Proposed Probability Density Function Research Articles

A probability density function of stress amplitude applicable to power spectral density curves of structural random vibration response

The power spectral density curves of the random vibration response of a plate specimen with different spectral width coefficients were obtained by selecting different frequency ranges, and the probability distributions of the stress amplitudes were obtained by applying Monte Carlo method and rainflow counting method. Each probability distribution was divided into three segments by the corresponding zero-order spectral moment value, and each segment was fitted using either a Rayleigh function or an exponential function. Finally, a probability density function of stress amplitude was proposed by combining two Rayleigh functions and an exponential function. The proposed probability density function can be used to obtain the stress amplitude probability distribution for the power spectral density curves of structural random vibration response with any bandwidth, and the accuracy of the proposed function was verified based on the response data of a specimen of 7075-T651 aluminum alloy material under the action of six typical power spectral curves.

Asteroid lightcurve inversion with Bayesian inference

Context.We assess statistical inversion of asteroid rotation periods, pole orientations, shapes, and phase curve parameters from photometric lightcurve observations, here sparse data from the ESAGaiaspace mission (Data Release 2) or dense and sparse data from ground-based observing programs.Aims.Assuming general convex shapes, we develop inverse methods for characterizing the Bayesian a posteriori probability density of the parameters (unknowns). We consider both random and systematic uncertainties (errors) in the observations, and assign weights to the observations with the help of Bayesian a priori probability densities.Methods.For general convex shapes comprising large numbers of parameters, we developed a Markov-chain Monte Carlo sampler (MCMC) with a novel proposal probability density function based on the simulation of virtual observations giving rise to virtual least-squares solutions. We utilized these least-squares solutions to construct a proposal probability density for MCMC sampling. For inverse methods involving triaxial ellipsoids, we update the uncertainty model for the observations.Results.We demonstrate the utilization of the inverse methods for three asteroids withGaiaphotometry from Data Release 2: (21) Lutetia, (26) Proserpina, and (585) Bilkis. First, we validated the convex inverse methods using the combined ground-based andGaiadata for Lutetia, arriving at rotation and shape models in agreement with those derived with the help of Rosetta space mission data. Second, we applied the convex inverse methods to Proserpina and Bilkis, illustrating the potential of theGaiaphotometry for setting constraints on asteroid light scattering as a function of the phase angle (the Sun-object-observer angle). Third, with the help of triaxial ellipsoid inversion as applied toGaiaphotometry only, we provide additional proof that the absoluteGaiaphotometry alone can yield meaningful photometric slope parameters. Fourth, for (585) Bilkis, we report, with 1-σuncertainties, a refined rotation period of (8.5750559 ± 0.0000026) h, pole longitude of 320.6° ± 1.2°, pole latitude of − 25.6° ± 1.7°, and the first shape model and its uncertainties from convex inversion.Conclusions.We conclude that the inverse methods provide realistic uncertainty estimators for the lightcurve inversion problem and that theGaiaphotometry can provide an asteroid taxonomy based on the phase curves.

Gearbox fault diagnosis using RMS based probability density function and entropy measures for fluctuating speed conditions

Fault diagnosis of gearbox which operates on low rotating speed with high fluctuations is highly important because its ignorance can led to a catastrophe. The uncertainty within the vibration signal of the gearbox can be identified by the entropy measures, on the basis of probability density function of a signal. But, under fluctuating speeds, entropies may show insignificant results, hence making them non-reliable. The aim of this article is to develop a reliable and stable technique for gear fault detection under such fluctuating speeds. Therefore, a root mean square–based probability density function is proposed to improve the efficiency of entropy measures. The fault detection capabilities of proposed technique were demonstrated experimentally. Various entropy measures, namely, Shannon entropy, Rényi entropy, approximate entropy, and sample entropy, were compared as well as evaluated for both Gaussian and proposed probability density function. The proposed technique was further validated using two condition indicators based on amplitude of probability density function. Results suggest the effective fault diagnosis using proposed method.

Application of the transitional Markov chain Monte Carlo algorithm to probabilistic site characterization

This paper applies the transitional Markov chain Monte Carlo (TMCMC) algorithm to probabilistic site characterization problems. The purpose is to characterize the statistical uncertainties in the spatial variability parameters based on the cone penetration test (CPT) dataset. The spatial variability parameters of interest include the trend function, standard deviation and scale of fluctuation for the spatial variability, and so on. In contrast to the Metropolis–Hastings (MH) algorithm, the TMCMC algorithm is a tune-free algorithm: it does not require the specification of the proposal probability density function (PDF), hence there is no need to tune the proposal PDF. Also, there is no burn-in period to worry about, and the convergence issue is mild for TMCMC because the samples spread widely. Moreover, it can estimate the model evidence, a quantity essential for Bayesian model comparison, without extra computation cost. The effectiveness for the TMCMC algorithm is demonstrated through simulated examples and a real case study.

Multi-events earthquake early warning algorithm using a Bayesian approach

Current earthquake early warning (EEW) systems lack the ability to appropriately handle multiple concurrent earthquakes, which led to many false alarms during the 2011 Tohoku earthquake sequence in Japan. This paper uses a Bayesian probabilistic approach to handle multiple concurrent events for EEW. We implement the theory using a two-step algorithm. First, an efficient approximate Bayesian model class selection scheme is used to estimate the number of concurrent events. Then, the Rao-Blackwellized Importance Sampling method with a sequential proposal probability density function is used to estimate the earthquake parameters, that is hypocentre location, origin time, magnitude and local seismic intensity. A real data example based on 2 months data (2011 March 9–April 30) around the time of the 2011 M9 Tohoku earthquake is studied to verify the proposed algorithm. Our algorithm results in over 90 per cent reduction in the number of incorrect warnings compared to the existing EEW system operating in Japan.

Probabilistic identification of simulated damage on the Dowling Hall footbridge through Bayesian finite element model updating

Summary This paper presents a probabilistic damage identification study on a full-scale structure, the Dowling Hall footbridge, through a Bayesian finite element (FE) model updating. The footbridge is located at Tufts University and is equipped with a continuous monitoring system that measures its ambient acceleration response. A set of data is recorded once every hour or when triggered by large vibrations. The modal parameters of the footbridge are extracted from each set of data and are used for FE model updating. In this study, effects of physical damage are simulated by loading a small segment of the footbridge deck with concrete blocks. The footbridge deck is divided into five segments in an FE model of the test structure, and the added mass on each segment is considered as an updating parameter. Overall, 72 sets of data are collected during the loading period, and different subsets of these data are used to find the location and extent of the damage (added mass). The adaptive Metropolis–Hastings algorithm with adaption on the proposal probability density function is successfully used to generate Markov Chains for sampling the posterior probability distributions of the five updating parameters. Effects of the number of data sets used in the identification process are investigated on the posterior probability distributions of the updating parameters. The probabilistic model updating framework accurately predicts the simulated damage and the level of confidence on the obtained results. The maximum a-posteriori estimates of damage in the probabilistic approach are found to be in good agreement with their corresponding deterministic counterparts. Copyright © 2014 John Wiley & Sons, Ltd.

Interference of Cognitive Wireless Networks on Rayleigh and Rice Fading Channels

We propose a mathematical interference model of cognitive wireless networks based on two channels suffering different fading. The primary channel between primary transmitter and primary receiver is under Rayleigh fading. The secondary interference channel between secondary transmitter and primary receiver is under Rice fading. By employing various factors, such as the spectrum sensing scheme and the spatial distribution of the secondary users, the proposed model gives an accurate expression of interference probability compared to the statistical models that just approximate the probability model of the interference or bounds models that aim at finding the bounds of the interference. The proposed probability density function (PDF), cumulative distribution function (CDF), and the mean and variance of interference can be extended to a number of applications including power control, spectrum sensing evaluation, secondary users density analysis, and setting interference toleration threshold. Moreover, the framework developed in this paper can be applied to the interference analysis for the nodes of cognitive networks distributed in irregular geographical shapes like finite or infinite zone. The simulation result verifies our theory and is in agreement with the expectation.

Probability density function method for Langevin equations with colored noise.

Understanding the mesoscopic behavior of dynamical systems described by Langevin equations with colored noise is a fundamental challenge in a variety of fields. We propose a new approach to derive closed-form equations for joint and marginal probability density functions of state variables. This approach is based on a so-called large-eddy-diffusivity closure and can be used to model a wide class of non-Markovian processes described by the noise with an arbitrary correlation function. We demonstrate the accuracy of the proposed probability density function method for several linear and nonlinear Langevin equations.

A generalization of the adaptive rejection sampling algorithm

Rejection sampling is a well-known method to generate random samples from arbitrary target probability distributions. It demands the design of a suitable proposal probability density function (pdf) from which candidate samples can be drawn. These samples are either accepted or rejected depending on a test involving the ratio of the target and proposal densities. The adaptive rejection sampling method is an efficient algorithm to sample from a log-concave target density, that attains high acceptance rates by improving the proposal density whenever a sample is rejected. In this paper we introduce a generalized adaptive rejection sampling procedure that can be applied with a broad class of target probability distributions, possibly non-log-concave and exhibiting multiple modes. The proposed technique yields a sequence of proposal densities that converge toward the target pdf, thus achieving very high acceptance rates. We provide a simple numerical example to illustrate the basic use of the proposed technique, together with a more elaborate positioning application using real data.

Probabilistic Evaluation of the Moment Resisting Frame Under Earthquake Loading

A Monte Carlo simulation for the failure criterion, which depends on the design capacity and the demand, is developed for the moment resisting frame subjected to earthquake loading. The probabilistic reliability has been modelled using a newly proposed probability density function [1]. Inelastic rotation is the main focus of the Monte Carlo simulation and is selected as the random variable for the Monte Carlo simulation. The elastoplastic model is employed to represent the inelastic behaviour of each member in the structure. While using the Monte Carlo simulation, it has been shown that probabilistic simulations of the existing earthquake records are needed to predict accurately the earthquake activities at a specified location. Assuming that the building of which this frame is a part has a useful life of 50 years, failure analysis based on the probabilistic approach is performed to predict the failure of the moment resisting frame for the next 50 years.

Ocean microwave backscatter distributions

A probability density function, p(a), for the instantaneous amplitude of the backscattered field at moderate incidence angles is proposed. The key step is in writing the (slowly varying) backscatter cross section, σ0, as an exponential function to second order in the long wave slope component, sx, along the scatterometer look direction; that is, σ0 ∼ exp (a1sx + a2sx2). The function p(σ0) is then computed by a straightforward transformation of the normally distributed slope component, while p(a) follows from Bayes' theorem assuming p(a|σ0) to be Rayleigh distributed. The proposed probability density function was compared to Ku band dual‐polarized data taken at 20° and 45° incidence during the Synthetic Aperture and X Band Ocean Nonlinearities experiments at the Chesapeake Light Tower and at Forschungsplatform Nordsee. The function p(a) was found to follow a Weibull distribution with slope parameter v ≅ 1.9 for small amplitudes. At large amplitudes the distribution lengthens but appears to be neither Weibull nor K distributed as has been reported for neargrazing incidence data.