX1,X2=MIDG∗X1,X2log2minr,s,where MIDG∗X1,X2 represents the maximum MI of the state feature variables X1andX2 for the dataset D under the network G.The maximum MI under differentr, s values is then calculated based on the above finally using the following equation to obtain the MIC under diverse network divisions:(12)MICD=minr∗s<n0.6MDGX1,X2,where n is the number of data samples.In this study, each prediction of the bearing’s remaining useful lifeRi corresponds to a set of state feature variables Xi, while different DLSTM training corresponds to various failure mode data. To determine the optimum predicted RUL value of the EDLSTM, the MIC of the currently predicted bearing and each independent DLSTM training sample is calculated as a weighted value w to achieve the optimum RUL estimation. The details have been shown.Step 1.
The MIC values of the current feature variable and the different training sample feature variables are computed. The current predicted bearing RUL feature variable isXo, the training sample feature variable is Ytk, t=1,...,n,k=1,...K, n is the length of training model features, and K is the number of training models. The MIC values for X1 and Ytk are calculated.(13)wtk=MICXo,Ytk,t=1,…,n.k=1,…,K,where t is the length of each training sample and K is the total number of different failure mode training samples for the EDLSTM.Step 2.
The mean MIC value of the current state variableXo and the training sample feature variable Ytk is computed.(14)wtok¯=1n∑t=1nMICXo,Ytk=1n∑t=1nwtk.Step 3.
Summingwtok¯ to get wall.(15)wall=1K∑k=1Kwtok¯.
The above equationK is the number of DLSTMs constituted in the EDLSTM.Step 4.
To accurately calculate the EDLSTM optimal RUL, a weight correlation factoru is proposed in this study to adjust the weighted value of the independent DLSTM predicted RUL.(16)Yk¯=1n∑t=1nYtk,u=MICXo,Yk¯,where Yk¯ is the mean of the state features for the different training sets.Step 5.
The weights obtained from the different DLSTM prediction state variables are multiplied by the corresponding RUL. Finally, obtained the optimal EDLSTM prediction RUL.(17)Rt¯=u1K∑k=1Kwtok¯wall∗Rtk,where Rt¯ is the optimal value of the predicted bearing RUL of the EDLSTM at time t and Rtk is the kth DLSTM predicted bearings RUL at time t.


## 2.4.1. MIC Ensemble

MIC is a way to measure the degree of correlation between two variables [27]. When calculating the degree of correlation, this method is independent of the data distribution, is not restricted to a specific form of the correlation function, and is fairer and more extensive than other methods. This process of computing MIC for two-state variables X1,X2 in a data set M is as follows.To compute the mutual information (MI) of the features, assume that the two-state feature variablesX1=x11,…,xn1andX2=x12,…,xn2, then the MI of the feature state variables is as follows [27]:(10)MIX1,X2=∑xi2∈X2∑xj1∈X1pxj1,xi2logpxj1,xi2pxj1pxi2.In the above equation,pxj1,xi2 is the joint probability density function between the state feature variables X1,X2, and pxj1 and pxi2 are the marginal probability density functions of X1andX2.The state featuresX1andX2 are divided into a network of r rows and s columns, denoted as G=r,s, and it will obtain a variety of ways G division schemes when given r and s. According to the mutual information formula (10), the MI of different divisions G is computed. Finally, the maximum MI value of a division scheme is selected. Meanwhile, the MI value is normalized to obtain the maximum mutual information value.(11)MDGX1,X2=MIDG∗X1,X2log2minr,s,where MIDG∗X1,X2 represents the maximum MI of the state feature variables X1andX2 for the dataset D under the network G.The maximum MI under differentr, s values is then calculated based on the above finally using the following equation to obtain the MIC under diverse network divisions:(12)MICD=minr∗s<n0.6MDGX1,X2,where n is the number of data samples.In this study, each prediction of the bearing’s remaining useful lifeRi corresponds to a set of state feature variables Xi, while different DLSTM training corresponds to various failure mode data. To determine the optimum predicted RUL value of the EDLSTM, the MIC of the currently predicted bearing and each independent DLSTM training sample is calculated as a weighted value w to achieve the optimum RUL estimation. The details have been shown.Step 1.
The MIC values of the current feature variable and the different training sample feature variables are computed. The current predicted bearing RUL feature variable isXo, the training sample feature variable is Ytk, t=1,...,n,k=1,...K, n is the length of training model features, and K is the number of training models. The MIC values for X1 and Ytk are calculated.(13)wtk=MICXo,Ytk,t=1,…,n.k=1,…,K,where t is the length of each training sample and K is the total number of different failure mode training samples for the EDLSTM.Step 2.
The mean MIC value of the current state variableXo and the training sample feature variable Ytk is computed.(14)wtok¯=1n∑t=1nMICXo,Ytk=1n∑t=1nwtk.Step 3.
Summingwtok¯ to get wall.(15)wall=1K∑k=1Kwtok¯.
The above equationK is the number of DLSTMs constituted in the EDLSTM.Step 4.
To accurately calculate the EDLSTM optimal RUL, a weight correlation factoru is proposed in this study to adjust the weighted value of the independent DLSTM predicted RUL.(16)Yk¯=1n∑t=1nYtk,u=MICXo,Yk¯,where Yk¯ is the mean of the state features for the different training sets.Step 5.
The weights obtained from the different DLSTM prediction state variables are multiplied by the corresponding RUL. Finally, obtained the optimal EDLSTM prediction RUL.(17)Rt¯=u1K∑k=1Kwtok¯wall∗Rtk,where Rt¯ is the optimal value of the predicted bearing RUL of the EDLSTM at time t and Rtk is the kth DLSTM predicted bearings RUL at time t.


## 3. Experimental Analyses

In this paper, a prediction framework was constructed to implement bearing RUL prediction, as shown in Figure5. First, the network is trained with different failure mode features of the bearings, and then the proposed method is validated using other failure mode data.Figure 5
The procedure of EDLSTM prognosis RUL for rolling bearing.


### 3.1. Data Set Description

To test the predictive capability of the proposed EDLSTM method, the rolling bearing full-life vibration signals used in this study were obtained from the XJTU-SY dataset of Xi’an Jiaotong University’s Joint Laboratory for Health Monitoring of Mechanical Equipment [23, 28]. The dataset includes full life-cycle vibration signals from 15 bearings with various failure modes under three operating conditions. Table 1 shows the specific operating conditions, Table 2 shows the concrete failure modes of the bearing under various operating conditions, and Figure 6 shows the accelerated deterioration experimental bed.Table 1
Bearing operating conditions.
Condition no.123Speed (r/min)210022502400Radial force (kN)121110Table 2
Information on the failure of different bearings.
Condition no.Bearing no.Failure location1Bearing 1-1Outer raceBearing 1-2Outer raceBearing 1-3Outer raceBearing 1-4CageBearing 1-5Inter race and outer race2Bearing 2-1Inter raceBearing 2-2Outer raceBearing 2-3CageBearing 2-4Outer raceBearing 2-5Outer race3Bearing 3-1Outer raceBearing 3-2Inter race, ball, cage and outer raceBearing 3-3Inter raceBearing 3-4Inter raceBearing 3-5Outer raceFigure 6
Testbed of rolling element bearings.The experiment bed consists of an AC motor, motor speed controller, rotating shaft, support bearings, acceleration sensors, hydraulic loading system, and test bearings. The AC motor is used for torque output to vary the operating conditions of the bearing. The hydraulic loading system load is used to accelerate the bearing deterioration. Two accelerometers have been mounted horizontally and vertically to collect the life cycle signal from the bearing at a frequency of 25.6 kHz, with a sampling interval of 1 minute and a duration of 1.28 seconds per sample.


### 3.2. Feature Extraction

Time-domain features, spectrum energy features, and Shannon entropy prediction features are calculated to reflect the bearing health state more accurately and predict the bearing RUL. The time-domain features are widely used in bearing life prediction research [29].Due to the large number of time-domain features calculated, some of them include little information on the health status of the bearing and taking them into the model will increase the computational effort of the computer. To reasonably arrange the processing speed of the computer operation, the correlation coefficient method is used to filter the time-domain features. In the study, the correlation coefficient is used in each time-domain feature with RMS because it can directly reflect the health status of the machinery. If the correlation coefficient value is greater than 0.5, the correlation is considered strong and can be used as a prediction feature quantity. The database contained horizontal and vertical vibration signals, which are separately calculated when calculating the features, with the specific correlation coefficients indicated in Figure7.Figure 7
Time-domain feature correlation coefficient. (a) Bearing 1-2. (b) Bearing 2-1.∗1: Mean. 2: Standard deviation. 3: Coefficient of variation. 4: Square root amplitude. 5: Root mean square. 6: Peak-to-peak. 7: Skewness. 8: Kurtosis. 9: Skewness factor. 10: Kurtosis factor. 11: Peak factor. 12: Margin factor. 13: Waveform factor. 14: Impulse index.
(a)(b)The selection criterion is that the correlation coefficient is greater than 0.5. Figure7 shows that the standard deviation, root mean square, peak-to-peak, and RMS correlation coefficients all exceed 0.9. The above four features are used as the predictive features. Combined with a correlation coefficient greater than 0.5, the results are consistent when considering the differences in the bearing input dimensions. There are distinctions in the fifth time-domain feature chosen for different bearings, with the fifth feature for bearing 1-2 being the cliff factor, but the bearing 2-1 being the waveform factor.The spectrum reflects the location and type of mechanical failure, and the signal spectrum is obtained using the FFT. The spectrum energy, which represents the state of the bearing at a particular time, is calculated based on the spectrum amplitude and is given in the second part. The energy characteristics of the spectrum calculated from the above are shown in Figure8.Figure 8
The bearing spectrum energy. (a) Bearing 1-2. (b) Bearing 2-1.
(a)(b)From Figure8, with the degree of degradation growing, the energy of the different failure modes bearing spectrum is gradually increasing. This rule is consistent with the actual energy change rule of the bearing degradation process.Shannon entropy is used to measure the uncertainty of a variable, and it can measure the uncertainty information of the bearing. The degree of degradation of a bearing in actual use grows with time. The uncertainty information of the bearing at the degradation stage also increases. Thus, the degeneration process can be represented by the Shannon entropy. As can be seen from Figure9, the Shannon entropy of the bearings in the different failure modes increases with the extent of degradation.Figure 9
Bearing Shannon entropy. (a) Bearing 1-2. (b) Bearing 2-1.
(a)(b)


### 3.3. Predictive Evaluation Indicators

In this study, it is necessary to construct different evaluation metrics to evaluate the prediction results. The root mean square error (RMSE) measures the deviation of the predicted values from the actual values. The cumulative relative accuracy (CRA) evaluates the degree of closeness between the predicted RUL and the actual RUL. Generally, these two metrics are used in prediction estimation to evaluate and predict model performance [30]. In this study, these two metrics are also selected as the evaluation metrics for the prediction model. The specific formulas for the above three evaluation metrics, RMSE, and CRA are as follows:(18)RMSE=1N∑i=1NRULiT−RULiP2,CRA=1N∑i=1N1−RULiT−RULiPRULiT.In the above equation,N is the length of the predicted sequences, RULiT is the true remaining life of the predicted sequences, and RULiP is the remaining life value of the series by the prediction model.In this study, to evaluate the prediction model with higher accuracy, the scoring function (19) is chosen to evaluate the prediction model, with higher scores meaning better prediction model performance.(19)Score=1N∑i=1NAi,(20)Ai=exp−ln0.5Eri5,Eri≤0,exp+ln0.5Eri20,Eri>0,Eri=RULiT−RULiPRULiT×100%.In the above equation,Eri is the percentage error between the ith predicted bearing RUL and the actual RUL.


### 3.4. RUL Prediction and Analysis

This study uses MIC to ensemble multiple DLSTM models to predict rolling bearing RUL. For the different DLSTM network structure parameters in EDLSTM and hyperparameter, the setup is shown in Table3 after several attempts. The normal stage vibration signal of the bearing does not contain degradation features. It is meaningless and wasteful of computer resources to perform RUL predictions for this stage of the bearing, so this study only makes RUL predictions for the degradation stage of the bearing. First, the initial degradation point of the bearing was determined according to the SRMS, and then the RUL prediction was performed for the degradation stage. Considering the different operating conditions and failure modes of the bearings, meanwhile, to verify the accuracy and stability of the established EDLSTM prediction model, this study divided the XJTU-SY data set into a training set and a test set, as shown in Table 4.Table 3
Network parameter setting.
NameDetailFull connected layer (FC0)in_features = 15, out_features = 128LSTM layerinput_size = 128, hidden_size = 256, dropout = 0.2, num_layers = 2Full connected layer (FC1)in_features = 256, out_features = 128Full connected layer (FC2)in_features = 128, out_features = 64Full connected layer (FC3)in_features = 64, out_features = 32Output layer (out)in_features = 32, out_features = 1Batch_size8OptimiserASGDLoss functionMSE lossActivation functionReLUNumber of iterations100Table 4
Training and testing set data division.
NameDetailTraining setsBearing 1-2, bearing 1-4, bearing 2-1, bearing 2-5Testing setsOthersIn this study, the selected data set contains five failure modes. It includes eight outer ring failure bearings, three inner ring failure bearings, one inner and one outer ring failure bearing, two cage failure bearings, and one multifailure mixed failure bearing. To verify the stability of the model to a maximum degree, bearing 1-2 for outer ring failure in condition 1, bearing 2-5 for outer ring failure in condition 2, bearing 2-1 for inner ring failure in condition 2, and bearing 1-4 for cage failure in condition 1 were selected. The remaining 11 bearing data were used as a test set to verify the accuracy and stability of the model prediction.The EDLSTM is trained through the training set and learns to give the model adequately expressive power. Figure10 shows the expressiveness of the EDLSTM prediction model.Figure 10
DELSTM training set training results. (a) Bearing 1-2. (b) Bearing 1-4. (c) Bearing 2-1. (d) Bearing 2-5.
(a)(b)(c)(d)From the training results of Figure10, the prediction results of the training are identical to the actual RUL of the bearings, indicating that the prediction model has a strong learning expression capability.In the study, a test set to verify the prediction accuracy and stability was used in the EDLSTM prediction model. The corresponding bearing prediction RUL results obtained are shown in Figure2.According to Figure2, it is shown that the EDLSTM bearing RUL prediction model developed in this paper can not only accurately predict the actual trend of bearing RUL but also has sufficient stability. Table 5 summarizes the performance evaluation results of prognostics approaches in reference [23] and the proposed prognostics approaches. To further illustrate the accuracy and stability of the prediction method, the evaluation metric statistics were conducted for the four bearing RUL prediction results from the training set and the 11 bearing RUL prediction results from the test set. The evaluation metric statistics for the training set are shown in Table 6. The detailed predicted results and relative errors for the test set are shown in Table 7.Table 5
Performance evaluation for the six prognostics approaches using CRA.
Operating conditionBearing datasetCRARVMDBNPFEKFHybrid prognostics [23]Proposed approachCondition 1Bearing 10.570.430.610.920.900.90Bearing 20.180.620.730.350.860.90Bearing 30.630.560.490.800.850.81Bearing 40.370.950.230.680.720.96Bearing 50.610.660.430.500.790.89Condition 2Bearing 10.570.550.400.510.860.93Bearing 20.180.200.260.430.650.94Bearing 30.620.900.740.880.960.82Bearing 40.070.530.460.500.630.43Bearing 50.260.070.180.480.630.83Condition 3Bearing 10.430.700.660.770.890.80Bearing 20.220.130.150.540.650.67Bearing 30.090.520.130.970.890.83Bearing 40.660.610.780.670.810.91Bearing 50.490.560.390.500.650.87Table 6
Training set evaluation metric statistics.
Bearing no.RMSECRAScoreBearing 1-2412.960.90127.4Bearing 1-447.390.9643.9Bearing 2-186.620.9333.7Bearing 2-5477.990.83184.2Table 7
Detailed prediction results and relative errors of bearing RUL.
Bearing 1-1Bearing 1-3Bearing 1-5Bearing 2-2Bearing 2-3Bearing 2-5Bearing 3-1Bearing 3-2Bearing 3-3Bearing 3-4Bearing 3-5APREAPREAPREAPREAPREAPREAPREAPREAPREAPREAPRE2817.542760.000.025724.395880.000.031168.671080.000.086243.186300.000.0120925.1322020.000.051722.421740.000.0110758.3011340.000.05123248.40139440.000.131727.531680.000.035715.795760.000.016137.726360.000.042025.132280.000.135332.094740.000.111007.76900.000.115103.725100.000.0019840.1617700.000.111681.031500.000.1110399.809180.000.12123620.00111660.000.101348.101380.000.024984.974620.000.075300.335160.000.031550.211740.000.124519.683540.000.22782.29660.000.163357.143840.000.1412339.8713320.000.081645.091200.000.278689.746960.000.2095817.2683820.000.13959.971080.000.133119.233420.000.104237.473900.000.081254.071200.000.042865.012340.000.18485.99420.000.142536.022580.000.0211036.708940.000.191514.57840.000.456447.145340.000.1769963.9055980.000.20632.39720.000.142021.712220.000.102701.672640.000.02701.35660.000.061010.901140.000.13256.71240.000.071252.381320.000.054733.054560.000.04630.74360.000.433473.503120.000.102660.6128140.009.58284.62360.000.26935.931020.000.09947.901380.000.461.4260.0041.17159.5660.000.6227.3760.001.19104.1260.000.42543.8760.000.892.4660.0023.3961.6860.000.031306.4260.000.9514.4260.003.1697.8660.000.3969.5960.000.14∗A: actual RUL. P: predicted RUL. R.E: relative error.According to Table5, the proposed approach has a higher prediction accuracy and stable robustness compared with the methods in reference [23]. The proposed method has yet to use the identical condition of bearing data to train the model. It is trained using full life-cycle data with failure modes including outer ring failure bearing 1-2 and bearing 2-5, inner ring failure mode bearing 2-1, and cage failure mode bearing 1-4. The model was validated using data from the other 11 bearings.From Table7, it can be concluded that for most bearings, the relative error in predicting RUL using the method proposed in this paper ranges from 0.2 to 0.0, and the predicted RUL is consistent with the actual RUL variation of the bearing. However, the prognosis RUL relative error is too large for a small number of bearings. After analysis, it was found that the reasons for the large deviations in the predictions were as follows. Bearing 3-2 is a compound fault consisting of multiple failure components, and the training prediction model has limited information to learn to represent this type of failure mode. Consequently, there are large deviations in its predicted RUL. For bearing 2-3, the analysis of the collected signals showed that the vibration amplitude of the full life-cycle data for this bearing during the failure stage differed significantly from the diagnostic amplitude of training bearings 1-4, having a greater vibration amplitude. The result was attributed to the fact that the variation in operating conditions reduced the transferability of the model to the extent that there was a large deviation in predicting RUL. The failure mode of bearing 2-4 is an outer ring failure; the training conditions include the same failure mode and operating conditions for this bearing, but with large prediction deviations. In Reference [23], the predicted RUL for this bearing deviates significantly from the actual. Analysis of the full life-cycle vibration signal revealed that the bearing had a different signal waveform to the same failed bearing. Bearings 2-4 had a smooth vibration in the normal phase, and a sudden failure increased the vibration amplitude until complete failure. The cause of the above was a manufacturing defect in the bearing, and it was this manufacturing defect in the bearing that led to a large deviation in predicted RUL.In this study, to compare the advantages of the proposed method, the bagging method was selected to ensemble multiple DLSTM independent prediction models, and the validation set prediction metrics are obtained as shown in Table8.Table 8
Bagging-DLSTMpredicted metric results.
Bearing no.CRAScoreBagging-DLSTMProposed approachBagging-DLSTMProposed approachBearing 1-10.330.9043.545.8Bearing 1-30.570.8192.896.1Bearing 1-50.860.8917.617.8Bearing 2-20.600.94103.1104.6Bearing 2-30.380.82348.9361.3Bearing 2-40.280.4326.827.1Bearing 3-10.430.80177.4184.0Bearing 3-20.280.672113.82265.6Bearing 3-30.550.8327.127.8Bearing 3-40.470.9192.7795.5Bearing 3-50.140.8796.6105.2Mean0.450.81285.49302.80Both the CRA value and score values are greater than the bagging-DLSTM of the proposed method, as shown in Table8. The mean CRA value for bagging-DLSTM is 0.45, whereas it is 0.81 for the proposed method, demonstrating the superiority of the latter.In summary, this paper proposes a highly accurate and robust EDLSTM method to predict the RUL of bearings. This method accurately predicts the RUL of the training bearing samples and the RUL of the nontraining samples. Based on the outcomes of all bearings, it can be concluded that the method has strong stability in predicting the RUL of untrained bearing failure modes.


## 3.1. Data Set Description

To test the predictive capability of the proposed EDLSTM method, the rolling bearing full-life vibration signals used in this study were obtained from the XJTU-SY dataset of Xi’an Jiaotong University’s Joint Laboratory for Health Monitoring of Mechanical Equipment [23, 28]. The dataset includes full life-cycle vibration signals from 15 bearings with various failure modes under three operating conditions. Table 1 shows the specific operating conditions, Table 2 shows the concrete failure modes of the bearing under various operating conditions, and Figure 6 shows the accelerated deterioration experimental bed.Table 1
Bearing operating conditions.
Condition no.123Speed (r/min)210022502400Radial force (kN)121110Table 2
Information on the failure of different bearings.
Condition no.Bearing no.Failure location1Bearing 1-1Outer raceBearing 1-2Outer raceBearing 1-3Outer raceBearing 1-4CageBearing 1-5Inter race and outer race2Bearing 2-1Inter raceBearing 2-2Outer raceBearing 2-3CageBearing 2-4Outer raceBearing 2-5Outer race3Bearing 3-1Outer raceBearing 3-2Inter race, ball, cage and outer raceBearing 3-3Inter raceBearing 3-4Inter raceBearing 3-5Outer raceFigure 6
Testbed of rolling element bearings.The experiment bed consists of an AC motor, motor speed controller, rotating shaft, support bearings, acceleration sensors, hydraulic loading system, and test bearings. The AC motor is used for torque output to vary the operating conditions of the bearing. The hydraulic loading system load is used to accelerate the bearing deterioration. Two accelerometers have been mounted horizontally and vertically to collect the life cycle signal from the bearing at a frequency of 25.6 kHz, with a sampling interval of 1 minute and a duration of 1.28 seconds per sample.


## 3.2. Feature Extraction

Time-domain features, spectrum energy features, and Shannon entropy prediction features are calculated to reflect the bearing health state more accurately and predict the bearing RUL. The time-domain features are widely used in bearing life prediction research [29].Due to the large number of time-domain features calculated, some of them include little information on the health status of the bearing and taking them into the model will increase the computational effort of the computer. To reasonably arrange the processing speed of the computer operation, the correlation coefficient method is used to filter the time-domain features. In the study, the correlation coefficient is used in each time-domain feature with RMS because it can directly reflect the health status of the machinery. If the correlation coefficient value is greater than 0.5, the correlation is considered strong and can be used as a prediction feature quantity. The database contained horizontal and vertical vibration signals, which are separately calculated when calculating the features, with the specific correlation coefficients indicated in Figure7.Figure 7
Time-domain feature correlation coefficient. (a) Bearing 1-2. (b) Bearing 2-1.∗1: Mean. 2: Standard deviation. 3: Coefficient of variation. 4: Square root amplitude. 5: Root mean square. 6: Peak-to-peak. 7: Skewness. 8: Kurtosis. 9: Skewness factor. 10: Kurtosis factor. 11: Peak factor. 12: Margin factor. 13: Waveform factor. 14: Impulse index.
(a)(b)The selection criterion is that the correlation coefficient is greater than 0.5. Figure7 shows that the standard deviation, root mean square, peak-to-peak, and RMS correlation coefficients all exceed 0.9. The above four features are used as the predictive features. Combined with a correlation coefficient greater than 0.5, the results are consistent when considering the differences in the bearing input dimensions. There are distinctions in the fifth time-domain feature chosen for different bearings, with the fifth feature for bearing 1-2 being the cliff factor, but the bearing 2-1 being the waveform factor.The spectrum reflects the location and type of mechanical failure, and the signal spectrum is obtained using the FFT. The spectrum energy, which represents the state of the bearing at a particular time, is calculated based on the spectrum amplitude and is given in the second part. The energy characteristics of the spectrum calculated from the above are shown in Figure8.Figure 8
The bearing spectrum energy. (a) Bearing 1-2. (b) Bearing 2-1.
(a)(b)From Figure8, with the degree of degradation growing, the energy of the different failure modes bearing spectrum is gradually increasing. This rule is consistent with the actual energy change rule of the bearing degradation process.Shannon entropy is used to measure the uncertainty of a variable, and it can measure the uncertainty information of the bearing. The degree of degradation of a bearing in actual use grows with time. The uncertainty information of the bearing at the degradation stage also increases. Thus, the degeneration process can be represented by the Shannon entropy. As can be seen from Figure9, the Shannon entropy of the bearings in the different failure modes increases with the extent of degradation.Figure 9
Bearing Shannon entropy. (a) Bearing 1-2. (b) Bearing 2-1.
(a)(b)


## 3.3. Predictive Evaluation Indicators

In this study, it is necessary to construct different evaluation metrics to evaluate the prediction results. The root mean square error (RMSE) measures the deviation of the predicted values from the actual values. The cumulative relative accuracy (CRA) evaluates the degree of closeness between the predicted RUL and the actual RUL. Generally, these two metrics are used in prediction estimation to evaluate and predict model performance [30]. In this study, these two metrics are also selected as the evaluation metrics for the prediction model. The specific formulas for the above three evaluation metrics, RMSE, and CRA are as follows:(18)RMSE=1N∑i=1NRULiT−RULiP2,CRA=1N∑i=1N1−RULiT−RULiPRULiT.In the above equation,N is the length of the predicted sequences, RULiT is the true remaining life of the predicted sequences, and RULiP is the remaining life value of the series by the prediction model.In this study, to evaluate the prediction model with higher accuracy, the scoring function (19) is chosen to evaluate the prediction model, with higher scores meaning better prediction model performance.(19)Score=1N∑i=1NAi,(20)Ai=exp−ln0.5Eri5,Eri≤0,exp+ln0.5Eri20,Eri>0,Eri=RULiT−RULiPRULiT×100%.In the above equation,Eri is the percentage error between the ith predicted bearing RUL and the actual RUL.


## 3.4. RUL Prediction and Analysis

This study uses MIC to ensemble multiple DLSTM models to predict rolling bearing RUL. For the different DLSTM network structure parameters in EDLSTM and hyperparameter, the setup is shown in Table3 after several attempts. The normal stage vibration signal of the bearing does not contain degradation features. It is meaningless and wasteful of computer resources to perform RUL predictions for this stage of the bearing, so this study only makes RUL predictions for the degradation stage of the bearing. First, the initial degradation point of the bearing was determined according to the SRMS, and then the RUL prediction was performed for the degradation stage. Considering the different operating conditions and failure modes of the bearings, meanwhile, to verify the accuracy and stability of the established EDLSTM prediction model, this study divided the XJTU-SY data set into a training set and a test set, as shown in Table 4.Table 3
Network parameter setting.
NameDetailFull connected layer (FC0)in_features = 15, out_features = 128LSTM layerinput_size = 128, hidden_size = 256, dropout = 0.2, num_layers = 2Full connected layer (FC1)in_features = 256, out_features = 128Full connected layer (FC2)in_features = 128, out_features = 64Full connected layer (FC3)in_features = 64, out_features = 32Output layer (out)in_features = 32, out_features = 1Batch_size8OptimiserASGDLoss functionMSE lossActivation functionReLUNumber of iterations100Table 4
Training and testing set data division.
NameDetailTraining setsBearing 1-2, bearing 1-4, bearing 2-1, bearing 2-5Testing setsOthersIn this study, the selected data set contains five failure modes. It includes eight outer ring failure bearings, three inner ring failure bearings, one inner and one outer ring failure bearing, two cage failure bearings, and one multifailure mixed failure bearing. To verify the stability of the model to a maximum degree, bearing 1-2 for outer ring failure in condition 1, bearing 2-5 for outer ring failure in condition 2, bearing 2-1 for inner ring failure in condition 2, and bearing 1-4 for cage failure in condition 1 were selected. The remaining 11 bearing data were used as a test set to verify the accuracy and stability of the model prediction.The EDLSTM is trained through the training set and learns to give the model adequately expressive power. Figure10 shows the expressiveness of the EDLSTM prediction model.Figure 10
DELSTM training set training results. (a) Bearing 1-2. (b) Bearing 1-4. (c) Bearing 2-1. (d) Bearing 2-5.
(a)(b)(c)(d)From the training results of Figure10, the prediction results of the training are identical to the actual RUL of the bearings, indicating that the prediction model has a strong learning expression capability.In the study, a test set to verify the prediction accuracy and stability was used in the EDLSTM prediction model. The corresponding bearing prediction RUL results obtained are shown in Figure2.According to Figure2, it is shown that the EDLSTM bearing RUL prediction model developed in this paper can not only accurately predict the actual trend of bearing RUL but also has sufficient stability. Table 5 summarizes the performance evaluation results of prognostics approaches in reference [23] and the proposed prognostics approaches. To further illustrate the accuracy and stability of the prediction method, the evaluation metric statistics were conducted for the four bearing RUL prediction results from the training set and the 11 bearing RUL prediction results from the test set. The evaluation metric statistics for the training set are shown in Table 6. The detailed predicted results and relative errors for the test set are shown in Table 7.Table 5
Performance evaluation for the six prognostics approaches using CRA.
Operating conditionBearing datasetCRARVMDBNPFEKFHybrid prognostics [23]Proposed approachCondition 1Bearing 10.570.430.610.920.900.90Bearing 20.180.620.730.350.860.90Bearing 30.630.560.490.800.850.81Bearing 40.370.950.230.680.720.96Bearing 50.610.660.430.500.790.89Condition 2Bearing 10.570.550.400.510.860.93Bearing 20.180.200.260.430.650.94Bearing 30.620.900.740.880.960.82Bearing 40.070.530.460.500.630.43Bearing 50.260.070.180.480.630.83Condition 3Bearing 10.430.700.660.770.890.80Bearing 20.220.130.150.540.650.67Bearing 30.090.520.130.970.890.83Bearing 40.660.610.780.670.810.91Bearing 50.490.560.390.500.650.87Table 6
Training set evaluation metric statistics.
Bearing no.RMSECRAScoreBearing 1-2412.960.90127.4Bearing 1-447.390.9643.9Bearing 2-186.620.9333.7Bearing 2-5477.990.83184.2Table 7
Detailed prediction results and relative errors of bearing RUL.
Bearing 1-1Bearing 1-3Bearing 1-5Bearing 2-2Bearing 2-3Bearing 2-5Bearing 3-1Bearing 3-2Bearing 3-3Bearing 3-4Bearing 3-5APREAPREAPREAPREAPREAPREAPREAPREAPREAPREAPRE2817.542760.000.025724.395880.000.031168.671080.000.086243.186300.000.0120925.1322020.000.051722.421740.000.0110758.3011340.000.05123248.40139440.000.131727.531680.000.035715.795760.000.016137.726360.000.042025.132280.000.135332.094740.000.111007.76900.000.115103.725100.000.0019840.1617700.000.111681.031500.000.1110399.809180.000.12123620.00111660.000.101348.101380.000.024984.974620.000.075300.335160.000.031550.211740.000.124519.683540.000.22782.29660.000.163357.143840.000.1412339.8713320.000.081645.091200.000.278689.746960.000.2095817.2683820.000.13959.971080.000.133119.233420.000.104237.473900.000.081254.071200.000.042865.012340.000.18485.99420.000.142536.022580.000.0211036.708940.000.191514.57840.000.456447.145340.000.1769963.9055980.000.20632.39720.000.142021.712220.000.102701.672640.000.02701.35660.000.061010.901140.000.13256.71240.000.071252.381320.000.054733.054560.000.04630.74360.000.433473.503120.000.102660.6128140.009.58284.62360.000.26935.931020.000.09947.901380.000.461.4260.0041.17159.5660.000.6227.3760.001.19104.1260.000.42543.8760.000.892.4660.0023.3961.6860.000.031306.4260.000.9514.4260.003.1697.8660.000.3969.5960.000.14∗A: actual RUL. P: predicted RUL. R.E: relative error.According to Table5, the proposed approach has a higher prediction accuracy and stable robustness compared with the methods in reference [23]. The proposed method has yet to use the identical condition of bearing data to train the model. It is trained using full life-cycle data with failure modes including outer ring failure bearing 1-2 and bearing 2-5, inner ring failure mode bearing 2-1, and cage failure mode bearing 1-4. The model was validated using data from the other 11 bearings.From Table7, it can be concluded that for most bearings, the relative error in predicting RUL using the method proposed in this paper ranges from 0.2 to 0.0, and the predicted RUL is consistent with the actual RUL variation of the bearing. However, the prognosis RUL relative error is too large for a small number of bearings. After analysis, it was found that the reasons for the large deviations in the predictions were as follows. Bearing 3-2 is a compound fault consisting of multiple failure components, and the training prediction model has limited information to learn to represent this type of failure mode. Consequently, there are large deviations in its predicted RUL. For bearing 2-3, the analysis of the collected signals showed that the vibration amplitude of the full life-cycle data for this bearing during the failure stage differed significantly from the diagnostic amplitude of training bearings 1-4, having a greater vibration amplitude. The result was attributed to the fact that the variation in operating conditions reduced the transferability of the model to the extent that there was a large deviation in predicting RUL. The failure mode of bearing 2-4 is an outer ring failure; the training conditions include the same failure mode and operating conditions for this bearing, but with large prediction deviations. In Reference [23], the predicted RUL for this bearing deviates significantly from the actual. Analysis of the full life-cycle vibration signal revealed that the bearing had a different signal waveform to the same failed bearing. Bearings 2-4 had a smooth vibration in the normal phase, and a sudden failure increased the vibration amplitude until complete failure. The cause of the above was a manufacturing defect in the bearing, and it was this manufacturing defect in the bearing that led to a large deviation in predicted RUL.In this study, to compare the advantages of the proposed method, the bagging method was selected to ensemble multiple DLSTM independent prediction models, and the validation set prediction metrics are obtained as shown in Table8.Table 8
Bagging-DLSTMpredicted metric results.
Bearing no.CRAScoreBagging-DLSTMProposed approachBagging-DLSTMProposed approachBearing 1-10.330.9043.545.8Bearing 1-30.570.8192.896.1Bearing 1-50.860.8917.617.8Bearing 2-20.600.94103.1104.6Bearing 2-30.380.82348.9361.3Bearing 2-40.280.4326.827.1Bearing 3-10.430.80177.4184.0Bearing 3-20.280.672113.82265.6Bearing 3-30.550.8327.127.8Bearing 3-40.470.9192.7795.5Bearing 3-50.140.8796.6105.2Mean0.450.81285.49302.80Both the CRA value and score values are greater than the bagging-DLSTM of the proposed method, as shown in Table8. The mean CRA value for bagging-DLSTM is 0.45, whereas it is 0.81 for the proposed method, demonstrating the superiority of the latter.In summary, this paper proposes a highly accurate and robust EDLSTM method to predict the RUL of bearings. This method accurately predicts the RUL of the training bearing samples and the RUL of the nontraining samples. Based on the outcomes of all bearings, it can be concluded that the method has strong stability in predicting the RUL of untrained bearing failure modes.


## 4. Conclusion

In this study, an EDLSTM method is proposed to predict the bearing RUL. First, SRMS is used to determine the degradation stage, and then RUL predictions are performed for the degradation stage. Multiple independent DLSTMs are used to build the predictive model. Each DLSTM comprises a feature extraction layer, an LSTM layer, and a fully connected layer. To ensure that the DLSTM has sufficient learning expression capability, all models are trained concurrently with data from various bearing failure modes. Then, the MIC is integrated for each individual DLSTM to calculate the MIC values of the predicted RUL features. Finally, based on the weight values, the optimal value of the predicted RUL for different DLSTMs is estimated to achieve the best prediction of the bearing RUL by the EDLSTM. This paper validated the proposed method experimentally using the XJTU-SY bearing dataset. The method has high prediction accuracy and robustness, with an average CRA of 81.6% for the RUL prediction of bearings in the untrained conditions, which is 44.2% better than the average CRA of 37.4% predicted by bagging-DLSTM. Compared to this method, the proposed method improved the CRA of all test set bearings by a minimum of 0.03% and a maximum of 73.0%. This demonstrates that this method can improve the predictability of bearing RUL and provide dedicated decision support for daily bearing maintenance.However, there is insufficient data to validate this method, and the accuracy of predicting the RUL of some bearings could be improved. We are currently unable to address this type of issue. This paper only demonstrates the applicability of the proposed method to the current data set. It does not provide evidence that the method is equally applicable when the study object is altered. To further validate the applicability of the proposed method, we will construct our accelerated bearing degradation testing apparatus employing various bearing types and increasing the number of bearing-specific data samples. We will increase the amount of data on the same bearing failures under the same operating conditions to test whether the predictive performance of the method can be improved. The predictive stability of the approach will be tested by performing the same accelerated degradation tests on replacement bearing types.



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*Source: 3742912-2023-03-20.xml*