# An Accurate Prediction Algorithm of RUL for Bearings: Time-Frequency Analysis Based on MRCNN

Authors: Dezhi Huang; Gang Yu; Jian Zhang; Jian Tang; Jian Su
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

Remaining useful life (RUL) prediction of mechanical components is of high research value in the field of prognostics and health management (PHM). However, RUL prediction problems are completely challenging due to the complicacy of bearings’ operating environment. In this paper, we transform the vibration acceleration signal collected by sensors into a time-frequency domain matrix through continuous wavelet transform (CWT) and then extract the features of the time-frequency domain matrix through the proposed multiscale residual convolutional neural network (MRCNN), which enables the model to extract more local and global features while constructing more accurate health indicators (HI). In order to highlight the degradation trend of mechanical components, the obtained health indicators are smoothed by exponential moving average (EMA). Finally, linear regression is exploited to predict the RUL of the bearing. Performance evaluations based on the public dataset PRONOSTIA demonstrate the effectiveness of our proposed algorithm, which is superior to existing data-driven algorithms in terms of prediction accuracy.


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## Body

## 1. Introduction

Bearing plays an irreplaceable role in the machinery industry, which always keeps running as the core component of the machinery, especially as a major equipment or a key basic component. However, maintaining their availability and reliability usually consumes significant resources, leading to increased enterprise costs. Consequently, the bearings’ PHM becomes increasingly important and attractive in the industrial field, where accurate prediction of RUL of bearings can improve production efficiency and maximize economic benefits. Therefore, RUL prediction is essential for determining bearing conditions and developing maintenance strategies [1–4]. Fortunately, due to the development of the Internet of Things, massive amounts of bearing operating data on PHM systems could be collected for further data processing to estimate bearing states. However, how to effectively extract features from the data and accurately predict the remaining useful life is still a considered challenge for PHM [5–9], due to the harsh industrial environment and complex fault causes.RUL prediction methods in bearings are mainly divided into two categories: model-based methods (also known as physics-based methods) and data-driven methods. Model-based methods accurately describe the degradation mechanism of the equipment by building mathematical-statistical models with a full understanding of the failure mechanism. Concurrently, the parameters of the model-based method can be estimated by the collected data [10]. However, due to the complex internal structure of the device and the distinct degradation mechanisms, it is impractical to rely solely on a large amount of a priori and expert knowledge to accurately predict the RUL [11–13]. Compared with model-based methods, data-driven methods do not require establishing complex mathematical models, which predict the degradation trend and the RUL of the equipment through using the data analysis method and finding out the intrinsic relationship of the degradation trend according to the historical data collected by the sensor. Furthermore, with the flourishment of the Internet of Things and the advent of the era of big data, massive amounts of useful data can be collected through monitoring sensors. In addition, owing to the advantages of low cost and high accuracy, data-driven methods are progressively becoming a current research hotspot. Generally, the data-driven prediction method consists of three steps: data collection, HI construction, and RUL prediction [14]. Because of its powerful feature extraction and prediction capabilities, deep learning is widely used in PHM HI construction to predict the RUL without physical domain knowledge. Gebraeel et al. [15] designed an artificial neural network data-driven model to predict the RUL of bearings, which is the initial successful attempt of deep learning in bearing RUL prediction. Zhao et al. [16] combined a convolutional neural network (CNN) with a bidirectional long short-term memory (Bi-LSTM) network to form a (CBLSTM) network to predict tool wear, where they utilized CNN to extract degraded features from the original signals as inputs for a Bi-LSTM network to construct HI. This improved deep learning model had good prediction accuracy; however, it also consumed more time. In [17], the Fourier transform for the original signal is exploited to obtain frequency information, and then, the deep neural network (DNN) based on stacked autoencoder (SAE) is employed for fault diagnosis of hydraulic pumps. Malhi et al. [18] adopted recurrent neural networks (RNN) to possess better prediction accuracy for long-term health condition prediction of machines. Peng et al. [19] proposed a Bayesian deep learning method to cope well with the RUL prediction uncertainty problem. From the relatively novel articles in recent years, data-driven methods have more significant advantages than model-based methods in terms of prediction accuracy, which is also a direction worthy of key research in the future.Due to massive amounts of noise, extracting effective degradation features is always a pivotal challenge for RUL prediction. Actually, the bearings have to keep operating in a harsh industrial environment and frequently aggravate complex faults. In this paper, the experimental data are employed from PRONOSTIA in IEEE PHM 2012 Data Challenge [20]. The bearing degradation experimental platform is shown in Figure 1.Figure 1
Bearing degradation test platform.The platform uses sensors to collect vibration acceleration signal information during bearing degradation, where two types of vibration acceleration signals are collected: horizontal vibration acceleration signals and vertical vibration acceleration signals. The raw vibration signals for the full life of Bearing1_1 and Bearing1_2 are shown in Figure2. The x-axis direction indicates the time in 10 s, and the y-axis direction indicates the amplitude. The two different types of degradation can be clearly seen in Figure 2. One of the Bearing1_1 is the slow degradation type, and its amplitude shows a gradual increase with time. Seen from Figure 2, the amplitude increases slowly and regularly from times 0 s to 27000 s. From 27000 s to 28030 s, the amplitude increases faster and the bearing is damaged finally when the bearing reaches the end of its life. Bearing1_2 illustrates the type of sudden degradation with massive amounts of noise influence, whose trend of increasing amplitude from time 0 s to 8300 s is not obvious. From 8300 s to 8710 s, the amplitude suddenly increases sharply, which implies that the bearing is damaged finally. Consequently, Figure 2 shows that the slow degradation type is favourable for predicting RUL with expectation based on the growth of amplitude, while it is challenging with sudden degradation type of amplitude.Figure 2
Raw vibration signal of Bearing1_1 and Bearing1_2.The slow degradation bearings have the characteristic of stationary signal type, and the bearing’s time-domain features can describe the corresponding variations well. However, many bearings are the nonstationary signal type, which forces researchers to find weak fault information in the background of strong noise. Fortunately, time-frequency analysis is an effective method for analyzing nonstationary and transient vibration signals. Consequently, a new deep learning method for RUL prediction based on continuous wavelet transform (CWT) and multiscale residual convolutional neural network (MRCNN) is proposed in this paper. Firstly, the vibration acceleration signals collected from the bearings are processed by continuous wavelet transform and converted into a time-frequency domain 2-D matrix. In order to cut down the input volume of the model, the size of the 2-D matrix is reduced by exploiting the nearest interpolation algorithm. Afterward, the resized matrix is imported into the MRCNN model for feature extraction, which possesses a multiscale hopping structure allowing the model to extract more local and global features. Consequently, the network of the MRCNN model is easier for constructing more reasonable health indicators by training the model with excellent performance. To highlight the degradation trend, the constructed health indicators are smoothed by utilizing the exponential moving average (EMA) algorithm. Finally, linear regression is exploited to predict the remaining useful life.The main contributions of this paper can be summarized as follows:(1)
The proposed MRCNN model especially constructs a multiscale hopping structure, which accurately predicts the RUL of the bearings(2)
The MRCNN model utilizes global average pooling instead of flatten layer to reduce the number of parameters, shorten the training time, and reduce the problem of overfitting(3)
The obtained HI is smoothed in the final stage by using an exponential moving average algorithm, which highlights the recessionary trend and makes the predicted RUL more accurateThe remainder of this paper is organized as follows: Section2 provides the details of the materials and methods of the MRCNN model. In Section 3, the specific experimental steps to verify the effectiveness of the MRCNN model using the PRONOSTIA dataset are introduced. The experimental results are compared with existing data-driven algorithms in terms of prediction accuracy. Finally, Section 4 concludes this paper.


## 2. Materials and Methods

### 2.1. Data Analysis

In this paper, the famous public dataset PRONOSTIA at the IEEE PHM 2012 Data Challenge is utilized to validate the accuracy of the MRCNN model while predicting bearing RUL. As shown in Figure1, the sensors on the platform collect vibration acceleration signals (horizontal vibration acceleration signals and vertical vibration acceleration signals) during bearing running. Consequently, the MRCNN model utilizes horizontal and vertical vibration acceleration signals datasets in this paper. The experimental platform tests a total of 17 bearings for degradation experiments, which are divided into three types according to the external forces and rotational speeds. Each type has two training datasets, including vibration signals with a complete life cycle, and the test set is truncated (the first part of the life cycle) vibration signals. Specifically, the first two types have the vibration signals of two bearings as the training dataset and five bearings as the test dataset (condition 1 and condition 2 in Table 1), and the third type has the vibration signals of two bearings as the training dataset and only one bearing as the test dataset (condition 3 in Table 1). Specifically, vibration signal data is collected by the sensor at 10 s intervals, with a collection duration of 0.1 s and a sampling frequency of 25600 Hz. Since this paper mainly studies the prediction of bearing degradation trends under normal use, the first working condition is the most eligible. Consequently, the experiment mainly utilizes the data under the first working condition. The other two working conditions exert too much external force and the unpredictable events of the bearings increase; thus, condition 2 and condition 3 can simulate the bearing degradation prediction under sudden events, which is challenging. The specific experimental data are shown in Table 1.Table 1
Experimental data specification.
Condition 1
Condition 2
Condition 3
Load (N)
4000
4200
5000
Speed (rpm)
1800
1650
1500
Training dataset
Bearing1_1Bearing1_2
Bearing2_1Bearing2_2
Bearing3_1Bearing3_2
Testing dataset
Bearing1_3Bearing1_4Bearing1_5Bearing1_6Bearing1_7
Bearing2_3Bearing2_4Bearing2_5Bearing2_6Bearing2_7
Bearing3_3Through analyzing the vibration signals of Bearing1_1 and Bearing1_2, Figure2 illustrates that accurately predicting the RUL of the bearings is unrealistic only by the amplitude of the vibration signal. Fortunately, the CWT is an effective signal processing method, which extracts time-frequency features by transforming the original time-domain vibration signal into a 2-D time-frequency matrix. The CWT algorithm can effectively demonstrate the different changes in the frequency domain as time grows. For a stationary signal, its time-domain characteristics can well describe its relative variation. As seen from Figure 2 (Bearing1_2), the vibration signals of the bearings are nonstationary and have a weak fault signal in the strong noise background. Time-frequency analysis is an effective method for analyzing nonstationary and transient vibration signals. This paper utilizes the CWT to generate the time-frequency matrix as the dataset for training, while other papers use time-frequency images as the dataset, but this operation will encounter two problems of uniform colour bars and loss of accuracy. Consequently, as shown in Figures 3 and 4, time-frequency diagrams are only used as a demonstration of time-frequency characteristics. Figure 3 shows the waveform and time-frequency images of the first and last samples of Bearing1_1. According to the increase of the high-frequency component in the time-frequency plot, the colour change becomes increasingly obvious and the degradation characteristics are clearly demonstrated. In order to show the degradation trend of bearings completely, Figure 4 shows the time-frequency diagrams of Bearing1_1 and Bearing1_2 with the display step of 25 percent of the complete life. Obviously, the bearing failure becomes more serious, and the colour change on the time-frequency diagram becomes more obvious with time. This shows that time-frequency analysis can well expose the fault characteristics of bearings.Figure 3
Bearing1_1: (a) waveform for the first sample; (b) time-frequency image for the first sample; (c) waveform for the last sample; and (d) time-frequency diagram for the last sample.
(a)
(b)
(c)
(d)Figure 4
Bearing1_1 and Bearing1_2 display time-frequency images every 25% in the whole life.


### 2.2. Proposed Framework

Model-based methods and common data-driven methods have the drawback of low accuracy in predicting RUL. Consequently, this paper proposes a more accurate RUL prediction method based on CWT and MRCNN models. The framework diagram of the proposed method is shown in Figure5 as follows:Figure 5
Framework of the proposed method.In this paper, the proposed prediction algorithm is generally divided into three stages: the first stage is feature extraction, where the vibration signal is converted into a 2-D time-frequency matrix by CWT to extract time-frequency features. Concurrently, the training dataset and test dataset are obtained by extracting the time-frequency features utilizing the CWT algorithm. In order to cut down the input volume of the model, the size of the 2-D matrix is reduced by utilizing the nearest interpolation algorithm. The second stage is the offline health indicators construction, where the resized time-frequency matrix is imported into MRCNN. Since CNN is a supervised learning method, the design of its labels largely affects its RUL prediction results. This paper assumes that the bearing degradation trend is committed to a linear trend, thus setting the labels to decrease linearly from 1 to 0 to learn each stage’s degradation characteristics. HIs are structured by importing the test dataset into the trained MRCNN model. Finally, the third stage is online RUL prediction. To highlight the degradation trend, the structured HIs are smoothed by an exponential moving average algorithm. The RUL was finally obtained by linear regression. The main techniques are detailed described below.


### 2.3. Continuous Wavelet Transform

With the development of signal analysis techniques, different time-frequency transform techniques have emerged accordingly. Fourier transform, as the most basic time-frequency transform method, can transform the signal from the time domain to the frequency domain. Consequently, the Fourier transform can extract much information about the essence of the signal. However, the Fourier transform has the shortcoming in reflecting the trend of frequency variation over time. The windowed Fourier transform comes out accordingly, which enables us to observe the frequency components on different time domains by choosing the corresponding windows. But the fixed-size window makes us feel inconvenient. Consequently, the continuous wavelet transform was invented to solve this problem. The continuous wavelet transform replaces the infinite length trigonometric basis with a finite length decaying wavelet basis, which makes us intuitively feel the changes of time-frequency characteristics [21]. The vibration signal of the faulty bearing contains periodic pulses whose shape is similar to Morlet wavelets; thus, the experiment in this paper utilizes the Morlet wavelet basis function to transform the original vibration signal. The Morlet wavelet basis function is shown as follows:
(1)ψt=expiω0texp−t22,where expiω0t is called the complex trigonometric function whose role is to identify the frequency and exp−t2/2 is called the decay function, which can guarantee finite support in its time domain. In order to get different frequency signals, the original signals are multiplied by different scales of wavelet basis functions ψt. Consequently, the different frequency signals can be obtained only by stretching the wavelet basis function. The continuous wavelet transform formula is as follows:
(2)Uα,β=xt,ψα,β=∫−∞∞xtψ¯α,βt=∫−∞∞t1αψ¯t−βαdt,where xt is the input original 1-D signal and ψt is the mother wavelet function, ψ¯α,βt is the complex conjugate function of ψα,βt, and the coefficients α,β, respectively, represent the scale coefficients (frequency variables) and translation coefficients (time variables). By setting the input 1-D signal xt and the mother wavelet function ψt, the time-frequency coefficient matrix can be obtained according to the above integral (formula (2)). Consequently, the CWT can identify a specific frequency β under a specific time α.


### 2.4. Multiscale Residual Convolutional Neural Network

CNN is a deep learning model inspired by the visual cortex of the brain [22], which extracts the location information of the original image by convolutional operations. Since the perceptual fields of convolution kernels with different sizes on the input image are different, the learned features are also different. Consequently, the multiscale convolution structure contributes to the network learning more local and global features, which has a good application in the field of RUL prediction [23]. CNN mainly consists of three structures: convolutional layer, pooling layer, and fully connected layer. Deep CNN achieves stronger data representation capability by stacking convolution and pooling layers, whose end generally adopts a fully connected layer for classification and regression. Convolution kernels, also known as weights, are slid over the input image by performing convolution operations on each corresponding local perceptual field. The weight sharing greatly reduces the number of parameters and avoids the overfitting problem. The input image is convolved with a convolution layer to obtain the feature map, which is a feature representation of the image, and its calculation formula is as follows:
(3)xjt=φ∑ixit−1∗kijt+bjt,where xjt is the feature map of the t-th layer and the j-th output, kijt is the convolution kernel of the t-th layer, bjt is the bias of the t-th layer, and φ is the nonlinear activation function. This means that all feature maps in layer t−1 do the convolution operation with the convolution kernel in layer t, and the convolution result is summed with the bias bjt. Finally, the feature map in layer t is obtained by the nonlinear activation function.A pooling layer is usually added between adjacent convolutional layers. The pooling operation can reduce the size of the parameter matrix, simplify the complexity of the network, speed up the computation, and prevent overfitting. The output feature map of the pooling layer has the same variance as the input feature map. As shown in Figure6, MRCNN model utilizes two pooling methods: maximum pooling and global average pooling. Maximum pooling takes the maximum value of the data within the perceptual field to achieve the effect of subsampling. The global average pooling layer takes the average of the entire feature map. In the MRCNN model, global average pooling is utilized instead of flatten layers to reduce the number of parameters, shorten the training time, and reduce the occurrence of overfitting problems.Figure 6
MRCNN structure.The first two layers of the MRCNN model are ordinary convolutional layers, which consist of convolution, BN (batch normalization), Relu activation function, and max pooling. The Relu activation function is formulated asRelux=max0,x, which serves to enable the network to perform nonlinear operations and to keep the gradient from decaying when the input x is greater than 0, alleviating the vanishing gradient problem. Since BN operation can maintain the input data in the activation function sensitive region (mean γ and variance β learned by BN), speed up the convergence speed during model training, and make the model training process more stable. Consequently, adding the BN operation before the activation function is necessary. The BN formula is as follows:
(4)yib=BNxib=γ⋅xib−μxiσxi2+E+β,where xi is a row vector of size one batch, which represents the value of the b-th input node of the layer at the i-th sample in the current batch. Where μ and σ, respectively, represent the mean and variance of the row, and a minimal amount E is introduced to prevent the denominator from being zero. The purpose of the BN operation is to obtain a data distribution that makes convergence faster during training yib. The training learning parameters γ and β, respectively, represent the mean and variance of the data after BN operation.In the MRCNN model, from the third layer to the sixth layer is the multiscale residual layer composed of four multiscale residual blocks. The specific multiscale residual block structure is shown in Figure7. The feature map of the upper layer is input into the multiscale residual block, which experiences four structures: the first structure has a convolution layer with a kernel size of 1×1, the second structure has two convolution layers with kernel sizes of 3×3 and 1×1, respectively, the third structure has both convolution and pooling layers, in which the convolution kernel size is 1×1, and the pooling layer kernel size is 3×3. The fourth structure has three convolutional layers, whose first two layers have a kernel size of 3×3 and whose last layer has a kernel size of 1×1. The purpose of adopting the above structure is to utilize multiscale convolution kernels with different perceptual fields on the upper feature map, which can extract a variety of different features. Since the convolutional layer with a kernel size of 3×3 has a perceptual field of 9 on the previous feature map, the superposition of two layers with a kernel size of 3×3 has a perceptual field of 25 on the previous feature map at a step size of 1. Consequently, the superposition of two layers with a kernel size of 3×3 can replace a convolutional layer with a kernel size of 5×5, and more information can be learned because of the deeper network. The maximum pooling layer can learn the edge and texture structure of the image. The 1×1 convolutional layer in the multiscale residual block can be seen as a fully connected layer, which can serve to change the number of output channels. Eventually, these four structures are stitched together. The residual structure of the multiscale residual block is learned from the ResNet [23] network, which solves the network degradation and deep network model vanishing gradient and exploding gradient problems. Since the residual structure learns the difference between the target value and the input value when training to the optimal model, the structure will train the redundant layers into the identity function. Therefore, when the number of network layers deepens, its accuracy will not decline. In the fifth layer of the MRCNN model, due to the change in the number of channels, the residual structure is represented by a dashed line. The multiscale convolution block is activated by Relu nonlinear unit. Finally, the MRCNN model adopts the full connection layer and sigmoid activation function to distribute the output HI in the region of 0 to 1.Figure 7
Multiscale residual block.


### 2.5. Exponential Moving Average

The HI generated by the MRCNN model is smoothed by exploiting an exponential moving average algorithm. Although the HI shows a clear degradation trend from the overall, there will exist some HI deviation from the overall trend due to the interference of some strong noise. Consequently, in this paper, HI is discrete in narrowing the specified period. The exponential moving average algorithm is utilized to perform the smoothing operation on HI. The exponential moving average equation is presented as follows:(5)vt=βvt−1+1−βθt,where vt denotes the current predicted value, vt−1 is the previous moment predicted value, θt refers to the current HI, and β is the decay factor. According to Equation (5), the current prediction value vt is obtained by weighting the previous prediction value vt−1 with the current θt. Therefore, Equation (5) can be expanded as follows:
(6)vt=1−βθt+β1−βθt−1+β21−βθt−2+⋯βt−11−βθ1.As seen from Equation (6), the current prediction value vt is associated with the values of all previous moments, where the last predicted value vt is obtained by a process called sliding backward from the first value θ1 to the last value θt. As a result, the relationship between the current predicted value vt and the previous value θn shows βn1−β exponential decay. Ultimately, the exponential moving average algorithm makes the recession trend more visible and enables a more accurate prediction of RUL.


### 2.6. Linear Regression Prediction

The final step of RUL prediction is a linear regression operation on the smoothed health indicators. Since the label design of MRCNN model is that the bearing decay follows a linear change, the label design is to reduce linearly from 1 to 0 to learn the degradation characteristics of each stage. Consequently, for accurate prediction of RUL, a linear regression prediction in the last step is essential. Formally, the linear equationy=at+b is exploited to fit the last 1/3 HI values of the life cycle of the bearings, and the final RUL is obtained by extension. Referring to the ordinary least squares, the slope a and the intercept b can be obtained by the following two formulas [24].
(7)a=∑i=0Ntiyi−N+1t¯y¯∑i=0Nti2−N+1t¯2,b=y¯−at¯,where ti and yi represent the time of i-th and its corresponding predicted HI. The final predicted RUL RUL^ can be obtained by intercepting b over the upper slope a and subtracting the truncated time point tN, which is shown as follows:
(8)RUL^=ba−tN.


## 2.1. Data Analysis

In this paper, the famous public dataset PRONOSTIA at the IEEE PHM 2012 Data Challenge is utilized to validate the accuracy of the MRCNN model while predicting bearing RUL. As shown in Figure1, the sensors on the platform collect vibration acceleration signals (horizontal vibration acceleration signals and vertical vibration acceleration signals) during bearing running. Consequently, the MRCNN model utilizes horizontal and vertical vibration acceleration signals datasets in this paper. The experimental platform tests a total of 17 bearings for degradation experiments, which are divided into three types according to the external forces and rotational speeds. Each type has two training datasets, including vibration signals with a complete life cycle, and the test set is truncated (the first part of the life cycle) vibration signals. Specifically, the first two types have the vibration signals of two bearings as the training dataset and five bearings as the test dataset (condition 1 and condition 2 in Table 1), and the third type has the vibration signals of two bearings as the training dataset and only one bearing as the test dataset (condition 3 in Table 1). Specifically, vibration signal data is collected by the sensor at 10 s intervals, with a collection duration of 0.1 s and a sampling frequency of 25600 Hz. Since this paper mainly studies the prediction of bearing degradation trends under normal use, the first working condition is the most eligible. Consequently, the experiment mainly utilizes the data under the first working condition. The other two working conditions exert too much external force and the unpredictable events of the bearings increase; thus, condition 2 and condition 3 can simulate the bearing degradation prediction under sudden events, which is challenging. The specific experimental data are shown in Table 1.Table 1
Experimental data specification.
Condition 1
Condition 2
Condition 3
Load (N)
4000
4200
5000
Speed (rpm)
1800
1650
1500
Training dataset
Bearing1_1Bearing1_2
Bearing2_1Bearing2_2
Bearing3_1Bearing3_2
Testing dataset
Bearing1_3Bearing1_4Bearing1_5Bearing1_6Bearing1_7
Bearing2_3Bearing2_4Bearing2_5Bearing2_6Bearing2_7
Bearing3_3Through analyzing the vibration signals of Bearing1_1 and Bearing1_2, Figure2 illustrates that accurately predicting the RUL of the bearings is unrealistic only by the amplitude of the vibration signal. Fortunately, the CWT is an effective signal processing method, which extracts time-frequency features by transforming the original time-domain vibration signal into a 2-D time-frequency matrix. The CWT algorithm can effectively demonstrate the different changes in the frequency domain as time grows. For a stationary signal, its time-domain characteristics can well describe its relative variation. As seen from Figure 2 (Bearing1_2), the vibration signals of the bearings are nonstationary and have a weak fault signal in the strong noise background. Time-frequency analysis is an effective method for analyzing nonstationary and transient vibration signals. This paper utilizes the CWT to generate the time-frequency matrix as the dataset for training, while other papers use time-frequency images as the dataset, but this operation will encounter two problems of uniform colour bars and loss of accuracy. Consequently, as shown in Figures 3 and 4, time-frequency diagrams are only used as a demonstration of time-frequency characteristics. Figure 3 shows the waveform and time-frequency images of the first and last samples of Bearing1_1. According to the increase of the high-frequency component in the time-frequency plot, the colour change becomes increasingly obvious and the degradation characteristics are clearly demonstrated. In order to show the degradation trend of bearings completely, Figure 4 shows the time-frequency diagrams of Bearing1_1 and Bearing1_2 with the display step of 25 percent of the complete life. Obviously, the bearing failure becomes more serious, and the colour change on the time-frequency diagram becomes more obvious with time. This shows that time-frequency analysis can well expose the fault characteristics of bearings.Figure 3
Bearing1_1: (a) waveform for the first sample; (b) time-frequency image for the first sample; (c) waveform for the last sample; and (d) time-frequency diagram for the last sample.
(a)
(b)
(c)
(d)Figure 4
Bearing1_1 and Bearing1_2 display time-frequency images every 25% in the whole life.


## 2.2. Proposed Framework

Model-based methods and common data-driven methods have the drawback of low accuracy in predicting RUL. Consequently, this paper proposes a more accurate RUL prediction method based on CWT and MRCNN models. The framework diagram of the proposed method is shown in Figure5 as follows:Figure 5
Framework of the proposed method.In this paper, the proposed prediction algorithm is generally divided into three stages: the first stage is feature extraction, where the vibration signal is converted into a 2-D time-frequency matrix by CWT to extract time-frequency features. Concurrently, the training dataset and test dataset are obtained by extracting the time-frequency features utilizing the CWT algorithm. In order to cut down the input volume of the model, the size of the 2-D matrix is reduced by utilizing the nearest interpolation algorithm. The second stage is the offline health indicators construction, where the resized time-frequency matrix is imported into MRCNN. Since CNN is a supervised learning method, the design of its labels largely affects its RUL prediction results. This paper assumes that the bearing degradation trend is committed to a linear trend, thus setting the labels to decrease linearly from 1 to 0 to learn each stage’s degradation characteristics. HIs are structured by importing the test dataset into the trained MRCNN model. Finally, the third stage is online RUL prediction. To highlight the degradation trend, the structured HIs are smoothed by an exponential moving average algorithm. The RUL was finally obtained by linear regression. The main techniques are detailed described below.


## 2.3. Continuous Wavelet Transform

With the development of signal analysis techniques, different time-frequency transform techniques have emerged accordingly. Fourier transform, as the most basic time-frequency transform method, can transform the signal from the time domain to the frequency domain. Consequently, the Fourier transform can extract much information about the essence of the signal. However, the Fourier transform has the shortcoming in reflecting the trend of frequency variation over time. The windowed Fourier transform comes out accordingly, which enables us to observe the frequency components on different time domains by choosing the corresponding windows. But the fixed-size window makes us feel inconvenient. Consequently, the continuous wavelet transform was invented to solve this problem. The continuous wavelet transform replaces the infinite length trigonometric basis with a finite length decaying wavelet basis, which makes us intuitively feel the changes of time-frequency characteristics [21]. The vibration signal of the faulty bearing contains periodic pulses whose shape is similar to Morlet wavelets; thus, the experiment in this paper utilizes the Morlet wavelet basis function to transform the original vibration signal. The Morlet wavelet basis function is shown as follows:
(1)ψt=expiω0texp−t22,where expiω0t is called the complex trigonometric function whose role is to identify the frequency and exp−t2/2 is called the decay function, which can guarantee finite support in its time domain. In order to get different frequency signals, the original signals are multiplied by different scales of wavelet basis functions ψt. Consequently, the different frequency signals can be obtained only by stretching the wavelet basis function. The continuous wavelet transform formula is as follows:
(2)Uα,β=xt,ψα,β=∫−∞∞xtψ¯α,βt=∫−∞∞t1αψ¯t−βαdt,where xt is the input original 1-D signal and ψt is the mother wavelet function, ψ¯α,βt is the complex conjugate function of ψα,βt, and the coefficients α,β, respectively, represent the scale coefficients (frequency variables) and translation coefficients (time variables). By setting the input 1-D signal xt and the mother wavelet function ψt, the time-frequency coefficient matrix can be obtained according to the above integral (formula (2)). Consequently, the CWT can identify a specific frequency β under a specific time α.


## 2.4. Multiscale Residual Convolutional Neural Network

CNN is a deep learning model inspired by the visual cortex of the brain [22], which extracts the location information of the original image by convolutional operations. Since the perceptual fields of convolution kernels with different sizes on the input image are different, the learned features are also different. Consequently, the multiscale convolution structure contributes to the network learning more local and global features, which has a good application in the field of RUL prediction [23]. CNN mainly consists of three structures: convolutional layer, pooling layer, and fully connected layer. Deep CNN achieves stronger data representation capability by stacking convolution and pooling layers, whose end generally adopts a fully connected layer for classification and regression. Convolution kernels, also known as weights, are slid over the input image by performing convolution operations on each corresponding local perceptual field. The weight sharing greatly reduces the number of parameters and avoids the overfitting problem. The input image is convolved with a convolution layer to obtain the feature map, which is a feature representation of the image, and its calculation formula is as follows:
(3)xjt=φ∑ixit−1∗kijt+bjt,where xjt is the feature map of the t-th layer and the j-th output, kijt is the convolution kernel of the t-th layer, bjt is the bias of the t-th layer, and φ is the nonlinear activation function. This means that all feature maps in layer t−1 do the convolution operation with the convolution kernel in layer t, and the convolution result is summed with the bias bjt. Finally, the feature map in layer t is obtained by the nonlinear activation function.A pooling layer is usually added between adjacent convolutional layers. The pooling operation can reduce the size of the parameter matrix, simplify the complexity of the network, speed up the computation, and prevent overfitting. The output feature map of the pooling layer has the same variance as the input feature map. As shown in Figure6, MRCNN model utilizes two pooling methods: maximum pooling and global average pooling. Maximum pooling takes the maximum value of the data within the perceptual field to achieve the effect of subsampling. The global average pooling layer takes the average of the entire feature map. In the MRCNN model, global average pooling is utilized instead of flatten layers to reduce the number of parameters, shorten the training time, and reduce the occurrence of overfitting problems.Figure 6
MRCNN structure.The first two layers of the MRCNN model are ordinary convolutional layers, which consist of convolution, BN (batch normalization), Relu activation function, and max pooling. The Relu activation function is formulated asRelux=max0,x, which serves to enable the network to perform nonlinear operations and to keep the gradient from decaying when the input x is greater than 0, alleviating the vanishing gradient problem. Since BN operation can maintain the input data in the activation function sensitive region (mean γ and variance β learned by BN), speed up the convergence speed during model training, and make the model training process more stable. Consequently, adding the BN operation before the activation function is necessary. The BN formula is as follows:
(4)yib=BNxib=γ⋅xib−μxiσxi2+E+β,where xi is a row vector of size one batch, which represents the value of the b-th input node of the layer at the i-th sample in the current batch. Where μ and σ, respectively, represent the mean and variance of the row, and a minimal amount E is introduced to prevent the denominator from being zero. The purpose of the BN operation is to obtain a data distribution that makes convergence faster during training yib. The training learning parameters γ and β, respectively, represent the mean and variance of the data after BN operation.In the MRCNN model, from the third layer to the sixth layer is the multiscale residual layer composed of four multiscale residual blocks. The specific multiscale residual block structure is shown in Figure7. The feature map of the upper layer is input into the multiscale residual block, which experiences four structures: the first structure has a convolution layer with a kernel size of 1×1, the second structure has two convolution layers with kernel sizes of 3×3 and 1×1, respectively, the third structure has both convolution and pooling layers, in which the convolution kernel size is 1×1, and the pooling layer kernel size is 3×3. The fourth structure has three convolutional layers, whose first two layers have a kernel size of 3×3 and whose last layer has a kernel size of 1×1. The purpose of adopting the above structure is to utilize multiscale convolution kernels with different perceptual fields on the upper feature map, which can extract a variety of different features. Since the convolutional layer with a kernel size of 3×3 has a perceptual field of 9 on the previous feature map, the superposition of two layers with a kernel size of 3×3 has a perceptual field of 25 on the previous feature map at a step size of 1. Consequently, the superposition of two layers with a kernel size of 3×3 can replace a convolutional layer with a kernel size of 5×5, and more information can be learned because of the deeper network. The maximum pooling layer can learn the edge and texture structure of the image. The 1×1 convolutional layer in the multiscale residual block can be seen as a fully connected layer, which can serve to change the number of output channels. Eventually, these four structures are stitched together. The residual structure of the multiscale residual block is learned from the ResNet [23] network, which solves the network degradation and deep network model vanishing gradient and exploding gradient problems. Since the residual structure learns the difference between the target value and the input value when training to the optimal model, the structure will train the redundant layers into the identity function. Therefore, when the number of network layers deepens, its accuracy will not decline. In the fifth layer of the MRCNN model, due to the change in the number of channels, the residual structure is represented by a dashed line. The multiscale convolution block is activated by Relu nonlinear unit. Finally, the MRCNN model adopts the full connection layer and sigmoid activation function to distribute the output HI in the region of 0 to 1.Figure 7
Multiscale residual block.


## 2.5. Exponential Moving Average

The HI generated by the MRCNN model is smoothed by exploiting an exponential moving average algorithm. Although the HI shows a clear degradation trend from the overall, there will exist some HI deviation from the overall trend due to the interference of some strong noise. Consequently, in this paper, HI is discrete in narrowing the specified period. The exponential moving average algorithm is utilized to perform the smoothing operation on HI. The exponential moving average equation is presented as follows:(5)vt=βvt−1+1−βθt,where vt denotes the current predicted value, vt−1 is the previous moment predicted value, θt refers to the current HI, and β is the decay factor. According to Equation (5), the current prediction value vt is obtained by weighting the previous prediction value vt−1 with the current θt. Therefore, Equation (5) can be expanded as follows:
(6)vt=1−βθt+β1−βθt−1+β21−βθt−2+⋯βt−11−βθ1.As seen from Equation (6), the current prediction value vt is associated with the values of all previous moments, where the last predicted value vt is obtained by a process called sliding backward from the first value θ1 to the last value θt. As a result, the relationship between the current predicted value vt and the previous value θn shows βn1−β exponential decay. Ultimately, the exponential moving average algorithm makes the recession trend more visible and enables a more accurate prediction of RUL.


## 2.6. Linear Regression Prediction

The final step of RUL prediction is a linear regression operation on the smoothed health indicators. Since the label design of MRCNN model is that the bearing decay follows a linear change, the label design is to reduce linearly from 1 to 0 to learn the degradation characteristics of each stage. Consequently, for accurate prediction of RUL, a linear regression prediction in the last step is essential. Formally, the linear equationy=at+b is exploited to fit the last 1/3 HI values of the life cycle of the bearings, and the final RUL is obtained by extension. Referring to the ordinary least squares, the slope a and the intercept b can be obtained by the following two formulas [24].
(7)a=∑i=0Ntiyi−N+1t¯y¯∑i=0Nti2−N+1t¯2,b=y¯−at¯,where ti and yi represent the time of i-th and its corresponding predicted HI. The final predicted RUL RUL^ can be obtained by intercepting b over the upper slope a and subtracting the truncated time point tN, which is shown as follows:
(8)RUL^=ba−tN.


## 3. Results and Discussion

### 3.1. MRCNN Model Construction HI

In this section, the structure of the MRCNN model and the experimental procedure of constructing HI is described in detail. Firstly, the original vibration signals are converted into a 2-D matrix of size2560×2560 by CWT. In order to cut down the input volume of the model, the size of the 2-D matrix is reduced to 224×224 by utilizing the nearest interpolation algorithm, which greatly saves massive amounts of training time. The PRONOSTIA public dataset collects vibration signals in horizontal and vertical directions. Consequently, the input of MRCNN model is set as two channel numbers, which, respectively, correspond to the time-frequency matrices in these two directions. Afterward, the first two layers of MRCNN model are ordinary convolution layers, which utilize the structures of convolution, BN, Relu, and maximum pool. The convolution kernel size of the first layer is 7×7 and the step size is 1. The first layer convolution operation will not change the size of the 2-D matrix. The 2-D matrix output from the first layer of MRCNN model is halved to the size of 112×112, and the number of output channels is 64. The second convolutional layer uses a similar structure, doubling the number of output channels to 128×128 and halving the 2-D matrix size to 56×56. The difference is that the convolution operation has a convolution kernel of 3×3, stride of 1 and padding of 1. Next is the multiscale residual layer, which is composed of four multiscale residual blocks, each of which has four multiscale feature mapping operations. Specifically, the first operation is 1×1 convolution to extract feature map. The second is 1×1 convolution plus 3×3 convolution, and the output ratio is 1 : 2. The third is max pooling plus 1×1 convolution, and the output ratio is 1 : 1. The fourth is 1×1 convolution plus two 3×3 convolutions, and the output ratio is 1 : 2 : 2. Overall, the output ratio of these four channels is 2 : 4 : 1 : 1. The above parameters are obtained through massive amounts of experimental tuning parameters, which have strong applicability. In particular, MRCNN uses the residual hopping structure from layer 3 to layer 6 but further doubles the number of channels from layer 4 to layer 5 to 256×256. Therefore, the residual hopping structure here is first realized by changing the number of channels through 1×1 convolution and then hopping. The seventh layer of MRCNN is a normal convolutional layer with a convolutional kernel size of 3×3, a stride of 1, a padding of 1, and an output channel of 64. The eighth layer of MRCNN is the global averaging pooling layer, which is implemented by taking the average of a 56×56 size 2-D matrix and outputting 64 parameters. The ninth to eleventh layers of MRCNN are full connection layers, and the number of neurons in each layer is 128, 64, and 1, respectively. Finally, the MRCNN model obtains HIs through sigmoid activation function, which makes the distribution of HIs from 0 to 1.After CWT algorithm transformation, the training datasets from PRONOSTIA public dataset are successively imported into MRCNN model for training, because an obvious advantage of data-driven methods is that it does not require massive amounts of a priori knowledge. The label is set to decrease linearly from 1 to 0. In this paper, the full life of the bearing is recorded asTa and the current operating time is recorded as Tc. The true label of HI can be obtained by the following formula:
(9)label=1−TcTa.The HI represents the health of bearings, 1 indicates complete health, and 0 indicates damage and no longer usable. The loss function of the MRCNN model utilizes mean square error (MSE), which is widely adopted as the loss function of regression model. MSE is shown as follows:
(10)Ly^;y=12y∧−y2,where y^ is the predicted value of the MRCNN model and y is the true value. The MRCNN model adjusts the parameters by back propagation and gradient descent to reduce the loss function and make the target value closer to the true value. Since the optimizer utilizes stochastic gradient descent (SGD), which approximates the gradient on the loss function of the entire dataset by using the gradient on the loss function of small batch data, therefore, it is easier to adjust the direction of gradient descent and achieve global optimization in the training process. The learning rate is adjusted by the exponential decay method, whose initial learning rate is 0.1 and the multiplication factor is set to 0.95. Dynamically, decaying the learning rate can quickly reduce the loss in the early iteration stage and slightly reduce the loss in the later stage. The learning rate decay is shown as follows:
(11)lr=lr⋅0.95n,where lr denotes the learning rate and n is the number of iterative rounds. The training dataset includes Bearing1_1 and Bearing1_2, and the test dataset includes Bearing1_3 to Bearing1_7. In this paper, a total of 100 training rounds were performed. Figure 8(a) illustrates that the learning rate decays exponentially with the increase of round, and the decline law of the loss function is the same as expected. In the early stage, the loss is rapidly reduced, and in the later stage, the loss is slightly reduced. However, as shown in Figure 8(b), when the 5th to 10th rounds of training are performed, the loss function on the training and test dataset increases instead, which is caused by the excessive learning rate at that time. By utilizing the decay operation of the learning rate, the loss can be correctly reduced in the subsequent rounds. In addition, the MRCNN model adopts global average pooling before the fully connected layer to reduce the problem of overfitting.Figure 8
(a) Learning rate. (b) Training loss and testing loss.
(a)
(b)The trained MRCNN model is used to test Bearing1_3. The GPU model used in the experimental platform is GTX1080ti, and the MRCNN model is implemented based on the deep learning framework Pytorch. The truncation time point of test dataset Bearing1_3 is 18010 s, and it spends 6.4 s for MRCNN model to construct HIs. The MRCNN model constructs HIs on the test dataset Bearing1_3, whose results are shown in the blue curve in Figure9. The degradation trend of bearings can be clearly seen from HI in Figure 9. Firstly, in the range from 0 s to 7500 s, the bearing degradation is slow, and the bearing can operate stably in the early stage of its service life. Afterward, the degradation trend suddenly becomes faster at 7500 s, 10500 s, 13000 s, 15700 s, and 17500 s. Due to the influence of external force, the degradation of bearings gradually accelerates until it is finally damaged. Although the HIs shows a clear degradation trend from the overall, there will exist some HI deviation from the overall trend due to the interference of some strong noise. Consequently, as shown in Figure 9, HI is discrete in narrowing the specified period.Figure 9
RUL prediction result of Bearing1_3.


### 3.2. RUL Prediction Results

After obtaining the HI through the MRCNN model, the next step is to perform RUL prediction based on the HIs. Since the obtained HI distribution is more discrete, this problem is solved by fitting HIs with an exponential sliding average algorithm. According to formulas (5) and (6), the current predicted value is associated with the values of all previous moments. The last predicted value is obtained by a process like sliding backwards in sequence from the first value to the last value, doing exponential decay according to the decay factor. The decay factor in the experiment takes the value of 0.95. Consequently, the exponential sliding average algorithm makes the decline trend more obvious and enables more accurate prediction of the RUL, as shown in the orange curve in Figure 9. In this paper, through massive amounts of experiments, the following operations are the most accurate prediction of RUL: take the last 30% of the smooth data and expand it through linear fitting. As shown by the green line in Figure 9, the predicted life can be obtained by intersecting with the line whose ordinate is equal to zero. Consequently, the RUL of bearing can be obtained by subtracting the current time from the predicted life. The results of RUL prediction from Bearing1_4 to Bearing1_7 are shown in Figure 10. Specifically, Bearing1_4 predicts a full life of 14740 s, Bearing1_5 predicts a full life of 24980 s, Bearing1_6 predicts a full life of 24300 s, and Bearing1_7 predicts a full life of 21880 s.Figure 10
RUL prediction result of Bearing1_3 to Bearing1_7.Regarding the performance evaluation metrics of RUL prediction, this paper adopts the method given in the IEEE PHM 2012 Prognostic Challenge: error percentage, evaluation metricAi, and mean value of evaluation metric Ai. The error percentage is shown in
(12)Eri=100×ActRULi−RU^LiActRULi,where ActRULi represents the true RUL of the i-th bearing and RU^Li represents the predicted RUL of the i-th bearing, since the error percentage Er represents the deviation of the predicted RUL from the true RUL. Consequently, the closer the predicted RUL is to the true RUL, the closer the error percentage is to 0. In addition, the evaluation index is shown in
(13)Aiexp−ln0.5⋅Eri/5ifEri≤0,expln0.5⋅Eri/20ifEri>0.The evaluation metricAi represents a measure of RUL predictive performance. If the Eri is 0, Ai is equal to 1. As the absolute value of Eri increases, the Ai becomes larger. However, the cost of overpredicting and underpredicting the RUL is different. In the field of RUL prediction, the cost of underestimating RUL is lower than overestimating RUL. Equation (13) captures this well. The final score is shown in Equation (14), which represents the mean value of the evaluation metrics.
(14)Score=1n∑i=1nAi.To verify the superiority of the method proposed in this paper, other data-driven methods using the same dataset are compared. The literature [24] is the champion model of IEEE 2012 Data Challenge. Lei et al. [5] proposed WMQE model, which has good prediction performance by fusing several weighted features to predict RUL through correlation clustering between 28 features of bearings. Chen Y et al. [25] extracted five bandpass energy values of the spectrum as features to construct a network based on code and decoding framework and attention mechanism for network prediction of RUL. Attention model based on signal decomposition brings us a new idea. Table 2 writes the final scores for each bearing current time, actual RUL, predicted RUL by this method, and the percentage of error compared to the other three methods. In addition, the absolute mean of the error percentage Er¯ is added in this paper, which judges the merits of the method by comparing the dispersion degree of the RUL prediction error. As seen from Table 2, Er¯ is 12.81 and the final score is 0.47. Both in terms of prediction accuracy and dispersion of prediction, the method in this paper is better than the other three methods.Table 2
Comparison results of this research and related researches in the PRONOSTIA dataset.
Testing dataset
Current time (s)
Actual RUL (s)
Predicted RUL (s)
Er
1
Er
2[24]
Er
3[5]
Er
4[25]
Bearing1_3
18010
5730
5470
4.53
37
-0.35
7.62
Bearing1_4
11380
2890
3360
-16.26
80
5.60
-157.71
Bearing1_5
23010
1610
1970
-22.36
9
100.00
-72.57
Bearing1_6
23010
1460
1290
11.64
-5
28.08
0.93
Bearing1_7
15010
7570
6870
9.24
-2
19.55
85.99
Er
¯
12.81
26.6
30.71
64.96
Score
0.47
0.46
0.45
0.35


### 3.3. Discussion

The effectiveness and advantage of the proposed MRCNN method have been verified by experimental data and point out the shortcomings of the MRCNN model and the direction of improvement in the future.(1)
The MRCNN model adopts the time-frequency matrix transformed from the vibration signal, which saves the training time by reducing its size. However, reducing the size of features will lose a lot of useful information. Therefore, it is a very important problem to achieve a balance between the cost of time and the input of enough features(2)
The structure of MRCNN model design is complex, and the optimal parameters are obtained through a large number of experiments. In the future, attention model will be used, which can enable neural network to learn more important information of the channels(3)
In addition to time and frequency features, other useful features can be used, such as temperature and humidity. If there is a suitable dataset, we will use these features


## 3.1. MRCNN Model Construction HI

In this section, the structure of the MRCNN model and the experimental procedure of constructing HI is described in detail. Firstly, the original vibration signals are converted into a 2-D matrix of size2560×2560 by CWT. In order to cut down the input volume of the model, the size of the 2-D matrix is reduced to 224×224 by utilizing the nearest interpolation algorithm, which greatly saves massive amounts of training time. The PRONOSTIA public dataset collects vibration signals in horizontal and vertical directions. Consequently, the input of MRCNN model is set as two channel numbers, which, respectively, correspond to the time-frequency matrices in these two directions. Afterward, the first two layers of MRCNN model are ordinary convolution layers, which utilize the structures of convolution, BN, Relu, and maximum pool. The convolution kernel size of the first layer is 7×7 and the step size is 1. The first layer convolution operation will not change the size of the 2-D matrix. The 2-D matrix output from the first layer of MRCNN model is halved to the size of 112×112, and the number of output channels is 64. The second convolutional layer uses a similar structure, doubling the number of output channels to 128×128 and halving the 2-D matrix size to 56×56. The difference is that the convolution operation has a convolution kernel of 3×3, stride of 1 and padding of 1. Next is the multiscale residual layer, which is composed of four multiscale residual blocks, each of which has four multiscale feature mapping operations. Specifically, the first operation is 1×1 convolution to extract feature map. The second is 1×1 convolution plus 3×3 convolution, and the output ratio is 1 : 2. The third is max pooling plus 1×1 convolution, and the output ratio is 1 : 1. The fourth is 1×1 convolution plus two 3×3 convolutions, and the output ratio is 1 : 2 : 2. Overall, the output ratio of these four channels is 2 : 4 : 1 : 1. The above parameters are obtained through massive amounts of experimental tuning parameters, which have strong applicability. In particular, MRCNN uses the residual hopping structure from layer 3 to layer 6 but further doubles the number of channels from layer 4 to layer 5 to 256×256. Therefore, the residual hopping structure here is first realized by changing the number of channels through 1×1 convolution and then hopping. The seventh layer of MRCNN is a normal convolutional layer with a convolutional kernel size of 3×3, a stride of 1, a padding of 1, and an output channel of 64. The eighth layer of MRCNN is the global averaging pooling layer, which is implemented by taking the average of a 56×56 size 2-D matrix and outputting 64 parameters. The ninth to eleventh layers of MRCNN are full connection layers, and the number of neurons in each layer is 128, 64, and 1, respectively. Finally, the MRCNN model obtains HIs through sigmoid activation function, which makes the distribution of HIs from 0 to 1.After CWT algorithm transformation, the training datasets from PRONOSTIA public dataset are successively imported into MRCNN model for training, because an obvious advantage of data-driven methods is that it does not require massive amounts of a priori knowledge. The label is set to decrease linearly from 1 to 0. In this paper, the full life of the bearing is recorded asTa and the current operating time is recorded as Tc. The true label of HI can be obtained by the following formula:
(9)label=1−TcTa.The HI represents the health of bearings, 1 indicates complete health, and 0 indicates damage and no longer usable. The loss function of the MRCNN model utilizes mean square error (MSE), which is widely adopted as the loss function of regression model. MSE is shown as follows:
(10)Ly^;y=12y∧−y2,where y^ is the predicted value of the MRCNN model and y is the true value. The MRCNN model adjusts the parameters by back propagation and gradient descent to reduce the loss function and make the target value closer to the true value. Since the optimizer utilizes stochastic gradient descent (SGD), which approximates the gradient on the loss function of the entire dataset by using the gradient on the loss function of small batch data, therefore, it is easier to adjust the direction of gradient descent and achieve global optimization in the training process. The learning rate is adjusted by the exponential decay method, whose initial learning rate is 0.1 and the multiplication factor is set to 0.95. Dynamically, decaying the learning rate can quickly reduce the loss in the early iteration stage and slightly reduce the loss in the later stage. The learning rate decay is shown as follows:
(11)lr=lr⋅0.95n,where lr denotes the learning rate and n is the number of iterative rounds. The training dataset includes Bearing1_1 and Bearing1_2, and the test dataset includes Bearing1_3 to Bearing1_7. In this paper, a total of 100 training rounds were performed. Figure 8(a) illustrates that the learning rate decays exponentially with the increase of round, and the decline law of the loss function is the same as expected. In the early stage, the loss is rapidly reduced, and in the later stage, the loss is slightly reduced. However, as shown in Figure 8(b), when the 5th to 10th rounds of training are performed, the loss function on the training and test dataset increases instead, which is caused by the excessive learning rate at that time. By utilizing the decay operation of the learning rate, the loss can be correctly reduced in the subsequent rounds. In addition, the MRCNN model adopts global average pooling before the fully connected layer to reduce the problem of overfitting.Figure 8
(a) Learning rate. (b) Training loss and testing loss.
(a)
(b)The trained MRCNN model is used to test Bearing1_3. The GPU model used in the experimental platform is GTX1080ti, and the MRCNN model is implemented based on the deep learning framework Pytorch. The truncation time point of test dataset Bearing1_3 is 18010 s, and it spends 6.4 s for MRCNN model to construct HIs. The MRCNN model constructs HIs on the test dataset Bearing1_3, whose results are shown in the blue curve in Figure9. The degradation trend of bearings can be clearly seen from HI in Figure 9. Firstly, in the range from 0 s to 7500 s, the bearing degradation is slow, and the bearing can operate stably in the early stage of its service life. Afterward, the degradation trend suddenly becomes faster at 7500 s, 10500 s, 13000 s, 15700 s, and 17500 s. Due to the influence of external force, the degradation of bearings gradually accelerates until it is finally damaged. Although the HIs shows a clear degradation trend from the overall, there will exist some HI deviation from the overall trend due to the interference of some strong noise. Consequently, as shown in Figure 9, HI is discrete in narrowing the specified period.Figure 9
RUL prediction result of Bearing1_3.


## 3.2. RUL Prediction Results

After obtaining the HI through the MRCNN model, the next step is to perform RUL prediction based on the HIs. Since the obtained HI distribution is more discrete, this problem is solved by fitting HIs with an exponential sliding average algorithm. According to formulas (5) and (6), the current predicted value is associated with the values of all previous moments. The last predicted value is obtained by a process like sliding backwards in sequence from the first value to the last value, doing exponential decay according to the decay factor. The decay factor in the experiment takes the value of 0.95. Consequently, the exponential sliding average algorithm makes the decline trend more obvious and enables more accurate prediction of the RUL, as shown in the orange curve in Figure 9. In this paper, through massive amounts of experiments, the following operations are the most accurate prediction of RUL: take the last 30% of the smooth data and expand it through linear fitting. As shown by the green line in Figure 9, the predicted life can be obtained by intersecting with the line whose ordinate is equal to zero. Consequently, the RUL of bearing can be obtained by subtracting the current time from the predicted life. The results of RUL prediction from Bearing1_4 to Bearing1_7 are shown in Figure 10. Specifically, Bearing1_4 predicts a full life of 14740 s, Bearing1_5 predicts a full life of 24980 s, Bearing1_6 predicts a full life of 24300 s, and Bearing1_7 predicts a full life of 21880 s.Figure 10
RUL prediction result of Bearing1_3 to Bearing1_7.Regarding the performance evaluation metrics of RUL prediction, this paper adopts the method given in the IEEE PHM 2012 Prognostic Challenge: error percentage, evaluation metricAi, and mean value of evaluation metric Ai. The error percentage is shown in
(12)Eri=100×ActRULi−RU^LiActRULi,where ActRULi represents the true RUL of the i-th bearing and RU^Li represents the predicted RUL of the i-th bearing, since the error percentage Er represents the deviation of the predicted RUL from the true RUL. Consequently, the closer the predicted RUL is to the true RUL, the closer the error percentage is to 0. In addition, the evaluation index is shown in
(13)Aiexp−ln0.5⋅Eri/5ifEri≤0,expln0.5⋅Eri/20ifEri>0.The evaluation metricAi represents a measure of RUL predictive performance. If the Eri is 0, Ai is equal to 1. As the absolute value of Eri increases, the Ai becomes larger. However, the cost of overpredicting and underpredicting the RUL is different. In the field of RUL prediction, the cost of underestimating RUL is lower than overestimating RUL. Equation (13) captures this well. The final score is shown in Equation (14), which represents the mean value of the evaluation metrics.
(14)Score=1n∑i=1nAi.To verify the superiority of the method proposed in this paper, other data-driven methods using the same dataset are compared. The literature [24] is the champion model of IEEE 2012 Data Challenge. Lei et al. [5] proposed WMQE model, which has good prediction performance by fusing several weighted features to predict RUL through correlation clustering between 28 features of bearings. Chen Y et al. [25] extracted five bandpass energy values of the spectrum as features to construct a network based on code and decoding framework and attention mechanism for network prediction of RUL. Attention model based on signal decomposition brings us a new idea. Table 2 writes the final scores for each bearing current time, actual RUL, predicted RUL by this method, and the percentage of error compared to the other three methods. In addition, the absolute mean of the error percentage Er¯ is added in this paper, which judges the merits of the method by comparing the dispersion degree of the RUL prediction error. As seen from Table 2, Er¯ is 12.81 and the final score is 0.47. Both in terms of prediction accuracy and dispersion of prediction, the method in this paper is better than the other three methods.Table 2
Comparison results of this research and related researches in the PRONOSTIA dataset.
Testing dataset
Current time (s)
Actual RUL (s)
Predicted RUL (s)
Er
1
Er
2[24]
Er
3[5]
Er
4[25]
Bearing1_3
18010
5730
5470
4.53
37
-0.35
7.62
Bearing1_4
11380
2890
3360
-16.26
80
5.60
-157.71
Bearing1_5
23010
1610
1970
-22.36
9
100.00
-72.57
Bearing1_6
23010
1460
1290
11.64
-5
28.08
0.93
Bearing1_7
15010
7570
6870
9.24
-2
19.55
85.99
Er
¯
12.81
26.6
30.71
64.96
Score
0.47
0.46
0.45
0.35


## 3.3. Discussion

The effectiveness and advantage of the proposed MRCNN method have been verified by experimental data and point out the shortcomings of the MRCNN model and the direction of improvement in the future.(1)
The MRCNN model adopts the time-frequency matrix transformed from the vibration signal, which saves the training time by reducing its size. However, reducing the size of features will lose a lot of useful information. Therefore, it is a very important problem to achieve a balance between the cost of time and the input of enough features(2)
The structure of MRCNN model design is complex, and the optimal parameters are obtained through a large number of experiments. In the future, attention model will be used, which can enable neural network to learn more important information of the channels(3)
In addition to time and frequency features, other useful features can be used, such as temperature and humidity. If there is a suitable dataset, we will use these features


## 4. Conclusions

In this paper, the CWT algorithm is utilized to convert the time-domain signal into a time-frequency domain 2-D matrix firstly. Afterword, HIs are successfully constructed by MRCNN model. The advantages of the MRCNN model are summarized as follows. The multiscale structure of MRCNN model can extract the local and global features of multiple scales. The residual structure of MRCNN model can make the network easier to train into an accurate model. The MRCNN model uses global average pooling instead of flatten layers to reduce the problem of overfitting. Concurrently, the EMA algorithm is used to highlight the degeneration trend. RUL was finally obtained by linear regression prediction. The RUL prediction results prove the validity of the method. Finally, we compare with the method in recent years that predicts RUL on the same dataset, and the results show that the method in this paper has the best prediction results in terms of prediction accuracy and dispersion of prediction.



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*Source: 2222802-2022-06-20.xml*