The Predictive Posterior Probability Density Function for the Rectangular Probability Model and its Application to EMC and RF Measurements

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.

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Inverse probability and sufficient statistics

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100024750 ◽

1949 ◽

Vol 45 (2) ◽

pp. 225-229

Author(s):

V. S. Huzurbazar

Keyword(s):

Probability Density Function ◽

Probability Density ◽

Density Function ◽

Statistical Estimation ◽

Sufficient Statistics ◽

Inverse Probability ◽

General Terms ◽

Fiducial Probability ◽

Posterior Probability Density ◽

Posterior Probability Density Function

1. It is an interesting fact that in many problems of statistical estimation the results given by the theory of inverse probability (as modified by Jeffreys) are indistinguishable from those given by the methods of ‘fiducial probability’ or ‘confidence intervals’. The derivation of some of the important inverse distributions by Jeffreys(3) arouses one's curiosity. It seems that when this agreement is noticed there are usually sufficient statistics for parameters in the distribution. The object of this note is to throw some light, in general terms, on the similarity in form between the posterior probability-density function of the parameters and the probability-density function of the distribution when it admits sufficient statistics. For convenience the following notation in Jeffreys's probability logic is used below:P(q | p) is the probability of a proposition q on data p.

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Probability Model of Beijing Residentials’ Fire Load

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.368-373.993 ◽

2011 ◽

Vol 368-373 ◽

pp. 993-1002 ◽

Cited By ~ 1

Author(s):

Yan Chao Liu ◽

Dong Dong Liu ◽

Jin Ping Wang ◽

Wei Hong Chen ◽

Bin Zhao

Keyword(s):

Probability Density Function ◽

Probability Density ◽

Density Function ◽

Probability Model ◽

Estimation Method ◽

Extreme Value Distribution ◽

Likelihood Estimation ◽

Generalized Extreme Value Distribution ◽

Fire Load ◽

Distribution Models

We analyses 418 of fire load data of Beijing residential in city subdivision and suburban district which collected by Beijing University of Civil Engineering and Architecture and Institute of Building Fire Research. Suppose the probability density function of several fire load, using maximum likelihood estimation method to obtain the parameters, and use the K-S test examine the probability density function model, the final selections of Generalized extreme value distribution and Log logistic fit better as a Beijing residential fire load distribution probability distribution models. Finally using these models, according to the JCSS rules, the fire load standard value of Beijing residential is put forward.

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An FEM-Based State Estimation Approach to Nonlinear Hybrid Positioning Systems

Mathematical Problems in Engineering ◽

10.1155/2013/175425 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Yu-Xin Zhao ◽

Li-Juan Chen ◽

Yan Ma

Keyword(s):

Probability Density Function ◽

State Estimation ◽

Probability Density ◽

Density Function ◽

Differential Model ◽

Positioning Systems ◽

Forward Equation ◽

Hybrid Positioning ◽

Posterior Probability Density ◽

Low Dimensional

For hybrid positioning systems (HPSs), the estimator design is a crucial and important problem. In this paper, a finite-element-method- (FEM-) based state estimation approach is proposed to HPS. As the weak solution of hybrid stochastic differential model is denoted by the Kolmogorov's forward equation, this paper constructs its interpolating point through the classical fourth-order Runge-Kutta method. Then, it approaches the solution with biquadratic interpolation function to obtain a prior probability density function of the state. A posterior probability density function is gained through Bayesian formula finally. In theory, the proposed scheme has more advantages in the performance of complexity and convergence for low-dimensional systems. By taking an illustrative example, numerical experiment results show that the new state estimator is feasible and has good performance than PF and UKF.

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