A Bayesian wavelet packet denoising criterion for mechanical signal with non-Gaussian characteristic

Accepted Manuscript A Bayesian wavelet packet denoising criterion for mechanical signal with nonGaussian characteristic Guo-dong Yue, Xiu-shi Cui, Yuan-yuan Zou, Xiao-tian Bai, Yu-Hou Wu, Huaitao Shi PII: DOI: Reference:

S0263-2241(19)30180-0 https://doi.org/10.1016/j.measurement.2019.02.066 MEASUR 6410

To appear in:

Measurement

Received Date: Revised Date: Accepted Date:

26 December 2018 3 February 2019 24 February 2019

Please cite this article as: G-d. Yue, X-s. Cui, Y-y. Zou, X-t. Bai, Y-H. Wu, H-t. Shi, A Bayesian wavelet packet denoising criterion for mechanical signal with non-Gaussian characteristic, Measurement (2019), doi: https:// doi.org/10.1016/j.measurement.2019.02.066

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A Bayesian wavelet packet denoising criterion for mechanical signal with non-Gaussian characteristic Guo-dong Yue, Xiu-shi Cui, Yuan-yuan Zou, Xiao-tian Bai, Yu-Hou Wu, Huai-tao Shi* School of Mechanical Engineering, Shenyang Jianzhu University, Shenyang 110168, China Abstract. Three aspects of the traditional Bayesian wavelet packet denoising method have been rarely discussed in the literature for mechanical signals: (1) how to reduce noise if a precise prior description is needed for non-Gaussian data; (2) how to programmatically select a reasonable decomposition level; and (3) how to evaluate the denoising effect from multiple factors. Such three aspects have restricted the enhancement ability of the Bayesian denoising method. To seek the potential of application, this paper tries to establish a set of criteria. It develops the general Bayesian wavelet packet method based on the minimum mean square error, designs two indexes to select the decomposition level, and initially presents how to judge the denoising effect by multi-factor analysis. This criterion has two advantages: (1) if needed, any prior distribution can be employed to match with the signals; and (2) the selected decomposition level can maximize the distinction between real data and noise data. A comparison test is expected to show this enhancement ability of the proposed criterion. Simulated and experimental data demonstrate its excellent performance. This novel method keeps interpretable time-domain features and shows better denoising performance for weak information recognition than the published methods. Keywords: Weak characteristic extraction; prior probability; decomposition level; denoising effect evaluation; track vibration

1

Introduction Track fasteners play an important role in the track structure, which fixes the rail firmly to the ground. Due to rail temperature stress and wheel–rail impact, this will make the rail fastener be in a loose state and threaten the safety of train driving if not found. If the loose fasteners are detected in their earliest stage, it is feasible to avoid an accident by opportune maintenance. However, the track is in a statically indeterminate state. The earliest loose signatures excited by a small change in fasteners not only are weakened by the transmission path but also are contaminated by the surrounding operating environment. Such signatures can be considered as weak information and are hard to detect. For this reason, the detection of fastener status has been an important issue in track maintenance. To tackle this issue, many researchers apply the data-driven monitoring method and get a satisfactory result. Many sensors are mounted on the structure and sample the vibration data to estimate the safety status [1]. But the sensors are usually in the open and harsh environment for a long time. And the collected vibration data will be inevitably contaminated by external interference [2]. For this reason, it is necessary to denoise the data for further analysis. There are two kinds of ideas to reduce noise effects. One is to transform noise into fault characteristics to amplify the incipient weak information, like stochastic resonance (SR) [3,4,5]. But one transformation can only judge a fault characteristic. It takes a lot of time to estimate a structure status. The other is to remove noise from the sampling data. Its idea is easier to understand and requires less work. One of the representative methods is the Bayesian wavelet packet denoising methodology. It is derived from maximum likelihood estimation and makes full use of the wavelet packet features which can subdivide data indefinitely. It mainly consists of Corresponding author E-mail: [email protected]

three aspects, including prior information, wavelet decomposition level and denoising evaluation. Now, most of the related research focuses on the construction of prior distribution on the decomposition coefficients. These prior distributions generally have clear mathematical expressions, including Besov distributions [6], generalized Gaussian distributions [7], Bernoulli-Gaussian distributions [8], mixture distributions with a point mass at zero [9] and spike-and-slab distributions [10]. The general objective of the research mentioned above is to retain detailed texture structures and reduce the computational complexity. It is known that the more accurately the prior distribution describes the distribution characteristics of the signal, the better the noise reduction effect. For some signals, its prior distribution cannot be described by the aforementioned distribution. This requires us to develop a general Bayesian wavelet packet denoising method. In addition, very few researchers pay attention to choosing the decomposition level. At first, most researchers adopt a trial-and-error process to select the wavelet decomposition level [11]. This approach, however, requires the researchers to have rich experience and cannot get rid of human help. Jiang and Adeli used the autocorrelation function method (ACF) to get a reasonable decomposition level in the wavelet packet transform. When the autocorrelation trend of wavelet decomposition coefficient coincides with the trend of the time series, the wavelet decomposition level can provide sufficient accuracy [12]. Sang, Wang and Wu put forward a wavelet energy entropy method to select the decomposition level, considering the ratio of energy per level and energy at all levels [13]. To our knowledge, how to programmatically select the decomposition level is very important and rarely discussed. Finally, most of the papers drive the data itself to judge the denoising effect. For example, Movchan and Shen compared the PSNR values of the proposed method with LASSC, BM3D, and the exemplar-based method to outperform its performance [14]. Kumar, Panigrahy and Sahu utilized percent root mean square difference, SNR improvement, and mean square error to compare the proposed method with conventional filtering, adaptive filtering, wavelet soft threshold denoising, non-local mean, and empirical model decomposition [15]. Meanwhile, some researchers also extract the data characteristics to estimate the denoising effect [16,17,18]. If the data characteristics can be kept, the denoising effect is satisfactory. Among the preceding research on prior information, wavelet decomposition level and denoising evaluation, however, there are three disadvantages as follows. (1) For a mechanical signal with non-Gaussian characteristic, the aforementioned research cannot provide the precise description. (2) For weak signal information, the aforementioned methods are not suitable for wavelet level selection when noise information seriously affects the expression of physical information. (3) It is not reasonable to estimate the denoising effect without considering the further analysis. For example, if the fault characteristics are analyzed from the perspective of energy and mode shape [19], time-domain data need to be used. In a word, until now, there has been no criterion of Bayesian wavelet packet denoising for the weak information extraction to facilitate the mechanical vibration signal. In order to provide the criterion, the paper uses the Bayesian wavelet packet denoising methodology to denoise the data and get the earliest fault characteristics which are submerged in mechanical signals. We summarize the major contributions as follows. (1) A general Bayesian wavelet packet denoising method is developed for the non-Gaussian signal. (2) The decomposition level should separate the real value and noise as much as possible, that is, their wavelet coefficient

probability distribution functions show the greatest difference. Based on this novel idea, two indexes are proposed for selecting the decomposition level. (3) The noise reduction effect can be judged by multi-factor analysis. In this paper, we first develop the Bayesian wavelet packet based on the minimum mean square error (MMSE), analyze how to select the reasonable decomposition level and propose a denoising evaluation method in Section 2. Section 3 presents the numerical simulation verification. In Section 4, we use the proposed method to identify the state of a track fastener of a high speed railway (HSR). The possibility of applying the Bayesian method to weak information classification is discussed, and the conclusions of this work are illustrated in Section 5. 2. Bayesian wavelet packet denoising methodology 2.1 MMSE The time series can consist of both of real experimental data and noise , as follows: , i = 1, 2, …, (1) The noise is a random vector with independent, identically distributed (IID) errors. The denoising method aims to obtain the real time series from contaminated measurements We want to estimate the original value x by the MMSE method, as follows: (2) where is an estimator of . Assuming to solve the following formula:

to be strictly differentiable, Eq. (2) can be converted

(3) Then Eq. (3) is equivalent to the following formula:

(4) We can obtain the

: (5)

The discrete Bayesian MMSE estimation

of

can be given by: (6)

where

and

denote the prior distributions of variant

and . In this paper,

represents

the kth equally spaced point with the total number of points between the interval [2*min( ), 2*max( )]. 2.2 Wavelet packet decomposition Wavelet packet coefficients can be divided into scaling functions and wavelet functions. For the discrete wavelet packet transform (DWPT) decomposition, scaling functions and wavelet functions can be denoted by the vectors and , respectively. By the inverse wavelet transform, the measured data

can be reconstructed as follows:

where and represent the scalable mother wavelet. The be uniformly represented by the decomposed coefficient . 2.3 Bayesian DWPT denoising approach It makes sense that we also use DWPT to decompose the and wavelet coefficients and , respectively. Thus, can be expressed by , j = 0, …, J−1; k = 0, 1, …, 2j−1

and

(7) can

into a series of (8)

and we want to estimate the original coefficients . Then the time series can be obtained by inverse wavelet transform. The measurement mechanical vibration data usually demonstrates non-Gaussian characteristics with a sparse distribution. According to Eq. (6) and Eq. (8), we develop the following formula to process the non-Gaussian data: , k = 0, 1, …, 2j−1

(9)

where represents the ith equally spaced point with the total number of points 2 times the length of the vector M between the interval [2*min( ) 2*max( )] at the jth decomposition level. If the prior distributions and of the decomposed coefficients are given, the posterior Bayesian MMSE estimations of the coefficients are derived. According to Eq. 7, the denoised data are reconstructed. For the orthogonal wavelet transform, we can adopt the probability density functions (pdfs) of the original and as the prior distributions and . 2.4 Decomposition level How to select a reasonable decomposition level of the time series is very important in wavelet analysis. If the selected level is lower, noise will be mixed with the real value; if the decomposition level is higher, the real value will be too dispersed to each wavelet coefficient. Both of them will cause insufficient or excessive noise reduction. Research shows that the wavelet coefficients demonstrate clustering characteristics [20]. A few of them hold most of the information and have large amplitudes with the others’ amplitudes almost equal to zero. Their pdf shows non-Gaussian characteristics. The reasonable decomposition level should make the number of wavelet coefficients of real value around zero the largest and the number of wavelet coefficients of noise around point zero smaller. This makes it possible to separate the wavelet coefficients affected by the real value from the wavelet coefficients affected by the noise as much as possible. By studying the pdf characteristics of the prior signal at each decomposition level, we can choose a reasonable decomposition level. According to this idea, two indicators can be provided as follows:

i = 1, 2, …, N

(10)

i = 1, 2, …, N

(11)

where and indicate the sample kurtosis and skewness, respectively, at the jth decomposition level. N denotes the coefficient length. and denote the mean value and variance of the absolute value of wavelet coefficients series mj, respectively. The case that the and become larger means that most of the coefficients are mostly equal to zero and some coefficients keep most of the information. For two random variables with the same statistical characteristics, their pdfs of wavelet coefficients at the same decomposition level are also similar. Before denoising, we can firstly select the ideal decomposition level by means of prior information. 2.5 Denoising evaluation Generally, the denoising effect is estimated by the signal-to-noise ratio (SNR) of the denoised time series. The SNR can be denoted as follows: (12) If the SNR value is larger, the original signal contains less noise. It means that if the SNR value is larger than the real SNR value, noise will be considered as the effective information; if the SNR value is smaller than the real SNR value, the effective information will be considered as the noise. Both of them will cause insufficient or excessive noise reduction. The reasonable evaluation should be that the closer the SNR value after the denoising is to the real SNR value, the better the denoising effect. But the real SNR value is unknown. This falls into a logical circle. We should adopt any reasonable methods to judge the denoising effect from multi-factors. For example, the physical phenomena possessed by the studied structure may be known. If the physical phenomena reflected by the denoising data can be explained, it indicates that the denoising effect is good and the denoising method is feasible. Fig.1 shows the denoising procedure. It has the following parts: (1) DWPT; (2) selecting the reasonable decomposition level; (3) getting the estimation values of wavelet coefficients; (4) Inverse DWPT; (5) denoising evaluation. In step 2, when and reach the peak value for the first time, the further decomposition can be stopped.

Signal with noise

Step 1: get the and of each decomposition level after DWPT (using Eq. (10) and (11)) Step 2: select the decomposition level and so that and have the largest values, respectively.

Step 3:

No

? Yes

Step 4: get estimation (using Eq.(9))

Get estimation and , respectively (step 4); reconstruct two sets of signals (step 5)

at the level Step 6: compare the SNRs of the two sets of signals; Select the signal whose SNR is larger (using Eq. (12)).

Step 5: inverse DWPT (using Eq.(7))

Denoising signal Fig.1 The denoising procedure of the proposed criterion 3. Simulation evaluation of mechanical vibration signal To compare the validity of the proposed Bayesian wavelet packet method, a simulation signal is analyzed. It consists of two harmonic frequencies modulating an exponentially attenuated pulse as follows: (13) where = 800 Hz, = mod(t,1/fm), fm = 100 Hz, carrier frequency f1 = 3000 Hz, f2 = 8000 Hz, and T = 1/25000 s. The sampling data with a cycle time (8 ms) are used, as shown in Fig. 2(a). Its corresponding frequency spectrum by Fourier transform is plotted in Fig. 2(b).

(a)

t(ms) (b)

f(Hz)

Fig. 2 (a) Simulation of pulse signal; (b) corresponding frequency spectrum 3.1 Prior information The pdf of the mechanical vibration signal can be characterized with a peak around point zero. Hyvarinen [21] provides the following mathematical expression for such a pdf. (14) where d denotes the standard deviation of the wanted non-Gaussian data and can control the sparseness of the pdf. With the parameter increasing, the pdf becomes sparser. In Fig. 2, the standard deviation d of the impulse is equal to 0.3. Its pdf can be plotted in Fig. 3. Besides, the pdfs with d = 0.3 and = 0.04, 0.05 and 0.06 are also plotted for comparison. The standard deviations between the pdf of the impulse and the sparse pdf are respectively equal to 0.2911, 0.2803 and 0.2829. Obviously, the one corresponding to = 0.05 is very close to the pdf of the impulse.

Fig. 3 Comparison between the pdf of the impulse and the sparse distributions Gaussian white noise is added into this impulse, as shown in Fig. 4(a). The mean and standard deviation of the noise are zero and 0.5, respectively. Obviously, the noise data completely cover the time-domain vibration characteristics. Fig. 4(b) shows the corresponding frequency spectrum. It is difficult to clearly confirm the carrier frequency. According to Eq. (12), the SNR for the generated signal is −0.75 dB.

Amplitude

(a)

t(ms) (b)

f(Hz)

Fig. 4 (a) An impulse with very heavy Gaussian noise (SNR = −0.75 dB); (b) corresponding frequency spectrum 3.2 Decomposition level If the basic wavelet is similar to the signal to be identified, the valid information of the signal would concentrate on fewer coefficients. Hence, the Db4 wavelet is chosen as the mother wavelet. The first step in choosing the decompression level is to determine the pdf value around zero. Fig. 5(a) shows the pdf curve of wavelet decomposition coefficients of the pulse signal at different decomposition levels. In the fifth decomposition level, the wavelet coefficients are mostly concentrated in the vicinity of zero. In Fig. 5(b), the pdf curve of wavelet decomposition coefficients of white noise N(0,0.5) is plotted. It shows that the pdf around zero becomes higher with the decomposition level increasing. But the maximum of the pdf in Fig. 5(b) is much smaller than the one in Fig. 5(a) at each level. This indicates that the wavelet decomposition coefficients of the pulse signal demonstrate more obvious cluster characteristics than the ones of white noise. The cluster characteristics of their mixture should also be dominated by the pulse signal. According to Eq. (10) and Eq. (11), the K and S values of data series in Fig. 4(a) can be obtained and shown in Fig. 6(a) and (b) at different decomposition levels. When the level is equal to the fourth, the K and S values reach the maximum values. This shows that the cluster characteristics of the mixture cannot be determined by any single signal, but by their coupling effect. Therefore, the proposed decomposition level is the fourth one. (a)

(b)

Fig. 5 Comparison between the pdf of the wavelet coefficients of (a) the impulse; (b) the white noise N(0,0.5)

(b)

S

K

(a)

Decomposition level

Decomposition level

Fig. 6 (a) K; (b) S values at different decomposition levels 3.3 Denoising evaluation a) Comparison between different methods To extract the characteristic frequency, the Bayesian DWPT denoising approach with the mixture prior distributions of the Bernoulli and Gaussian distributions [8] and the traditional MMSE approach (according to Eq. (6)) are firstly used to process the impulse signal with noise. The probability of independent Bernoulli distribution of the mixture prior distributions is 0.4 with the standard deviation of Gaussian distribution equal to the maximum value of the wavelet coefficients at the corresponding level. And the prior distributions of the traditional MMSE approach are given according to Eq. (14) with d = 0.3 and = 0.15. Fig. 7(b) and Fig. 8(b) show the characteristic frequencies extracted from the impulse signal with noise. However, the denoised signal is still strongly interfered. And the signal shape after noise reduction is completely different from the shape of the original signal. Moreover, the output SNR using the traditional MMSE is −2.92 dB, far smaller than the real SNR −0.75 dB. For the Bayesian approach with the mixture prior distributions, the unsatisfactory performance can also be explained by the fact that the prior probability distribution cannot accurately describe non-Gaussian vibration signals.

(a)

t(ms) (b)

f(Hz) Fig.7 Using Bayesian DWPT denoising approach with the mixture prior distributions of the

Bernoulli and Gaussian distributions (a) the denoising signal (SNR = −1.19 dB); (b) corresponding frequency spectrum

(a)

t(ms)

(b)

f(Hz) Fig.8 Using the traditional MMSE (a) the denoising signal (SNR = −2.92 dB); (b) corresponding frequency spectrum The same impulse signal with noise is denoised by the proposed Bayesian DWPT denoising approach with the same prior distribution as the traditional MMSE approach used. Fig. 9 shows the denoised result. The characteristic frequencies dominate the whole frequency spectrum in Fig. 8(b) and are easily identified. The signal shape after noise reduction is close to the shape of the original signal. The SNR is equal to −0.98 dB, which is also close to the real SNR −0.75 dB. Comparing the final three denoising results, the proposed method is able to obtain better shape and indentify the characteristic frequency easily.

(a)

t(ms) (b)

f(Hz) Fig. 9 Using the proposed Bayesian DWPT denoising approach at the fourth decomposition level (a) the denoising signal (SNR = −0.98 dB); (b) corresponding frequency spectrum b) Comparison between different decomposition levels

As a comparison, Fig. 10 and Fig. 11 show the denoising effects at the third and fifth decomposition levels. The result shows that their SNRs (−1.56 dB and −1.28 dB, respectively) are lower than the SNR (−0.98 dB) at the fourth decomposition level. Their time-domain waveforms are not closer to real waveforms than at the fourth decomposition level. In Fig. 11(b), the characteristic frequency can be indentified easily. This is because the pulse signal has the maximum kurtosis at the fifth level. But in practice, we can only sample the coupling information. The proposed decomposition level has maximized the differences between the wavelet coefficients of the real values and noise as much as possible. And we have achieved the satisfactory denoising result.

(a)

t(ms) (b)

f(Hz) Fig. 10 Using the proposed Bayesian DWPT denoising approach at the third decomposition level (a) the denoising signal (SNR = −1.56 dB); (b) corresponding frequency spectrum

(a)

t(ms) (b)

f(Hz) Fig. 11 Using the proposed Bayesian DWPT denoising approach at the fifth decomposition level (a) the denoising signal (SNR = −1.28 dB); (b) corresponding frequency spectrum 4. State identification of the HSR track fastener

A hammer experiment is taken to study the track vibration characteristics under different fastener fastening states, as shown in Fig. 12. The noise sources result from the high track temperature and track electric current, which can seriously interfere with the acceleration sensor work. The hammering frequency is 0.1 Hz, and the interval between two consecutive hammerings is 10 s. The acceleration sensor is mounted on the track at measuring point to obtain the vertical vibration data. The sampling frequency is 10.24 kHz. When the fastener torque is 0 or 120 N∙m, the fastener is respectively in a loose or tight status [22]. When the test fastener is in the tight state with track temperature 10 ℃ and without electromagnetic interference, the original vibration signal is obtained from the measuring point, as shown in Fig. 13(a). Its corresponding frequency spectrum is shown in Fig. 13(b). When doing the hammer experiment with electromagnetic interference and track temperature 70 ℃, the test fasteners are in the loose state with the other fasteners in a tight state. The vertical acceleration signal of the track at the measuring point is shown in Fig. 14(a). Its frequency spectrum is displayed in Fig. 14(b). The characteristic frequency of the loose fastener cannot be easily indentified from the frequency spectrum because of heavy noise.

(a) Hammer point Test fastener Measuring point

(b)

(c)

Fig. 12 Identification experiment of track fastener status (a) relative position map; (b) test equipment collecting data; (c) rail comprehensive experimental platform for HSR

(a)

t(s)

(b)

f(Hz) Fig. 13 (a) Original vibration signal without noise; (b) corresponding frequency spectrum from the

measuring point with test fastener in tight state Since the rail is firmly restrained by the fasteners, rail vibration energy will soon weaken until becoming zero. This means that some acceleration signal value should be equal to zero. In Fig. 14(a), the real vibration value should be around zero in the interval between 0 and 0.03 s. Because of the noise, the absolute values of the sampling values are larger than zero. Therefore, the value in the interval [0, 0.03] can be considered as the white noise. The mean and variance of the noise can also be estimated as zero and 2g, respectively.

(a)

t(s)

(b)

f(Hz) Fig. 14 (a) Original vibration signal with noise; (b) corresponding frequency spectrum from the measuring point with test fastener in loose state 4.1 Comparison between different methods The proposed method is used to remove noise and highlight the characteristic frequency from the signal. The output denoising result with the optimal parameters d = 2.19, = 0.09 and the fourth decomposition level is shown in Fig. 15 and Fig. 16. Meanwhile, the two popular methods

are used again for comparison. The probability of independent Bernoulli distribution is 0.4, the standard deviation of Gaussian distribution is equal to the maximum value of the wavelet coefficients at each level, and the decomposition level is the seventh. The parameters of the traditional MMSE are the same with the aforementioned proposed method. The denoising result using the proposed approach shows that the vibration value in the interval between 0 and 0.02 s is around zero. It reveals that most of the noise should be removed. The characteristic frequency induced from the loose fastener is 330 Hz from Fig. 15(b) and Fig. 16(b). Using the two popular approaches, the results show that strong interference still exists there. In Fig. 15(a), the vibration values in the interval between 0 and 0.02 s do not fall near zero after the denoising. In Fig. 16(b), we cannot easily extract the characteristic frequency. The comparison results show that the proposed method shows better ability to regain the time-domain waveform and extract the hidden characteristics embedded in the vibration signals than the known two methods.

(a)

t(s) 330Hz (b)

f(Hz) Fig. 15 Using the proposed Bayesian DWPT denoising approach and Bayesian DWPT denoising approach with the mixture prior distributions of the Bernoulli and Gaussian distributions (a) the denoising signal (proposed approach with SNR = −0.14 dB and mixture prior distributions approach with SNR = 3.37 dB); (b) corresponding frequency spectrum

(a)

t(s) 330Hz

(b)

f(Hz) Fig. 16 Using the proposed Bayesian DWPT denoising approach and the traditional MMSE (a) the denoising signal (proposed approach with SNR = −0.14 dB and the traditional MMSE with SNR =

−0.61 dB); (b) corresponding frequency spectrum 4.2 Comparison between different decomposition levels According to Eq. (10) and Eq. (11), the proposed decomposition level can be programmatically selected. For the data in Fig. 14(a), the selected level is the fourth one. As a comparison, we also show the denoising effect at the third and fifth level in Fig. 17 and Fig. 18. The SNRs of the denoising signal are equal to −0.22 dB, which is less than the SNR 0.14 dB at the fourth level. Fig. 17(a) and Fig. 18(a) also demonstrate that some noise data are kept as the useful information. These show that the fourth level is the best level.

(a)

t(s)

(b)

f(Hz) Fig. 17 Comparison between the third and fourth decomposition levels (a) the denoising signal (level 3 with SNR = −0.22 dB and level 4 with SNR = −0.14 dB); (b) corresponding frequency

spectrum

(a)

t(s) (b)

f(Hz) Fig.18 Comparison between the fourth and fifth decomposition levels (a) the denoising signal (level 4 with SNR = −0.14 dB and level 5 with SNR = −0.22 dB); (b) corresponding frequency spectrum 5. Conclusion In this research, a novel denoising criterion about the Bayesian wavelet packet approach is developed for the weak information classification of mechanical equipment monitoring. It refers to the prior probability distribution, decomposition level and evaluation of denoising result. At first, the proposed denoising method is validated by a set of simulated noisy data. Then, the robustness of the proposed method is demonstrated by the experimental data from a hammering test of the HSR track fastener. In both the simulation and the engineering application, two existing denoising methods are employed as the comparison objects to highlight the better denoised effect of the proposed method. Several comparison results from the numerical examples have showed that the proposed denoising method performs better than the other two methods. This may be due to the following two reasons. (1) The proposed prior distributions can better describe the characteristics of the real signal. (2) The choice of the reasonable decomposition level makes full use of the clustering characteristics of wavelet coefficients, which can separate the wavelet coefficients of real signal and noise as much as possible. This proposed criterion has advantages as follows: (1) if needed, any prior distribution model can be adopted; and (2) the process of selecting the decomposition level can be completed by a computer program out of a person’s control. Therefore, the proposed methodology can be employed to effectively remove the noise of the measurements for further analysis of mechanical equipment monitoring. Acknowledgements The authors gratefully acknowledge support from Natural Science Foundation of China (NSFC, Grant No. 51705340), Natural Fund in Liaoning Province (No. 20170540745). References

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The denoising criterion is developed for the signal with weak information The denoising criterion develops a general Bayesian wavelet packet denoising method. The denoising criterion provides two indexes to select the decomposition level The denoising criterion make the denoising procedure automated without any manual intervention.