Some Improvements on a General Particle Filter Based Bayesian Approach for Extracting Bearing Fault Features

2015 ◽  
Vol 137 (4) ◽  
Author(s):  
Dong Wang ◽  
Qiang Miao
Keyword(s):  
Random Walk ◽  
Probability Density ◽  
Posterior Probability ◽  
Bayesian Method ◽  
Probability Density Functions ◽  
Density Functions ◽  
Bearing Fault ◽  
Fault Features ◽  
Posterior Probability Density ◽  
Gaussian Random Walk

In our previous work, a general particle filter based Bayesian method was proposed to derive the graphical relationship between wavelet parameters, including center frequency and bandwidth, and to posteriorly find optimal wavelet parameters so as to extract bearing fault features. In this work, some improvements on the previous Bayesian method are proposed. First, the previous Bayesian method strongly depended on an initial uniform distribution to generate random particles. Here, a random particle represented a potential solution to optimize wavelet parameters. Once the random particles were obtained, the previous Bayesian method could not generate new random particles. To solve this problem, this paper introduces Gaussian random walk to joint posterior probability density functions of wavelet parameters so that new random particles can be generated from Gaussian random walk to improve optimization of wavelet parameters. Besides, Gaussian random walk is automatically initialized by the famous fast kurtogram. Second, the previous work used the random particles generated from the initial uniform distribution to generate measurements. Because the random particles used in the previous work were fixed, the measurements were also fixed. To solve this problem, the first measurement used in this paper is provided by the fast kurtogram, and its linear extrapolations are used to generate monotonically increasing measurements. With the monotonically increasing measurements, optimization of wavelet parameters is further improved. At last, because Gaussian random walk is able to generate new random particles from joint posterior probability density functions of wavelet parameters, the number of the random particles is not necessarily set to a high value that was used in the previous work. Two instance studies were investigated to illustrate how the Gaussian random walk based Bayesian method works. Comparisons with the famous fast kurtogram were conducted to demonstrate that the Gaussian random walk based Bayesian method can better extract bearing fault features.

Fast Bayesian Inference on Optimal Wavelet Parameters for Bearing Fault Diagnosis

2015 ◽  
Author(s):  
Qiang Miao ◽  
Dong Wang
Keyword(s):  
Wavelet Transform ◽  
Bayesian Inference ◽  
Fault Diagnosis ◽  
Posterior Probability ◽  
Low Frequency ◽  
Probability Density Functions ◽  
Bearing Fault ◽  
Bearing Fault Diagnosis ◽  
Low Frequency Vibration ◽  
Posterior Probability Density

Rolling element bearings are widely used in various machinery to support rotation shaft and they are prone to failures. Once a bearing fails, it accelerates failures of other adjacent components and results in unexpected machine breakdown. To prevent machine breakdown and reduce unnecessary economic loss, bearing fault must be detected as early as possible. Besides spectral kurtosis, empirical mode decomposition, cyclostationarity, etc., wavelet transform has proven to be an effective method for identification of different bearing faults because it aims to highlight the inner product between an artificial wavelet function and a signal to be analyzed. In the application of wavelet transform, optimization of wavelet parameters attracts much attention because proper selection of wavelet parameters can maximize performance of wavelet transform and extract impulses caused by bearing faults in the case of interruption from other strong low-frequency vibration components and heavy noises. Compared with other optimization methods, such as genetic algorithm, particle swarm optimization, etc., an analytic and fast Bayesian inference on optimal wavelet parameters for an optimal wavelet filtering for bearing fault diagnosis is proposed in this paper. Prior to Bayesian inference, a state space model of wavelet parameters should be constructed to reflect the relationship between wavelet parameters and measurements. Here, measurements are monotonically increasing kurtosis values, which are able to quantify bearing fault signals. The first kurtosis value and initial wavelet parameters are provided by the fast kurtogram, which is a fast algorithm that can be used to locate one of resonant frequency bands for further demodulation with envelope analysis. For other measurements, they are generated by artificial extrapolations of the first kurtosis value. To iteratively infer posterior probability density functions of wavelet parameters and track the artificial measurements, an unscented transform based Bayesian method is introduced. As the iteration number increases, posterior probability density functions of wavelet parameters converge. Then, the optimal wavelet parameters can be found to conduct an optimal wavelet filtering so as to isolate bearing fault signals from other strong low-frequency vibration components. At last, squared envelope analysis and Fourier transform are utilized to demodulate bearing fault signals enhanced by the proposed method and to identify bearing fault characteristic frequencies, respectively. One real case study is used to illustrate how the proposed method works and to demonstrate that the proposed method can be effectively and efficiently used to extract bearing fault signatures. Additionally, a comparison with the fast kurtogram is conducted to show the proposed method is better than the fast kurtogram for bearing fault diagnosis.

A Bayesian Framework of Computing and Updating Wavelet Parameter Distributions for Identifying Early Bearing Faults

2014 ◽  
Author(s):  
Peter W. Tse ◽  
Dong Wang
Keyword(s):  
Probability Density Function ◽  
Wavelet Analysis ◽  
Probability Density ◽  
Density Function ◽  
Posterior Probability ◽  
Fault Features ◽  
Bearing Failures ◽  
Posterior Probability Density ◽  
Posterior Probability Density Function ◽  
Parameter Distributions

Rolling element bearings are widely used in machines to support rotation shafts. Bearing failures may result in machine breakdown. In order to prevent bearing failures, early bearing faults are required to be identified. Wavelet analysis has proven to be an effective method for extracting early bearing fault features. Proper selection of wavelet parameters is crucial to wavelet analysis. In this paper, a Bayesian framework is proposed to compute and update wavelet parameter distributions. First, a smoothness index is used as the objective function because it has specific upper and lower bounds. Second, a general sequential Monte Carlo method is introduced to analytically derive the joint posterior probability density function of wavelet parameters. Last, approximately optimal wavelet parameters are inferred from the joint posterior probability density function. Simulated and real case studies are investigated to demonstrate that the proposed framework is effective in extracting early bearing fault features.

scholarly journals Bayesian model checking: A comparison of tests

2018 ◽  
Vol 614 ◽  
pp. A25 ◽  
Author(s):  
L. B. Lucy
Keyword(s):  
Bayesian Model ◽  
Test Problem ◽  
Goodness Of Fit ◽  
Posterior Probability ◽  
Probability Density Functions ◽  
Density Functions ◽  
Simple Test ◽  
Hubble Expansion ◽  
Posterior Probability Density ◽  
Comparison Of Tests

Two procedures for checking Bayesian models are compared using a simple test problem based on the local Hubble expansion. Over four orders of magnitude,p-values derived from a global goodness-of-fit criterion for posterior probability density functions agree closely with posterior predictivep-values. The former can therefore serve as an effective proxy for the difficult-to-calculate posterior predictivep-values.

Anticoagulant Control with Chromogenic Measurements of Factors X and VII

1979 ◽  
Author(s):  
A.A. Famodu ◽  
G.I.C. Ingram ◽  
S.C. Darby
Keyword(s):  
Posterior Probability ◽  
Colour Change ◽  
Probability Density Functions ◽  
Factor Vii ◽  
Density Functions ◽  
Similar Data ◽  
Factor X ◽  
Coefficients Of Variation ◽  
Posterior Probability Density ◽  
Residual Errors

Prothrombin time ratios (Manchester reagent) and parallel assays for factor X with Russell’s viper venom by coagulation (charcoal-filtered ox plasma) and chromogenic (S2222, pH8.6) methods were performed on plasma samples from coumarin-treated patients, following strict biometrical designs. Clotting times were read manually and colour change by spectrophotometer. Duplicate reaction tubes were read on each plasma in PTs and on each dilution tested in assays. Residual errors were calculated in logs. to homogenize the variance and expressed as coefficients of variation; the precision of assays was measured as λ=(residual SD)/slope. Readings of the S2222 end points were highly reproducible, but this did not yield better agreement between replicate reactions nor a significantly higher precision in assays . For stabilized patients, similar correlations were obtained between both assays and the PT ratio: for coagulation X, r31 = 0.897; for S2222 X, r36=0.902. Posterior probability density functions have also been calculated. Similar data for factor VII assays in newly anticoagulated and in stabilized patients will also be given.

scholarly journals Modeling Diameter Distributions with Six Probability Density Functions in Pinus halepensis Mill. Plantations Using Low-Density Airborne Laser Scanning Data in Aragón (Northeast Spain)

2021 ◽  
Vol 13 (12) ◽  
pp. 2307
Author(s):  
J. Javier Gorgoso-Varela ◽  
Rafael Alonso Ponce ◽  
Francisco Rodríguez-Puerta
Keyword(s):  
Probability Density ◽  
Linear Models ◽  
Pinus Halepensis ◽  
Beta Function ◽  
Probability Density Functions ◽  
Lidar Data ◽  
Density Functions ◽  
Minimum Diameter ◽  
Location Parameters ◽  
Diameter Distributions

The diameter distributions of trees in 50 temporary sample plots (TSPs) established in Pinus halepensis Mill. stands were recovered from LiDAR metrics by using six probability density functions (PDFs): the Weibull (2P and 3P), Johnson’s SB, beta, generalized beta and gamma-2P functions. The parameters were recovered from the first and the second moments of the distributions (mean and variance, respectively) by using parameter recovery models (PRM). Linear models were used to predict both moments from LiDAR data. In recovering the functions, the location parameters of the distributions were predetermined as the minimum diameter inventoried, and scale parameters were established as the maximum diameters predicted from LiDAR metrics. The Kolmogorov–Smirnov (KS) statistic (Dn), number of acceptances by the KS test, the Cramér von Misses (W2) statistic, bias and mean square error (MSE) were used to evaluate the goodness of fits. The fits for the six recovered functions were compared with the fits to all measured data from 58 TSPs (LiDAR metrics could only be extracted from 50 of the plots). In the fitting phase, the location parameters were fixed at a suitable value determined according to the forestry literature (0.75·dmin). The linear models used to recover the two moments of the distributions and the maximum diameters determined from LiDAR data were accurate, with R2 values of 0.750, 0.724 and 0.873 for dg, dmed and dmax. Reasonable results were obtained with all six recovered functions. The goodness-of-fit statistics indicated that the beta function was the most accurate, followed by the generalized beta function. The Weibull-3P function provided the poorest fits and the Weibull-2P and Johnson’s SB also yielded poor fits to the data.

scholarly journals A synergy of the velocity gradients technique and the probability density functions for identifying gravitational collapse in self-absorbing media

2021 ◽  
Vol 502 (2) ◽  
pp. 1768-1784
Author(s):  
Yue Hu ◽  
A Lazarian
Keyword(s):  
Magnetic Fields ◽  
Probability Density ◽  
Column Density ◽  
Mass Density ◽  
Probability Density Functions ◽  
Emission Lines ◽  
Density Functions ◽  
Star Forming ◽  
Velocity Gradients ◽  
Self Gravity

ABSTRACT The velocity gradients technique (VGT) and the probability density functions (PDFs) of mass density are tools to study turbulence, magnetic fields, and self-gravity in molecular clouds. However, self-absorption can significantly make the observed intensity different from the column density structures. In this work, we study the effects of self-absorption on the VGT and the intensity PDFs utilizing three synthetic emission lines of CO isotopologues 12CO (1–0), 13CO (1–0), and C18O (1–0). We confirm that the performance of VGT is insensitive to the radiative transfer effect. We numerically show the possibility of constructing 3D magnetic fields tomography through VGT. We find that the intensity PDFs change their shape from the pure lognormal to a distribution that exhibits a power-law tail depending on the optical depth for supersonic turbulence. We conclude the change of CO isotopologues’ intensity PDFs can be independent of self-gravity, which makes the intensity PDFs less reliable in identifying gravitational collapsing regions. We compute the intensity PDFs for a star-forming region NGC 1333 and find the change of intensity PDFs in observation agrees with our numerical results. The synergy of VGT and the column density PDFs confirms that the self-gravitating gas occupies a large volume in NGC 1333.

scholarly journals Interactive design of probability density functions for shape grammars

2015 ◽  
Vol 34 (6) ◽  
pp. 1-13 ◽  
Author(s):  
Minh Dang ◽  
Stefan Lienhard ◽  
Duygu Ceylan ◽  
Boris Neubert ◽  
Peter Wonka ◽  
...  
Keyword(s):  
Probability Density ◽  
Probability Density Functions ◽  
Interactive Design ◽  
Density Functions ◽  
Shape Grammars
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