Bayesian Markov chain Monte Carlo inversion for anisotropy of PP-and PS-wave in weakly anisotropic and heterogeneous media

A single set of vertically aligned cracks embedded in a purely isotropic background may be con- sidered as a long-wavelength effective transversely iso- tropy （HTI） medium with a horizontal symmetry axis. The crack-induced HTI anisotropy can be characterized by the weakly anisotropic parameters introduced by Thomsen. The seismic scattering theory can be utilized for the inversion for the anisotropic parameters in weakly aniso- tropic and heterogeneous HTI media. Based on the seismic scattering theory, we first derived the linearized PP- and PS-wave reflection coefficients in terms of P- and S-wave impedances, density as well as three anisotropic parameters in HTI media. Then, we proposed a novel Bayesian Mar- kov chain Monte Carlo inversion method of PP- and PS- wave for six elastic and anisotropic parameters directly. Tests on synthetic azimuthal seismic data contaminated by random errors demonstrated that this method appears more accurate, anti-noise and stable owing to the usage of the constrained PS-wave compared with the standards inver- sion scheme taking only the PP-wave into account.

Minimizing transmission loss using inspired ant colony optimization and Markov Chain Monte Carlo in underwater WSN environment

In Underwater Wireless Sensor Networks(UWSNs),the most important challenging issues are propagation delay,high error probability,high latency,high communication cost,limited bandwidth,limited memory,low packet delivery ratio,and transmission loss.In our proposed work,the various efforts are taken to minimize the propagation delay and transmission loss during data transmission in an underwater environment.A hybrid mechanism is implemented to improve energy efficiency for faster data transmission in underwater WSN.In the underwater environment of acoustic channel condition,propagation delay and transmission loss lead to high complexity in accessing the information and also it is difficult to obtain the Channel Status Information(CSI).To address this problem,Ant Colony Optimization(ACO)routing with Markov Chain Monte Carlo(MCMC)algorithm is used and to capture the transmission loss in the MCMC approach,CSI Forecast Prediction(FP)algorithm is used.The experimental simulations are evaluated by utilizing the performance evaluation metrics such as Transmission Loss,Probability Density Function,Average Delay and Throughput.From the simulation results it is evident that the proposed algorithm,ACO-MCMC has recorded the minimum transmission loss,increase in probability density function,minimum average delay and maximum throughput of the network when compared to the existing algorithms.

On Stochastic Error and Computational Efficiency of the Markov Chain Monte Carlo Method

InMarkov ChainMonte Carlo(MCMC)simulations,thermal equilibria quantities are estimated by ensemble average over a sample set containing a large number of correlated samples.These samples are selected in accordance with the probability distribution function,known from the partition function of equilibrium state.As the stochastic error of the simulation results is significant,it is desirable to understand the variance of the estimation by ensemble average,which depends on the sample size(i.e.,the total number of samples in the set)and the sampling interval(i.e.,cycle number between two consecutive samples).Although large sample sizes reduce the variance,they increase the computational cost of the simulation.For a given CPU time,the sample size can be reduced greatly by increasing the sampling interval,while having the corresponding increase in variance be negligible if the original sampling interval is very small.In this work,we report a few general rules that relate the variance with the sample size and the sampling interval.These results are observed and confirmed numerically.These variance rules are derived for theMCMCmethod but are also valid for the correlated samples obtained using other Monte Carlo methods.The main contribution of this work includes the theoretical proof of these numerical observations and the set of assumptions that lead to them.

Multilevel Markov Chain Monte Carlo Method for High-Contrast Single-Phase Flow Problems

In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems.It is based on the generalized multiscale finite element method(GMsFEM)and multilevel Monte Carlo(MLMC)methods.The former provides a hierarchy of approximations of different resolution,whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels.The number of basis functions in the online GMsFEM stage can be varied to determine the solution resolution and the computational cost,and to efficiently generate samples at different levels.In particular,it is cheap to generate samples on coarse grids but with low resolution,and it is expensive to generate samples on fine grids with high accuracy.By suitably choosing the number of samples at different levels,one can leverage the expensive computation in larger fine-grid spaces toward smaller coarse-grid spaces,while retaining the accuracy of the final Monte Carlo estimate.Further,we describe a multilevel Markov chain Monte Carlo method,which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on fine grids,while combining the samples at different levels to arrive at an accurate estimate.The framework seamlessly integrates the multiscale features of the GMsFEM with the multilevel feature of the MLMC methods following the work in[26],and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates.

Weighted Markov chains for forecasting and analysis in Incidence of infectious diseases in jiangsu Province,China

This paper first applies the sequential cluster method to set up the classification standard of infectious disease incidence state based on the fact that there are many uncertainty characteristics in the incidence course.Then the paper presents a weighted Markov chain,a method which is used to predict the future incidence state.This method assumes the standardized self-coefficients as weights based on the special characteristics of infectious disease incidence being a dependent stochastic variable.It also analyzes the characteristics of infectious diseases incidence via the Markov chain Monte Carlo method to make the long-term benefit of decision optimal.Our method is successfully validated using existing incidents data of infectious diseases in Jiangsu Province.In summation,this paper proposes ways to improve the accuracy of the weighted Markov chain,specifically in the field of infection epidemiology.

Statistical Inversion Based on Nonlinear Weighted Anisotropic Total Variational Model and Its Application in Electrical Impedance Tomography

Electrical impedance tomography (EIT) aims to reconstruct the conductivity distribution using the boundary measured voltage potential. Traditional regularization based method would suffer from error propagation due to the iteration process. The statistical inverse problem method uses statistical inference to estimate unknown parameters. In this article, we develop a nonlinear weighted anisotropic total variation (NWATV) prior density function based on the recently proposed NWATV regularization method. We calculate the corresponding posterior density function, i.e., the solution of the EIT inverse problem in the statistical sense, via a modified Markov chain Monte Carlo (MCMC) sampling. We do numerical experiment to validate the proposed approach.

A Probabilistic Description of the Impact of Vaccine-Induced Immunity in the Dynamics of COVID-19 Transmission

The recent outbreak of COVID-19 has caused millions of deaths worldwide and a huge societal and economic impact in virtually all countries. A large variety of mathematical models to describe the dynamics of COVID-19 transmission have been reported. Among them, Bayesian probabilistic models of COVID-19 transmission dynamics have been very efficient in the interpretation of early data from the beginning of the pandemic, helping to estimate the impact of non-pharmacological measures in each country, and forecasting the evolution of the pandemic in different potential scenarios. These models use probability distribution curves to describe key dynamic aspects of the transmission, like the probability for every infected person of infecting other individuals, dying or recovering, with parameters obtained from experimental epidemiological data. However, the impact of vaccine-induced immunity, which has been key for controlling the public health emergency caused by the pandemic, has been more challenging to describe in these models, due to the complexity of experimental data. Here we report different probability distribution curves to model the acquisition and decay of immunity after vaccination. We discuss the mathematical background and how these models can be integrated in existing Bayesian probabilistic models to provide a good estimation of the dynamics of COVID-19 transmission during the entire pandemic period.

DSMC: Fast direct simulation Monte Carlo solver for the Boltzmann equation by Multi-Chain Markov Chain and multicore programming

Direct Simulation Monte Carlo(DSMC)solves the Boltzmann equation with large Knudsen number.The Boltzmann equation generally consists of three terms:the force term,the diffusion term and the collision term.While the first two terms of the Boltzmann equation can be discretized by numerical methods such as the finite volume method,the third term can be approximated by DSMC,and DSMC simulates the physical behaviors of gas molecules.However,because of the low sampling efficiency of Monte Carlo Simulation in DSMC,this part usually occupies large portion of computational costs to solve the Boltzmann equation.In this paper,by Markov Chain Monte Carlo(MCMC)and multicore programming,we develop Direct Simulation Multi-Chain Markov Chain Monte Carlo(DSMC3):a fast solver to calculate the numerical solution for the Boltzmann equation.Computational results show that DSMC3 is significantly faster than the conventional method DSMC.

Active Kriging-Based Adaptive Importance Sampling for Reliability and Sensitivity Analyses of Stator Blade Regulator

The reliability and sensitivity analyses of stator blade regulator usually involve complex characteristics like highnonlinearity,multi-failure regions,and small failure probability,which brings in unacceptable computing efficiency and accuracy of the current analysismethods.In this case,by fitting the implicit limit state function(LSF)with active Kriging(AK)model and reducing candidate sample poolwith adaptive importance sampling(AIS),a novel AK-AIS method is proposed.Herein,theAKmodel andMarkov chainMonte Carlo(MCMC)are first established to identify the most probable failure region(s)(MPFRs),and the adaptive kernel density estimation(AKDE)importance sampling function is constructed to select the candidate samples.With the best samples sequentially attained in the reduced candidate samples and employed to update the Kriging-fitted LSF,the failure probability and sensitivity indices are acquired at a lower cost.The proposed method is verified by twomulti-failure numerical examples,and then applied to the reliability and sensitivity analyses of a typical stator blade regulator.Withmethods comparison,the proposed AK-AIS is proven to hold the computing advantages on accuracy and efficiency in complex reliability and sensitivity analysis problems.

Dynamic Spatio-Temporal Modeling in Disease Mapping

Spatio-temporal models are valuable tools for disease mapping and understanding the geographical distribution of diseases and temporal dynamics. Spatio-temporal models have been proven empirically to be very complex and this complexity has led many to oversimply and model the spatial and temporal dependencies independently. Unlike common practice, this study formulated a new spatio-temporal model in a Bayesian hierarchical framework that accounts for spatial and temporal dependencies jointly. The spatial and temporal dependencies were dynamically modelled via the matern exponential covariance function. The temporal aspect was captured by the parameters of the exponential with a first-order autoregressive structure. Inferences about the parameters were obtained via Markov Chain Monte Carlo (MCMC) techniques and the spatio-temporal maps were obtained by mapping stable posterior means from the specific location and time from the best model that includes the significant risk factors. The model formulated was fitted to both simulation data and Kenya meningitis incidence data from 2013 to 2019 along with two covariates;Gross County Product (GCP) and average rainfall. The study found that both average rainfall and GCP had a significant positive association with meningitis occurrence. Also, regarding geographical distribution, the spatio-temporal maps showed that meningitis is not evenly distributed across the country as some counties reported a high number of cases compared with other counties.

Hierarchical Bayesian Calibration and On-line Updating Method for Influence Coefficient of Automatic Dynamic Balancing Machine

Measurement error of unbalance＇s vibration response plays a crucial role in calibration and on-line updating of influence coefficient（IC）. Focusing on the two problems that the moment estimator of data used in calibration process cannot fulfill the accuracy requirement under small sample and the disturbance of measurement error cannot be effectively suppressed in updating process, an IC calibration and on-line updating method based on hierarchical Bayesian method for automatic dynamic balancing machine was proposed. During calibration process, for the repeatedly-measured data obtained from experiments with different trial weights, according to the fact that measurement error of each sensor had the same statistical characteristics, the joint posterior distribution model for the true values of the vibration response under all trial weights and measurement error was established. During the updating process, information obtained from calibration was regarded as prior information, which was utilized to update the posterior distribution of IC combined with the real-time reference information to implement online updating. Moreover, Gibbs sampling method of Markov Chain Monte Carlo（MCMC） was adopted to obtain the maximum posterior estimation of parameters to be estimated. On the independent developed dynamic balancing testbed, prediction was carried out for multiple groups of data through the proposed method and the traditional method respectively, the result indicated that estimator of influence coefficient obtained through the proposed method had higher accuracy; the proposed updating method more effectively guaranteed the measurement accuracy during the whole producing process, and meantime more reasonably compromised between the sensitivity of IC change and suppression of randomness of vibration response.

Target tracking in glint noise using a MCMC particle filter

In radar target tracking application, the observation noise is usually non-Gaussian, which is also referred as glint noise. The performances of conventional trackers degra de severely in the presence of glint noise. An improved particle filter, Markov chain Monte Carlo particle filter (MCMC-PF), is applied to cope with radar target tracking when the measurements are perturbed by glint noise. Tracking performance of the filter is demonstrated in the present of glint noise by computer simulation.

Bayesian Reliability Modeling and Assessment Solution for NC Machine Tools under Small-sample Data

Although Markov chain Monte Carlo（MCMC） algorithms are accurate, many factors may cause instability when they are utilized in reliability analysis; such instability makes these algorithms unsuitable for widespread engineering applications. Thus, a reliability modeling and assessment solution aimed at small-sample data of numerical control（NC） machine tools is proposed on the basis of Bayes theories. An expert-judgment process of fusing multi-source prior information is developed to obtain the Weibull parameters＇ prior distributions and reduce the subjective bias of usual expert-judgment methods. The grid approximation method is applied to two-parameter Weibull distribution to derive the formulas for the parameters＇ posterior distributions and solve the calculation difficulty of high-dimensional integration. The method is then applied to the real data of a type of NC machine tool to implement a reliability assessment and obtain the mean time between failures（MTBF）. The relative error of the proposed method is 5.8020×10-4 compared with the MTBF obtained by the MCMC algorithm. This result indicates that the proposed method is as accurate as MCMC. The newly developed solution for reliability modeling and assessment of NC machine tools under small-sample data is easy, practical, and highly suitable for widespread application in the engineering field; in addition, the solution does not reduce accuracy.

Bayesian zero-failure reliability modeling and assessment method for multiple numerical control(NC) machine tools

A new problem that classical statistical methods are incapable of solving is reliability modeling and assessment when multiple numerical control machine tools(NCMTs) reveal zero failures after a reliability test. Thus, the zero-failure data form and corresponding Bayesian model are developed to solve the zero-failure problem of NCMTs, for which no previous suitable statistical model has been developed. An expert-judgment process that incorporates prior information is presented to solve the difficulty in obtaining reliable prior distributions of Weibull parameters. The equations for the posterior distribution of the parameter vector and the Markov chain Monte Carlo(MCMC) algorithm are derived to solve the difficulty of calculating high-dimensional integration and to obtain parameter estimators. The proposed method is applied to a real case; a corresponding programming code and trick are developed to implement an MCMC simulation in Win BUGS, and a mean time between failures(MTBF) of 1057.9 h is obtained. Given its ability to combine expert judgment, prior information, and data, the proposed reliability modeling and assessment method under the zero failure of NCMTs is validated.

Updating the models and uncertainty of mechanical parameters for rock tunnels using Bayesian inference

Rock mechanical parameters and their uncertainties are critical to rock stability analysis,engineering design,and safe construction in rock mechanics and engineering.The back analysis is widely adopted in rock engineering to determine the mechanical parameters of the surrounding rock mass,but this does not consider the uncertainty.This problem is addressed here by the proposed approach by developing a system of Bayesian inferences for updating mechanical parameters and their statistical properties using monitored field data,then integrating the monitored data,prior knowledge of geotechnical parameters,and a mechanical model of a rock tunnel using Markov chain Monte Carlo(MCMC)simulation.The proposed approach is illustrated by a circular tunnel with an analytical solution,which was then applied to an experimental tunnel in Goupitan Hydropower Station,China.The mechanical properties and strength parameters of the surrounding rock mass were modeled as random variables.The displacement was predicted with the aid of the parameters updated by Bayesian inferences and agreed closely with monitored displacements.It indicates that Bayesian inferences combined the monitored data into the tunnel model to update its parameters dynamically.Further study indicated that the performance of Bayesian inferences is improved greatly by regularly supplementing field monitoring data.Bayesian inference is a significant and new approach for determining the mechanical parameters of the surrounding rock mass in a tunnel model and contributes to safe construction in rock engineering.

A statistical inference for generalized Rayleigh model under Type-Ⅱ progressive censoring with binomial removals

This paper considers the parameters and reliability characteristics estimation problem of the generalized Rayleigh distribution under progressively Type-Ⅱ censoring with random removals,that is,the number of units removed at each failure time follows the binomial distribution.The maximum likelihood estimation and the Bayesian estimation are derived.In the meanwhile,through a great quantity of Monte Carlo simulation experiments we have studied different hyperparameters as well as symmetric and asymmetric loss functions in the Bayesian estimation procedure.A real industrial case is presented to justify and illustrate the proposed methods.We also investigate the expected experimentation time and discuss the influence of the parameters on the termination point to complete the censoring test.

Bayesian geological and geophysical data fusion for the construction and uncertainty quantification of 3D geological models

Traditional approaches to develop 3D geological models employ a mix of quantitative and qualitative scientific techniques,which do not fully provide quantification of uncertainty in the constructed models and fail to optimally weight geological field observations against constraints from geophysical data.Here,using the Bayesian Obsidian software package,we develop a methodology to fuse lithostratigraphic field observations with aeromagnetic and gravity data to build a 3D model in a small(13.5 km×13.5 km)region of the Gascoyne Province,Western Australia.Our approach is validated by comparing 3D model results to independently-constrained geological maps and cross-sections produced by the Geological Survey of Western Australia.By fusing geological field data with aeromagnetic and gravity surveys,we show that 89%of the modelled region has>95%certainty for a particular geological unit for the given model and data.The boundaries between geological units are characterized by narrow regions with<95%certainty,which are typically 400-1000 m wide at the Earth's surface and 500-2000 m wide at depth.Beyond~4 km depth,the model requires geophysical survey data with longer wavelengths(e.g.,active seismic)to constrain the deeper subsurface.Although Obsidian was originally built for sedimentary basin problems,there is reasonable applicability to deformed terranes such as the Gascoyne Province.Ultimately,modification of the Bayesian engine to incorporate structural data will aid in developing more robust 3D models.Nevertheless,our results show that surface geological observations fused with geophysical survey data can yield reasonable 3D geological models with narrow uncertainty regions at the surface and shallow subsurface,which will be especially valuable for mineral exploration and the development of 3D geological models under cover.

Joint optimization of sampling interval and control for condition-based maintenance using availability maximization criterion

Most of the maintenance optimization models in condition-based maintenance(CBM) consider the cost-optimal criterion, but few papers have dealt with availability maximization for maintenance applications. A novel optimal Bayesian control approach is presented for maintenance decision making. The system deterioration evolves as a three-state continuous time hidden semi-Markov process. Considering the optimal maintenance policy, the multivariate Bayesian control scheme based on the hidden semi-Markov model(HSMM) is developed, the objective is to maximize the long-run expected average availability per unit time. The proposed approach can optimize the sampling interval and control limit jointly. A case study using Markov chain Monte Carlo(MCMC)simulation is provided and a comparison with the Bayesian control scheme based on hidden Markov model(HMM), the age-based replacement policy, Hotelling’s T2, multivariate exponentially weihted moving average(MEWMA) and multivariate cumulative sum(MCUSUM) control charts is given, which illustrates the effectiveness of the proposed method.

Study of photometric phase curve:assuming a cellinoid ellipsoid shape for asteroid(106)Dione

We carried out new photometric observations of asteroid （106） Dione at three apparitions （2004, 2012 and 2015） to understand its basic physical properties. Based on a new brightness model, new photometric observational data and published data of （106） Dione were analyzed to characterize the morphology of Dione＇s photometric phase curve. In this brightness model, a cellinoid ellipsoid shape and three-parameter （H, G1, G2） magnitude phase function system were involved. Such a model can not only solve the phase function system parameters of （106） Dione by considering an asymmetric shape of an asteroid, but also can be applied to more asteroids, especially those without enough photometric data to solve the convex shape. Using a Markov Chain Monte Carlo （MCMC） method, Dione＇s absolute magnitude of H = 7.66＋0.03-0.03 mag, and phase function parameters G1 = 0.682＋0.077-0.077 and G2 = 0.081＋0.042-0.042 were obtained. Simultaneously, Dione＇s simplistic shape, orientation of pole and rotation period were also determined preliminarily.

Bayesian Study Using MCMC of Three-Parameter Frechet Distribution Based on Type-I Censored Data

Type-I censoring mechanism arises when the number of units experiencing the event is random but the total duration of the study is fixed. There are a number of mathematical approaches developed to handle this type of data. The purpose of the research was to estimate the three parameters of the Frechet distribution via the frequentist Maximum Likelihood and the Bayesian Estimators. In this paper, the maximum likelihood method (MLE) is not available of the three parameters in the closed forms;therefore, it was solved by the numerical methods. Similarly, the Bayesian estimators are implemented using Jeffreys and gamma priors with two loss functions, which are: squared error loss function and Linear Exponential Loss Function (LINEX). The parameters of the Frechet distribution via Bayesian cannot be obtained analytically and therefore Markov Chain Monte Carlo is used, where the full conditional distribution for the three parameters is obtained via Metropolis-Hastings algorithm. Comparisons of the estimators are obtained using Mean Square Errors (MSE) to determine the best estimator of the three parameters of the Frechet distribution. The results show that the Bayesian estimation under Linear Exponential Loss Function based on Type-I censored data is a better estimator for all the parameter estimates when the value of the loss parameter is positive.