Incipient Fault Detection of Rolling Element Bearings Based on Deep EMD-PCA Algorithm

. Due to the relatively weak early fault characteristics of rolling bearings, the diﬃculty of early fault detection increases. For unsolving this problem, an incipient fault detection method based on deep empirical mode decomposition and principal component analysis (Deep EMD-PCA) is proposed. In this method, multiple data processing layers are created to extract weak incipient fault features, and EMD is used to decompose the vibration signal. This method establishes an accurate data mode, which can improve the incipient fault detection capability. It overcomes the diﬃculties of incipient fault detection, in which weak fault features can be extracted from the background of strong noise. From a theoretical point of view, this paper proves that the Deep EMD-PCA method can retain more variance information and has a good early fault detection ability. The experiment results indicate that the detection rate of Deep EMD-PCA is about 85%, and the failure detection delay time is almost zero. The incipient faults of rolling element bearings can be detected accurately and timely by Deep EMD-PCA. The method eﬀectively improves the accuracy and timeliness of fault detection under actual working conditions and has good practical application value.


Introduction
Rolling element bearings are one of the common mechanical components in rotating machinery, and their operating conditions often directly affect the performance of the entire machine [1][2][3]. It is vital to detect the incipient faults of rolling element bearings accurately and timely, which can prevent them from developing into a more serious fault, resulting in huge losses [4]. Vibration signal analysis is the main technique to diagnose a rolling bearing fault [5,6]. e main challenges of incipient fault detection for rolling element bearings are as follows [7,8]: (1) e incipient features of the device are weak and insignificant, and the fault signal is often buried in a strong noise background. During the running process of the rolling element bearings, there is a large amount of amplitude interference noise in the signal (2) Due to the harsh operating environment and limited field measurement conditions, the obtained signal has an impact interference signal ere are many incipient fault detection methods for rolling element bearings, which can be roughly divided into two methods based on model-driven and data-driven methods [9]. An incipient fault detection method based on analytical models requires modeling the system to analyze system faults [10,11]. Due to the inaccuracy of the modeling, the detection performance of the system failure is reduced. Data-driven methods can effectively overcome these problems. e data collected by the system is analyzed by data-driven methods, to achieve the effect of incipient fault detection [12][13][14].
Signal decomposition methods are commonly used for vibration signals [15], including wavelet transform methods [16,17] and empirical mode decomposition (EMD) methods [18][19][20]. e wavelet transform method cannot achieve adaptive decomposition because it is limited by the basis function. e EMD method is a common method for analyzing nonstationary, nonlinear vibration signals. However, due to the inclusion of some redundant information in the feature extraction process, the data dimension is high and the calculation efficiency is low.
Deep learning is one of the popular incipient detection methods [21,22]. e commonly used deep learning incipient fault detection methods include Support Vector Machine (SVM) [23] and Convolutional Neural Networks (CNN) [24]. ese methods can compress the signal to achieve dimensionality reduction. However, the computational strength of the model becomes larger, and the training speed of the system decreases in a large network structure.
A common fault detection method for removing data redundancy and reducing the data dimension is the principal component analysis (PCA) method [25][26][27]. e main idea is to project the high-dimensional data space into the low-dimensional principal component space [28]. However, due to the PCA method in the process of extracting the feature values, only a small number of principal elements are selected, resulting in the loss of information, and it is impossible to diagnose some weak fault features. e study [29] proposed a new data-driven method based on Deep PCA for incipient fault diagnosis. e method can decompose the fault information in a more detailed manner, obtain weak fault features, and achieve the effect of incipient fault diagnosis. But the PCA fault detection method cannot detect the vibration signal.
In order to solve the above problems, this paper proposes an incipient fault diagnosis method based on Deep EMD-PCA. Firstly, the EMD-PCA fault detection method is used to extract the effective information in the vibration signal and remove the noise information in the vibration signal. Secondly, the data is decomposed to obtain multiple subdatasets. Finally, each subdataset is analyzed, redundant information is removed, and weaker fault features will be mined to detect incipient faults in rolling element bearings. Deep EMD-PCA can mine the small fault information in the vibration signal and better detect the incipient fault problem compared with the traditional EMD-PCA fault diagnosis method. In the third part and the fourth part, the fault detection capability of the method is proved from both theoretical and experimental aspects. e main structure of this paper is as follows. e second part introduces the basic theoretical knowledge of EMD and PCA. e third part presents the fault detection method based on Deep EMD-PCA and proves its feasibility. e fourth part verifies the accuracy and feasibility of the method through five experiments. e fifth part summarizes the fault detection method of Deep EMD-PCA.

EMD Method.
EMD is a method to analyze the decomposition of nonlinear and nonstationary signals. It defines the time delay between the adjacent peak points of the signal to be decomposed as a time scale and decomposes the nonstationary signals into several Intrinsic Mode Function (IMF) and residual terms according to the fluctuation or trend of different scales [30]. Each basic pattern component reflects the local characteristic information of the signal. e remainder indicates the central trend of the signal. e flowchart of EMD signal decomposition is shown in Figure 1.
For a group of signals X(t), the steps of EMD are as follows.
Step 1. Find all maximum and minimum points of signal X(t), respectively, and synthesize the upper and lower envelopes of the original data sequence by a cubic spline function.
Step 2. Calculate the mean value between upper and lower envelopes, denoted as m 1 (t). Subtract the mean value from the original data sequence to get a new data sequence h 1 with the low frequency being removed: (1) If h 1 satisfies the following two conditions: (a) the number of crossing zero and extreme points must be equal or at most one difference in the whole dataset; (b) at any time, the average value of the local maximum and minimum envelopes should be zero, then, h 1 is the first IMF component.
If the conditions of IMF are not met, h 1 is taken as the original data, and the screening process (1) is repeated to obtain the difference h 11 between the average value of the Start Input X (t) Determine local maximum and minimum values of X (t) Draw the upper and lower envelopes E1 E2 Determine if the IMF condition is met Determine if the terminal condition is met upper and lower envelopes and h 1 and to determine whether the criteria of IMF are satisfied, until h 1k meets the above criteria of IMF, then h 1k is the first IMF component. Let us call the first IMF component IMF 1 .
Step 3. Separate IMF 1 from X(t) to obtain residual term r 1 : r 1 is taken as the original data. en, repeat steps (1) and (2) to obtain the second IMF component of X(t)−IMF 2 . Repeat such screening process for n times to obtain n IMF components and residual term r n of signal X(t). When the residual term r n is less than the preset value or becomes a monotone function, the components that meet the IMF conditions can no longer be extracted from it, and the whole screening process is finished. After the above steps, signal X(t) is decomposed into the sum of n IMF components and residual terms r n : eoretically, IMF components divide a series of locally high-order and low-order narrow-band stationary sequences strictly according to different characteristic scales of the sequence.

PCA Method.
PCA method is a multivariate statistical method. It can achieve dimensionality reduction of complex samples; that is, it can generate reduced variables through feature extraction without losing important information carried in samples. Since the low-dimensional principal component space can retain most of the variance information of the original data space, and the orthogonality of the principal component variables can remove the redundant information of the original data space, principal component analysis has gradually become an effective method of data compression and information extraction. e basic theory of PCA is as follows.
Let the observation matrix X ∈ R m×n be obtained through m times of independent sampling by n sensors. In order to eliminate the influence of false principal components caused by different dimensions, the sample x i (k) collected at the k moment should be standardized: where i � 1, 2, . . . , n, k � 1, 2, . . . , m. E(x i ) is the sample mean and var(x i ) is the sample variance, as follows: e normalized sample matrix is X * ∈ R n * m : e matrix X * is decomposed into the cross product sum of n vectors: where p i ∈ R n is the load vector; P � [p 1 , p 2 , . . . , p n ] is the load matrix. t i ∈ R n is the goal vector and also is the principal component of X * ; T � [t 1 , t 2 , . . . , t n ] is the goal matrix and represents the projection of the original data matrix on the load direction. e load vector of X * can be obtained by Singular Value Decomposition (SVD), as follows: where U ∈ R m×n and V ∈ R n×n is the unitary matrix. Σ ∈ R n×n is the diagonal matrix, and its diagonal elements are decreasing nonnegative eigenvalues. Formula (12) is equal to the covariance matrix of X * : Singular Value Decomposition is where the diagonal elements of Σ ∈ R n×n are equal to the variance of X * in the corresponding direction of the load matrix P. e projection of data X * on the following load vector directions is often caused by noise, so in order to effectively remove the correlation of the original data and reduce the influence of interference, we retain the first load vector corresponding to the first maximum eigenvalue to compose the load matrix P ∈ R n×a . e original space is divided into principal component subspace X and residual subspace E: As for the selection method of principal components, we adopted the Cumulative Percentage Variance (CPV) method: where n is the number of variables in the data and λ i is the i th eigenvalue of the covariance matrix of X. e threshold percentage is usually set at 85%.

Deep EMD-PCA
e commonly used EMD-PCA combined fault detection method can analyze the vibration signal for fault detection. However, due to the limitations of the PCA method, some fault features are discarded during the process of extracting Shock and Vibration the principal elements, which may result in PCA not being able to accurately detect weak incipient failure message. erefore, this paper proposes a fault detection method based on Deep EMD-PCA.

e Fundamental Deep EMD-PCA.
Firstly, the EMD method is used to decompose the vibration signal to obtain n IMF components, then the n IMF components are formed into a matrix, and the eigenvalue decomposition of the matrix is obtained by the PCA method to obtain the principal element space and the residual space. en, the principal space and the residual space are decomposed multiple times to obtain the j-layer subspace, the statistics and control limits of each subspace are calculated, and a comparison is made to determine whether a fault has occurred. e data structure of Deep EMD-PCA is shown in Figure 2.
We obtain the principal element space and the residual space of dataset IMF by EMD-PCA method: where IMF 11 is the first-order principal component subspace of original data IMF and IMF 12 is the first-order residual subspace of original data IMF. ey can be obtained by Where P 11 is the is the extracted principal component feature vector matrix of IMF.
To obtain more information about the second-order principal component space and residual space, we repeat the above process and then obtain IMF 11 � IMF 21 + IMF 22 , Suppose that P 21 and P 23 are the principal vector of IMF 11 and IMF 12 , respectively, and P 22 and P 24 are the residual vector of IMF 11 and IMF 12 , respectively. en, the original data IMF can be obtained by where IMF 21 � P 21 P T 21 IMF 11 , By parity of reasoning, the original data IMF can be represented as the sum of 2 j subspace, and j is the order. e structure of Deep EMD-PCA is shown in Figure 2. e subspace IMF jk can be expressed as Suppose that IMF jk is the k th subspace of j th order; then, IMF � 2 j k�1 IMF jk , and P jk is the extracted principal component feature vector matrix of IMF (j−1(k+1)/2) . rough the increase in the order of the Deep EMD-PCA method, extremely weak fault information can be detected, thereby achieving incipient fault detection.

Determination of Statistics and Control Limits.
We extend the method of statistical control limit to Deep EMD-PCA. Considering all subprincipal spaces, T 2 jk of each subdataset X jk can be expressed as Λ jk is principal component space characteristic value of matrix.
For the control limit of T 2 jk , it can be calculated through the preset significance level α and the probability density function of T 2 jk . e probability density function of T 2 jk can be expressed as where T b jk represents T 2 jk of column b. en, the probability density function is integrated to make it less than the significance level alpha, namely, SPE statistics reflect the deviation degree of data, which can be expressed as Similar to the control limit of T b jk , the probability density function of SPE jk is expressed as where SPE b jk represents SPE jk of column b, according to the probability density function and significance level a, namely, If the statistics is larger than the control limit, a fault will occur. Otherwise, the data are in a normal state.

eoretical Proof.
is section demonstrates the incipient fault detection capabilities of Deep EMD-PCA from a theoretical perspective. e covariance of Deep EMD-PCA can be expressed as 4 Shock and Vibration e size of the covariance represents the amount of information retained by the method, so the larger the covariance, the more information retained.
According to formula (14), we can obtain that For the j th subspace, the covariance can be expressed as We simplify formula (28): Suppose that S jk � (1/(N − 1))IMF T jk IMF jk ; then, the covariance matrix can be simplified as So, we can obtain that 2 j −1 k�1 S jk > . . . > 3 k�1 S 2k > S 11 . Because the covariance matrix describes all the information about the relationships among the dimensions, it can be concluded that as the order increases, the information retained by Deep EMD-PCA increases. erefore, the Deep EMD-PCA method can retain more sample information compared with the traditional EMD-PCA method, so as to mine the hidden fault information in the sample abandoned by EMD-PCA to achieve the purpose of incipient fault detection.
We can project the dataset IMF in normal operation into the third-order principal space and the residual space based on equations (14)- (18), as shown in Figure 3. For the conventional EMD-PCA method, the residual is IMF 12 . e residual of the second order of the Deep EMD-PCA method is IMF 24 . e residual of the third order of the Deep EMD-PCA method is IMF 38 . According to Figure 3, it can be seen that the residual amount IMF 12 is significantly larger than IMF 24 , and IMF 24 is larger than IMF 38 . erefore, we can conclude that with the increase of the order, the residual of the Deep EMD-PCA method is getting smaller and smaller, that is, the more variance information saved, so that the weak fault information can be diagnosed.
However, with the increase in order, the amount of calculation increases gradually. Considering the complexity of the calculation, it is usually calculated to the third order to mine the weak fault information in the dataset.
As can be seen from Figure 3, all subspaces and subdatasets are orthogonal. So, the 2 j subdatasets of original data are nonoverlapping, there is no duplicate information, and it can contain all information on the dataset.

3.4.
e Process of Deep EMD-PCA. e fault detection method based on Deep EMD-PCA is divided into two parts: offline modeling and online monitoring. e Deep EMD-PCA approach is defined by Algorithm 1.

Experimental Results and Analysis
is section uses the experimental data of Case Western Reserve University and the self-built mechanical failure  (1) e inner ring failure: since the inner ring rotates with the rolling bearing during the operation of the rolling bearing, when a certain failure occurs in the inner ring of the rolling bearing, the failure point and the magnitude of the pulse intensity have a certain consistency, and the amplitude will also show periodic changes. In the process of collecting vibration signals, the signal reflecting the fault information of the inner ring will be affected by the impact vibration, resulting in the loss of high-frequency vibration components, resulting in a certain energy loss, so the detection of the inner ring fault of the rolling bearing is more difficult (2) e outer ring failure: due to a certain failure in the outer ring of the rolling bearing, the vibration signal will show a periodic pulse force whenever the operation passes the failure point. Different from the inner ring failure, since the outer ring is generally fixed, the pulse intensity of each cycle is constant. erefore, the detection of the outer ring failure is relatively simple compared to the detection of the inner ring failure of the rolling bearing

4.1.
e Experimental Data of Case Western Reserve University. In this section, the experimental data of Case Western Reserve University is used to verify the Deep EMD-PCA method. In this experiment, three groups of experimental data were collected, namely, vibration signal data under normal condition, fault data of bearing inner ring with fault diameter of 0.007 inches, and fault data of bearing outer ring with fault diameter of 0.007 inches. e fault diameter is 0.007 inches, which is very small. At this time, the fault characteristics are relatively weak. erefore, this fault is considered to be an incipient fault. e vibration acceleration signal of the bearing is collected by the acceleration sensor, and the motor speed is 1797 RPM. e abscissa is the number of samples, and the ordinate is the statistics and the control limit. Among them, the blue line indicates the statistics, and the red line indicates the control limit. When the statistics exceeds the control limit, it indicates that the data is detected as faulty. From Figures 4(a) and 4(b), we obtain that T 2 01 and SPE 01 are less affected by this fault. Figure 5 shows the results of the PCA-SVM method of detecting the failure of the inner ring of the rolling bearing. Label 0 means normal data, and label 1 means inner ring fault data. e blue circle represents the predicted result, and the red cross represents the actual diagnosis result. We can see from the image that the detection rate is about 60%, and the incipient fault detection results are not satisfactory. is very large, and the rate of detection is about 85%. erefore, the fault detection of the third layer of Deep EMD-PCA is better. Figure 7 depicts the results of a fault detection based on the EMD-PCA fault detection method for an outer ring fault with a fault diameter of 0.007 inches. From Figures 7(a) and 7(b), we can obtain that the detection rate of the fault detection result of the first layer of Deep EMD-PCA exceeds 50%. Figure 8 shows the results of the PCA-SVM method of detecting the failure of the inner ring. Label 0 means normal data, and label 1 means inner ring fault data. e blue circle represents the predicted result, and the red cross represents the actual diagnosis result. We can see from the image that the detection rate is about 45%, and the incipient fault detection results are not satisfactory.  e detection results show that T 2 24 is most affected by this fault, and the detection rate is about 85%. erefore, the third layer of Deep EMD-PCA has good fault detection performance.
We compare the experimental results of the Deep EMD-PCA proposed in this paper, EMD-PCA method, and PCA-SVM method by analyzing the detection rate and detection delay as shown in Table 1.
e detection rate refers to the percentage of the number of accurate sampling points in the detection process to the total number of sampling points. e detection delay time refers to the interval from the start of the failure to the detection of the failure. Table 1 describes the comparison results of the detection rate and detection delay based on the EMD-PCA, PCA-SVM, and Deep EMD-PCA methods. From Table 1, for the inner ring fault, the detection rate of EMD-PCA method and PCA-SVM method is 49.44% and 58.71%, respectively, and the detection rate of Deep EMD-PCA method is 86.92%; for outer ring fault, the detection rate of EMD-PCA and PCA-SVM is about 50%, and the detection rate of Deep EMD-PCA method is 86.77%, so we can obtain that the detection rate of Deep EMD-PCA method is significantly increased. On the other hand, from the comparison of detection delay, Deep EMD-PCA can be detected from the beginning of the fault. So, the Deep EMD-PCA method has good fault detection performance.

Mechanical Failure Comprehensive Simulation Experiment.
e mechanical failure comprehensive simulation experimental device includes a mechanical fault comprehensive simulation test bench, AIC9916FS equipment fault comprehensive simulation diagnostic analysis system software, FL6816L16 channel sensor, BSZ800D-16 vibration signal acquisition instrument, and storage device computer.
e mechanical failure comprehensive simulation test bench is shown in Figure 10.
rough this test bench, the fault data of the outer ring, inner ring, and rolling element of the bearing are collected and analyzed. Figure 11 depicts the results of a fault detection based on the EMD-PCA fault detection method for an outer ring fault. From Figures 9(a) and 9(b), we can obtain that the false detection rate of the detection result of the first layer of Deep EMD-PCA is more than 80%. Figure 12 depicts the mechanical failure comprehensive simulation experiment bearing outer ring failure results based on PCA-SVM. Label 0 means normal data, and label 1 means outer ring fault data. e blue circle represents the predicted result, and the red cross represents the actual diagnosis result. We can see from the image that the detection rate is about 80%, and the incipient fault detection results are not satisfactory. Figure 13 depicts the results of a fault detection based on the Deep EMD-PCA fault detection method for an outer ring fault. Figures 13(a)-13(d) describe the result of the second-layer datasets dataset IMF 11 and IMF 12 . Figures 13(e)-13(l) describe the third-layer datasets IMF 21 -IMF 24 . In Figure 13(b), the result of SPE 11 is greatly affected by this fault. In Figures 13(e) and 13(j)-13(l), T 2 21 , SPE 23 , T 2 24 , and SPE 24 are more affected by this fault. erefore, the fault detection effect of the third-layer Deep EMD-PCA is better, which proves that the method has better incipient fault detection capability. Figure 14 depicts the results of a fault detection based on the Deep EMD-PCA fault detection method for an inner    Figures 14(a) and 14(b), we can obtain that the detection rate of the fault detection result of the first layer of Deep EMD-PCA is more than 60%. Figure 15 depicts the mechanical failure comprehensive simulation experiment bearing inner ring failure results based on PCA-SVM. Label 0 means normal data, and label 1 means inner ring fault data. e blue circle represents the predicted result, and the red cross represents the actual diagnosis result. We can see from the image that the detection rate is about 85%.   (1) Offline modeling: Perform EMD decomposition to obtain n IMF components and residual Rn (4) Group n IMF components into a new matrix IMF (5) Standardization processing for IMF (6) Select the appropriate order j (7) Calculate the corresponding IMF jk (8) Calculate the number of principals of each IMF jk , the eigenvalue matrix Λ jk , and the eigenvector matrix P jk (9) Calculate the statistics T 2 jk and SPE jk (10) Calculate the probability density function of T 2 jk and SPE jk (11) Calculate the control limits of T 2 jk and SPE jk (12) Online Monitoring: (13) Xt(t) is the test dataset (14) Perform EMD decomposition to obtain n IMF components and residual Rn (15) Group n IMF components into a new matrix IMFt (16) Standardization processing for IMFt (17) Obtain the principal element space and the residual space of the subdataset IMFt jk (18) Calculate the statistics T 2 jk and SPE jk (19) If the statistics > the control limits (20) en fault (21) Else normal (22) End if ALGORITHM 1:Deep EMD-PCA.   Actual test set classification Predict the test set classification   erefore, the fault detection effect of the third-layer Deep EMD-PCA is better, which proves that the method has better incipient fault detection capability.
We compare the experimental results of the Deep EMD-PCA proposed in this paper, the traditional EMD-PCA method, and the PCA-SVM method by analyzing the detection rate and detection delay as shown in Table 2. Table 2 describes the comparison results of the detection rate and detection delay based on the EMD-PCA, PCA-SVM, and Deep EMD-PCA methods. For the outer ring fault, the detection rate of EMD-PCA method and PCA-SVM method is about 75%, and the detection rate of the Deep EMD-PCA method is 90.87%; for outer ring fault, the detection rate of EMD-PCA and PCA-SVM is 54.29% and 86.72%, respectively, and the detection rate of Deep EMD-PCA method is 91.36%, so we can obtain that the detection rate of Deep EMD-PCA method is significantly increased.
is is because the method can retain more variance information and weak fault features can be extracted quickly.
e false detection rate of the Deep EMD-PCA method is significantly reduced. On the other hand, from the comparison of the detection delay, Deep EMD-PCA can be detected from the beginning of the fault. So, the Deep EMD-PCA method has good fault detection performance.

Conclusions
In this paper, a novel incipient fault detection method based on Deep EMD-PCA is proposed. e method uses EMD-PCA to decompose the vibration signal and divide the signal into components of different frequency segments from high to low, reduce the data dimension, and realize fault detection. en, we divide the dataset into multiple subdatasets for analysis, so that the weak fault information in the system is analyzed and incipient fault detection is realized. It is proved by theoretical analysis that this method has good performance in incipient fault detection. e method has the following advantages: (1) Deep EMD-PCA method can extract weak fault features compared with EMD-PCA method and PCA-SVM method through multi-data processing layer, which is helpful for incipient fault detection (2) It establishes accurate data models to improve incipient fault detection capability Data Availability e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.