Bearing Performance Degradation Assessment Using Lifting Wavelet Packet Symbolic Entropy and SVDD

Bearing performance degradation assessment is of great significance for proactive maintenance and near-zero downtime. For this purpose, a novel assessmentmethod is proposed based on liftingwavelet packet symbolic entropy (LWPSE) and support vector data description (SVDD). LWPSE is presented for feature extraction by jointing use of lifting wavelet packet transform and symbolic entropy. Firstly, the LWPSEs of bearing signals from normal bearing condition are extracted to train an SVDD model by fitting a tight hypersphere around normal samples. Then, the relative distance from the LWPSEs of testing signals to the hypersphere boundary is calculated as a quantitative index for bearing performance degradation assessment.The feasibility and efficiency of the proposed method were validated by the life-cycle data obtained from NASA’s prognostics data repository and the comparison with Hidden Markov Model (HMM). Finally, the assessment results were verified by the envelope spectrum analysis method based on empirical mode decomposition and Hilbert envelope demodulation.


Introduction
Bearings play an important role in rotating machinery.The performance of bearings generally affects the operation reliability of whole equipment directly [1].Actually, bearings always undergo a degenerative process from normal states to failure states.There is possibility to make some proper maintenance strategies to prevent the performance deterioration of bearings, if we can monitor the degenerative process of bearings in time [2].Therefore, degradation assessment of bearings is significant to help reduce production downtime and save maintenance costs [3].
Vibration signals are always being used to monitor the running state of mechanical system because they carry a lot of information which indicates the health condition of mechanical equipment [4].Based on vibration signal analysis, lots of fault diagnosis methods combining advanced signal processing and pattern recognition techniques have been proposed for bearings [5].Although bearing fault diagnosis can be helpful for the indication of condition-based maintenance (CBM), they cannot reveal the degenerative trends of bearings.Given this problem, bearing performance degradation assessment has been a subject of extensive research in recent years, and more and more attention has been received for its benefits in implementing CBM strategies [6].
Generally speaking, performance degradation assessment of bearings mainly includes two steps, namely, feature extraction and degradation assessment.Until now, a lot of work covering these two aspects has been done.Conventional monitor indexes, such as root mean square (RMS) and kurtosis, have been frequently used as features for degradation assessment of bearings [7,8].Pan et al. proposed spectral entropy as a complementary index to evaluate the degradation state of bearings and the results of both simulations and experiments showed that spectral entropy can effectively reflect the degradation process of bearings [9].Yu proposed a method of locality preserving projection for feature extraction and used a complementary index, namely, negative loglikelihood probability-based exponential weighted moving average statistic (NLLP-EWMA) to assess the performance degradation process of bearings [10].Hong et al. proposed a method of wavelet packet-empirical mode decomposition for 2 Shock and Vibration feature extraction and then used self-organization mapping (SOM) for performance degradation assessment of bearings [3].Nelwamondo et al. utilized GMM and HMM to diagnose fault in rolling bearings, based on extracted features using Multiscale Fractal Dimension (MFD), Mel Frequency Cepstral Coefficients, and kurtosis.However, the major drawback of HMM classifier is that it is costly and complex [11].Guo et al. investigated the Hilbert envelope spectrum and SVM method to diagnose REB with ball fault, inner race fault, and outer race fault [12].The results show that the proposed method provides accurate diagnosis and good diagnostic resolution.Zhang et al. utilized particle swarm optimizationsupport vector machine (PSO-SVM) to realize the classification of fault location and degradation degree of rolling bearings [13].Dong and Luo used principal component analysis (PCA) to reduce the dimension of original features and then constructed the LS-SVM model for degradation evaluation of bearings [14].Sun et al. employed a kernel locality preserving projection-based method to obtain an index for evaluating the degradation degree of a bearing [15].Liu et al. extracted zero crossing features of vibration signals and then employed a coupled Hidden Markov Model for assessing the performance degradation of bearings quantitatively [16].Wang and Chen used bilateral spectrum for feature extraction as it could reveal nonlinearities and nonstationary characteristic and then employed a support vector data description (SVDD) model to assess the degradation process of bearings [17].
Though a lot of work has been done, there are still many challenges in assessing the performance degradation of bearings effectively.One of the challenges is how to extract more consistent features for degradation assessment since different features may only be useful at certain service stage or be applicable for specific degradation modes of bearings.For example, RMS can correlate well with the fault development of bearings but it is not sensitive to incipient faults, while Kurtosis Factor is sensitive to impulse faults (especially incipient faults) but it shows the poor stability as the damage grows [7,8].Therefore, a composite index which can satisfy both the conditions of sensitivity and stability simultaneously is necessary.Another challenge is how to build an intelligent assessment model based on the extracted features.
To solve the abovementioned problems, a novel assessment method combining lifting wavelet packet symbolic entropy (LWPSE) and support vector data description (SVDD) is proposed in this paper.LWPSE is the combination of lifting wavelet packet transform (LWPT) and symbolic entropy.LWPT has some advantages over classical wavelet packet transform, including performing integer-tointeger wavelet transform and less computation and memory.Besides, LWPT can be reconstructed no matter how the prediction and update operators are designed.Recently, the applications of LWPT in fault diagnosis have achieved some good results [18,19].Symbolic time series analysis (STSA) provides a coarse grained description of a dynamical system based on a set of symbols.It is not sensitive to the measure noise, and hence it has a good robustness.Recently, it has been used for motoring degradation and fatigue damage using ultrasonic signals [20,21].SVDD is a single value classification method developed from SVM and the model can be established using normal data only.Moreover, it possesses the advantages of high computing efficiency and robustness.Therefore, SVDD has been widely used in degradation assessment of bearings [8,17,22].
The following paper is organized as follows.Sections 2 and 3 are dedicated to the theories of lifting wavelet packet symbolic entropy and SVDD, respectively.Section 4 presents the degradation assessment method based on LWPSE and SVDD and HMM.Experimental validation and the related analysis are provided in Section 5.The conclusions are provided in Section 6.

Lifting Wavelet Packet Symbolic Entropy
2.1.Lifting Wavelet Packet Transform.Lifting wavelet packet transform (LWPT) is realized through the lifting scheme.It no longer relies on Fourier transform, and all computations are implemented in time domain.Moreover, it is possible to obtain the wavelets with some special properties through the design of predictor and updater [23].The forward transform process of LWPT is shown in Figure 1.We can run the lifting scheme backwards to derive the inverse transform from the forward transform.The process of lifting scheme mainly includes three steps: split, predict, and update.The detail procedure is as follows [24].
(2) Predict and Update.In this step, each subband coefficient is calculated at level : where  is a prediction operator and  is an update operator.Here,  and  are designed by interpolation subdivision principle [25].In the context of interpolation subdivision principle, the orders of  and  decide their values.Therefore, selecting different orders is equivalent to selecting different bilateral orthogonal wavelet filters with different vanishing moments.In the current study, the orders of  and  are both selected as 12, since this can structure a wavelet function which is closely similar to mechanical impulses [24].Then, we can obtain the values of  and .

Symbolic Entropy.
Symbolic time series analysis (STSA) aims at symbolizing the time series data.The basic process of STSA is converting the original time series signals into sequences of discrete symbols via partition function, and then we can use statistical features of the symbols to describe the dynamic statuses of a system [26].STSA includes the symbolization of original time series and the quantitative analysis of symbolic series.
Consider an original time series  = {(),  = 1, 2, . . ., }.Converting  into a binary symbol series  = {(),  = 1, 2, . . ., } is the most commonly used method in the symbolization of original time series.In this process, partitioning is the first step, and it is also a key step.Here, the partition function is obtained by finite difference method.It is defined as follows: where () ( = 1, 2, . . ., ) is the element of original time series signals  and () is the binary symbol series which is equivalent to 0 or 1.After the symbolization, a binary-coded symbol series  that only contains "0" and "1" can be got.
And the next thing to do is dividing  into decimal sequences with length , namely, words.When the delay time  and the length of word  are given,  can be cut into short symbol sequences: where  is the length of .Then, () needs to be transformed to the word ().
Based on the work above, a series of words can be obtained.Here, in order to reveal the intrinsic complexity of the original time series, the histogram of symbol sequence as the basis for quantitative statistics is introduced to measure the occurrence number of each word in all the words [27].For example, a series of words derived from a binary-coded symbol series  (the length of  is 124) are 0, 1, 2, . . ., 7, and the occurrence numbers of each word are 27, 16, 2, 18, 15, 4,  and 17; then, the histogram of symbol sequence can be shown in Figure 2, and the probability of each word (the normalization of Figure 2) can be shown in Figure 3. From the histogram of symbol sequence, we can use the modified Shannon entropy to describe the complexity of symbol sequence.The modified Shannon entropy is defined as where   is the probability of the th word and  is the number of all the symbol sequences.In the end, symbolic entropy is equivalent to the modified Shannon entropy of STSA.Symbolic entropy is a complexity measurement; it can measure the complexity of signals.The vibration signals will be more and more complex with the increase of fault severity, thus causing the increase of the entropy.Therefore, symbolic entropy can reveal the complexity of vibration signals.

Feature Extraction Using Lifting Wavelet Packet Symbolic
Entropy.The basic steps in lifting wavelet packet symbolic entropy (LWPSE) based feature extraction are as follows: (1) Employ LWPT to decompose bearing vibration signals.It has been illustrated in Section 2.1 that the best choices of the orders of  and  are both 12.And in order to describe the features of bearing vibration signals more subtly, the decomposition level is defined as 4.Then, the decomposition coefficients can be obtained.
(2) Choose the decomposition coefficients of each junction in the last level to reconstruct the original signals.
Then, we can obtain the reconstructed signals.
Here, the values of  and  are suggested to be 5 and 15 based on a lot of trials since there is no theoretical rule to determine them.Then, the symbolization of each reconstructed signal can be done and the histogram of symbol sequence can be obtained.

Support Vector Data Description
Support vector data description (SVDD) proposed by Tax and Duin is inspired by the theory of support vector machine (SVM) proposed by Vapnik [28].The main idea of SVDD is to find an optimal hypersphere with minimal volume containing all or most targets, as shown in Figure 4. Consider a training set {  ,  = 1, 2, . . ., },  is the total number of samples.We try to find the optimal hypersphere which contains all or most normal samples.This hypersphere is described by center  and radius  and satisfies the following optimization function: where  is a penalty parameter which controls the tradeoff between the volume of hypersphere and errors and   are slack variables which permit a few training data to be outside the hypersphere.
Generally speaking, ( 6) is solved by introducing Lagrange multipliers and it can be transformed into the following maximizing function  with respect to the Lagrange multipliers Since the data in the input space are not always linearly predicted, we introduce a kernel function (  ,   ) = (Φ(  )⋅ Φ(  )) to replace the inner product (  ⋅   ), where  is a Mercer kernel.The kernel function (  ,   ) can map the data into a high-dimensional feature space and transform the nonlinear problem to a linear model.Any function meeting Mercer's theorem can be employed as kernel function, but not all of them are useful for SVDD.Gaussian kernel is the most commonly used function.It is defined as follows: where  is the width parameter.Since Gaussian kernel can restrain the growing distances for large feature spaces for describing the target data more compactly, we employ it as (  ,   ).Then, (7) becomes All   are got by solving (9) and only a few of them are nonzero.The samples with   > 0 are called support vectors.Then, the radius  is obtained by any support vector  sv : For a new sample   , its distance to the center  can be described as follows: Here, we introduce the theory of SVDD into bearing performance degradation assessment.Then, the relative distance between the new sample   and the hypersphere boundary can be used as the degradation value (DV) of   .It is defined as follows: If DV ≤ 0,   is accepted as a target which indicates that the bearing runs in a normal state.Otherwise, it is an outlier which indicates that the bearing runs in a degradation state.

Degradation Assessment Based on LWPSE and SVDD.
In this paper, we proposed a bearing performance degradation assessment method based on LWPSE and SVDD.The framework for performance degradation assessment is shown in Figure 5, which includes two parts, namely, offline modeling and online assessment.The steps of the proposed method are illustrated as follows: (1) The historical normal signals of a rolling bearing are collected and their feature vectors are extracted using LWPSE.
(2) The feature vectors of normal state obtained from step (1) are used as training samples to establish an SVDD model.Then, the radius  can be obtained by (10).
(3) As for the new testing signal   , its feature vectors are extracted using LWPSE.Then, the generalized distance is calculated using (11) which is related to the model established in step ( 2).
(4) The degradation value (DV) of   is calculated using (12).Then, the degradation index with a series of DVs of testing signals can be obtained.Here, we set all the values of DV to 0 when DV ≤ 0. From the degradation index, we can know that the bearing runs in a normal state while DV ≤ 0. Otherwise, it runs in a degradation state.Moreover, DV also reflects the degree of fault severity of a bearing, namely, the larger DV means the larger degree of fault severity.

Degradation Assessment Based on Hidden Markov Model (HMM).
Hidden Markov model can be written as  = (, , , , ) [29].Where  is the state number of Markov chain.Normal, initial failure, deterioration, deep degradation, and failure are the five states of bearings. is observation that may occur in each state.In the paper,  is 11 and  is the vector of initial probability distribution. is the model of the state transition probability matrix.Parameters  and  are generally uniform or selected randomly. is the model of observation probability matrix.
After model initialization is completed, this paper uses the first 200 sets of data to train model.It is set when two adjacent output of log-likelihood probability value are less than 10 −3 and model training is completed.Then, enter the 986 group under testing data into the model trained above.When the program stops iterating, the likelihood probability output value of rolling bearing is got.Because the similar probability of their output is relatively small, this paper uses the log-likelihood probability log  to represent the index of performance degradation.After getting its loglikelihood probability value, the paper uses plot function to draw performance degradation curve of rolling bearing.

Description of the Experiment.
The experimental data were obtained from NASA's prognostics data repository [30].The schematic of bearing run-to-failure test rig is shown in Figure 6.During the experiment, four Rexnord ZA-2115 double row bearings were tested on one shaft.The bearings have 16 rollers in each row.The roller diameter is 8.407 mm.The pitch diameter is 71.501 mm.And the tapered contact angle is 15.17 ∘ .The shaft was driven by a motor.The rotating speed was kept constant at 2000 rpm, and a 6000 lb radial load was added to the shaft and bearings.All the bearings were lubricated.The test stopped when the debris adhered to the magnetic plug exceeded a certain level.On each bearing, two accelerometers PCB 353B33 were installed to collect the accelerations generated by the vibration signals, with a sampling rate of 20 kHz.The interval time of each collection was 10 minutes and the data length of each collection was 20480 points.The test was carried out for about 163 hours and it ended up with an outer race defect in bearing 1.Therefore, the life-cycle data of bearing 1 was employed to assess the performance degradation of a bearing in this paper.From the geometric parameters of ZA-2115 bearings, we can know that its ball pass frequency of the outer race (BPFO) is about 236.4 Hz.

Assessment Results
Using the Proposed Method.The first 200 sets of the life-cycle data of bearing 1 were used as normal signals, and their feature vectors were extracted using the feature extraction method proposed in Section 2.3.Then, we obtained a 200 * 16 matrix, and the matrix was used to establish an SVDD model.After the model was established, all the life-cycle data were used as testing signals and their feature vectors were extracted.Then, the feature vectors were inputted to the trained SVDD model.Finally, the degradation index DV of bearing 1 was obtained.From the theory of time series analysis, we can know that to certain extent the output of a system at any time is determined by the output of the previous few moments.According to this theory, we can deal with the degradation index by five-point smoothing, namely, DV() = mean(DV( − 4 : )), where mean mean(DV) means getting mean values.The DV after five-point smoothing is shown in Figure 7.It is seen that the bearing's performance degradation process is clearly revealed by the DV.Before 5330 (the sample number is 533) minutes, the DV is approximately equal to 0 which indicates that the rolling bearing runs in the normal stage.At 5330 minutes, the DV has an obvious increase and it indicates that the initial fault begins to occur.After 6960 minutes, the increasing trend of the DV is more obvious and it indicates that the rolling bearing runs in the fault progression stage.After 9600 minutes, the DV has no more dramatic increase which indicates that the fault of rolling bearing is developing to failure.Also, sometimes the DV has some abrupt fluctuation since the edge of crack may be smoothed and rounded rapidly after their occurrence [31].
In sum, the performance degradation of bearing 1 could be divided into four periods successfully: normal stage from 0 to 5330 minutes, slight fault stage from 5330 to 6960 minutes, fault progression stage from 6960 to 9600 minutes, and developing-to-failure stage from 9600 minutes to the end.
In the proposed feature extraction method, there are two parameters to be predetermined, namely, the delay time  and the length of word  which is described in the third step of Section 2.3.Figure 8 plots the DV when the parameter  is equal to different values.It can be seen that the increasing trend of the DV is the best when  is equal to 5 as it can describe the degradation process more clearly.Figure 9 plots the DV when the parameter  is equal to different values.It can be observed that the DV is the steadiest when  is equal to 15.Therefore, suitable  and  are meaningful for feature extraction.

Assessment Results
Using RMS.RMS is one of the frequently used monitoring indexes.The RMS of bearing 1 is shown in Figure 10.It can be seen that the increasing trend of RMS amplitudes is not obvious before 6990 minutes which indicates the difference of RMS in the normal stage and slight fault stage is not evident, while the proposed degradation index is evident.In addition, it decreases after 7040 minutes which is not consistent with the degradation process.

Assessment Results
Using Hidden Markov Model.Performance degradation curve of the rolling bearing using HMM is shown in Figure 11.
The figure shows that probably in the 539th sample, the performance of the rolling bearing began to decline and early failure appeared.Probably in the 699th sample, the rolling bearing degraded.After the 820th sample, the rolling bearing exacerbated deeply.After the 963rd sample, the rolling bearing failed completely.We also can see that after about the 737th sample the performance degradation curve is inconsistent with its degree of fault.The overall trend of degradation curve is falling, but the curve still has relatively large fluctuations.LWPE extracts the energy entropy of each reconstructed signal obtained by lifting wavelet packet transform [32].The assessment result based on LWPE and SVDD is shown in Figure 12.It is observed that the DV has an obvious increase at 5350 minutes and the increasing trend is more obvious after 7000 minutes, which indicates the description of normal stage and slight fault stage is similar to the assessment result based on LWPSE and SVDD.But the curve increases after 7930 minutes, which is not consistent with the degradation process.By comparison among Figures 7, 10, and 12, it is obvious that the degradation index proposed in this paper can reflect the bearing performance degradation process more effectively than the RMS and the degradation index based on LWPE and SVDD.

Correctness Validation of Assessment Results.
To validate the correctness of assessment results that the initial fault occurs at 5330 minutes, the envelope spectrum analysis method based on empirical mode decomposition (EMD) and Hilbert envelope demodulation was employed in this paper.Here, we analyze the 533rd sample and the 986th sample.It was decomposed into several simple intrinsic mode functions (IMFs) by EMD first of all [33].Then, we applied the Hilbert envelope demodulation analysis on IMF1 due to the fact that it is the highest frequency signal which includes the most detailed information of vibration signals [34].The demodulation result is shown in Figure 13.It can be seen that there is an obvious spectrum peak at the frequency with 231 Hz, which is close to the BPFO with 236.4 Hz.Also, there exists obvious harmonic frequency characteristic.Moreover, the envelope spectrum of 532nd sample depicted in Figure 14 shows no obvious spectrum peak at the frequency close to the BPFO (the samples before the 532nd sample show the same results).The envelope spectrum of 986th sample depicted in Figure 15 shows that there is an obvious spectrum peak at the frequency with 231 Hz, which is close to the BPFO with 236.4 Hz.Also, there exists obvious harmonic frequency characteristic.It thus can be inferred that the initial fault of the outer race occurs at 5330 minutes.The analysis results are consistent with the previous assessment results.

Conclusions
In this paper, the degradation index based on LWPSE and SVDD is proposed for bearing performance degradation assessment.LWPSE is used to extract feature vectors and SVDD is employed to obtain the assessment results.The efficiency and validity of the proposed method are verified by the life-cycle data obtained from NASA's prognostics data repository.Analysis results show that, compared with the RMS and the degradation index based on LWPE and SVDD, the proposed degradation index is more sensitive to initial fault and it has a consistent increasing trend with the development of bearing fault.The method of HMM can show the overall performance degradation but its consistency of performance degradation is relatively poor.Further analysis shows that the degradation index is affected by the parameters  and  of the symbolic entropy; thus, suitable  and  are meaningful for feature extraction.Moreover, the correctness of assessment results is verified by the envelope spectrum analysis method based on EMD and Hilbert envelope demodulation.It may be concluded that the proposed method is better than HMM in the paper and it is of great significance in guiding the maintenance of rotating machinery.

Figure 1 :
Figure 1: Block diagram of the forward transform of LWPT.

Figure 3 :
Figure 3: Probability of each word.

Figure 5 :
Figure 5: Procedure of performance degradation assessment.

Figure 8 :Figure 9 :
Figure 8: Assessment result with different values of .