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.

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.

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.

Analysis of Ozone Behaviour in the City of Puebla-Mexico Using Non-Homogeneous Poisson Models with Multiple Change-Points

In this work, some non-homogeneous Poisson models are considered to study the behaviour of ozone in the city of Puebla, Mexico. Several functions are used as the rate function for the non-homogeneous Poisson process. In addition to their dependence on time, these rate functions also depend on some parameters that need to be estimated. In order to estimate them, a Bayesian approach will be taken. The expressions for the distributions of the parameters involved in the models are very complex. Therefore, Markov chain Monte Carlo algorithms are used to estimate them. The methodology is applied to the ozone data from the city of Puebla, Mexico.

Non-Homogeneous Poisson Processes Applied to Count Data:A Bayesian Approach Considering Different Prior Distributions

This article discusses the Bayesian approach for count data using non-homogeneous Poisson processes, considering different prior distributions for the model parameters. A Bayesian approach using Markov Chain Monte Carlo (MCMC) simulation methods for this model was first introduced by [1], taking into account software reliability data and considering non-informative prior distributions for the parameters of the model. With the non-informative prior distributions presented by these authors, computational difficulties may occur when using MCMC methods. This article considers different prior distributions for the parameters of the proposed model, and studies the effect of such prior distributions on the convergence and accuracy of the results. In order to illustrate the proposed methodology, two examples are considered: the first one has simulated data, and the second has a set of data for pollution issues at a region in Mexico City.

Stochastic Volatility Modeling based on Doubly Truncated Cauchy Distribution and Bayesian Estimation for Chinese Stock Market

In order to measure the uncertainty of financial asset returns in the stock market, this paper presents a new model, called SV-dt C model, a stochastic volatility(SV) model assuming that the stock return has a doubly truncated Cauchy distribution, which takes into account the high peak and fat tail of the empirical distribution simultaneously. Under the Bayesian framework, a prior and posterior analysis for the parameters is made and Markov Chain Monte Carlo(MCMC) is used for computing the posterior estimates of the model parameters and forecasting in the empirical application of Shanghai Stock Exchange Composite Index(SSECI) with respect to the proposed SV-dt C model and two classic SV-N(SV model with Normal distribution)and SV-T(SV model with Student-t distribution) models. The empirical analysis shows that the proposed SV-dt C model has better performance by model checking, including independence test(Projection correlation test), Kolmogorov-Smirnov test(K-S test) and Q-Q plot. Additionally, deviance information criterion(DIC) also shows that the proposed model has a significant improvement in model fit over the others.

The influence of ambient air pollution on the transmission of tuberculosis in Jiangsu,China

In this paper,based on the statistical data,we investigate the effects of long-term exposure to ambient particulate air pollution on the transmission dynamics of tuberculosis(TB)in Jiangsu,China by studying the threshold dynamics of the TB epidemic model via the statistical data analytically and numerically.The basic reproduction number R_(0)>1 reveals that TB in Jiangsu,China is an endemic disease and will persist for a long time.And the numerical results show that,in order to control the TB in Jiangsu effectively,we must decrease the depuration coefficient of PM10 in the body,the proportion of TB symptomatic infectious by direct transmission,the reactivation rate of the pre-symptomatic infectious and the effect coefficient of PM10 and MTB inhaled of TB transmission,and increase the uptake coefficient,the recovery rate of the symptomatic/pre-symptomatic infectious and the influence coefficient of PM10 on the body of mortality.Our study shows that PM10 is closely related to the incidence of TB,and the effective control efforts are suggested to focus on increasing close-contact distance and wearing protective mask to decrease the influence of PM10 on the TB transmission,which may shed a new light on understanding the environmental drivers to TB.

Conditional inversion to estimate parameters from eddy-flux observations

Aims Data assimilation is a useful tool to extract information from large datasets of the net ecosystem exchange(NEE)of CO_(2) obtained by eddy-flux measurements.However,the number of parameters in ecosystem models that can be constrained by eddy-flux data is limited by conventional inverse analysis that estimates parameter values based on one-time inversion.This study aimed to improve data assimilation to increase the number of constrained parameters.Methods In this study,we developed conditional Bayesian inversion to maximize the number of parameters to be constrained by NEE data in several steps.In each step,we conducted a Bayesian inversion to constrain parameters.The maximum likelihood estimates of the constrained parameters were then used as prior to fix parameter values in the next step of inversion.The conditional inversion was repeated until there were no more parameters that could be further constrained.We applied the conditional inversion to hourly NEE data from Harvard Forest with a physiologically based ecosystem model.Important Findings Results showed that the conventional inversion method constrained 6 of 16 parameters in the model while the conditional inversion method constrained 13 parameters after six steps.The cost function that indicates mismatch between the modeled and observed data decreased with each step of conditional Bayesian inversion.The Bayesian information criterion also decreased,suggesting reduced information loss with each step of conditional Bayesian inversion.A wavelet analysis reflected that model performance under conditional Bayesian inversion was better than that under conventional inversion at multiple time scales,except for seasonal and half-yearly scales.In addition,our analysis also demonstrated that parameter convergence in a subsequent step of the conditional inversion depended on correlations with the parameters constrained in a previous step.Overall,the conditional Bayesian inversion substantially increased the number of parameters to be constrained by NEE data and can be a powerful tool to be used in data assimilation in ecology.

Uncertainty analysis of forest carbon sink forecast with varying measurement errors:a data assimilation approach

Aims Accurate forecast of ecosystem states is critical for improving natural resourcemanagement and climate change mitigation.Assimilating observed data into models is an effective way to reduce uncertainties in ecological forecasting.However,influences ofmeasurement errors on parameter estimation and forecasted state changes have not been carefully examined.This study analyzed the parameter identifiability of a process-based ecosystem carbon cycle model,the sensitivity of parameter estimates and model forecasts to the magnitudes of measurement errors and the information contributions of the assimilated data to model forecasts with a data assimilation approach.Methods We applied a Markov Chain Monte Carlo method to assimilate eight biometric data sets into the Terrestrial ECOsystemmodel.The data were the observations of foliage biomass,wood biomass,fine root biomass,microbial biomass,litter fall,litter,soil carbon and soil respiration,collected at the Duke Forest free-air CO_(2)enrichment facilities from 1996 to 2005.Three levels ofmeasurement errorswere assigned to these data sets by halving and doubling their original standard deviations.Important Findings Results showed that only less than half of the 30 parameters could be constrained,though the observations were extensive and themodelwas relatively simple.Highermeasurement errors led to higher uncertainties in parameters estimates and forecasted carbon(C)pool sizes.The longterm predictions of the slow turnover pools were affected less by the measurement errors than those of fast turnover pools.Assimilated data contributed less information for the pools with long residence times in long-term forecasts.These results indicate the residence times of C pools played a key role in regulating propagation of errors from measurements to model forecasts in a data assimilation system.Improving the estimation of parameters of slowturnover C pools is the key to better forecast long-term ecosystem C dynamics.

A system for determining maximum tolerated dose in clinical trial

Toxicity study,especially in determining the maximum tolerated dose(MTD)in phase I clinical trial,is an important step in developing new life-saving drugs.In practice,toxicity levels may be categorised as binary grades,multiple grades,or in a more generalised case,continuous grades.In this study,we propose an overall MTD framework that includes all the aforementioned cases for a single toxicity outcome(response).The mechanism of determining MTD involves a function that is predetermined by user.Analytic properties of such a system are investigated and simu-lation studies are performed for various scenarios.The concept of the continual reassessment method(CRM)is also implied in the framework and Bayesian analysis,including Markov chain Monte Carlo(MCMC)methods are used in estimating the model parameters.