Search results for: MCMC

Authors: Suparman

Abstract:

This paper addresses the problem of the signal segmentation within a Bayesian framework by using reversible jump MCMC algorithm. The signal is modelled by piecewise constant Moving-Average (MA) model where the numbers of segments, the position of change-point, the order and the coefficient of the MA model for each segment are unknown. The reversible jump MCMC algorithm is then used to generate samples distributed according to the joint posterior distribution of the unknown parameters. These samples allow calculating some interesting features of the posterior distribution. The performance of the methodology is illustrated via several simulation results.

Keywords: piecewise, moving-average model, reversible jump MCMC, signal segmentation

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Authors: Suparman

Abstract:

Piecewise linear regression models are very flexible models for modeling the data. If the piecewise linear regression models are matched against the data, then the parameters are generally not known. This paper studies the problem of parameter estimation of piecewise linear regression models. The method used to estimate the parameters of picewise linear regression models is Bayesian method. But the Bayes estimator can not be found analytically. To overcome these problems, the reversible jump MCMC algorithm is proposed. Reversible jump MCMC algorithm generates the Markov chain converges to the limit distribution of the posterior distribution of the parameters of picewise linear regression models. The resulting Markov chain is used to calculate the Bayes estimator for the parameters of picewise linear regression models.

Keywords: regression, piecewise, Bayesian, reversible Jump MCMC

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Authors: Suparman

Abstract:

Piecewise polynomial regression model is very flexible model for modeling the data. If the piecewise polynomial regression model is matched against the data, its parameters are not generally known. This paper studies the parameter estimation problem of piecewise polynomial regression model. The method which is used to estimate the parameters of the piecewise polynomial regression model is Bayesian method. Unfortunately, the Bayes estimator cannot be found analytically. Reversible jump MCMC algorithm is proposed to solve this problem. Reversible jump MCMC algorithm generates the Markov chain that converges to the limit distribution of the posterior distribution of piecewise polynomial regression model parameter. The resulting Markov chain is used to calculate the Bayes estimator for the parameters of piecewise polynomial regression model.

Keywords: piecewise regression, bayesian, reversible jump MCMC, segmentation

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Authors: Suparman Suparman

Abstract:

A white noise in autoregressive (AR) model is often assumed to be normally distributed. In application, the white noise usually do not follows a normal distribution. This paper aims to estimate a parameter of AR model that has a exponential white noise. A Bayesian method is adopted. A prior distribution of the parameter of AR model is selected and then this prior distribution is combined with a likelihood function of data to get a posterior distribution. Based on this posterior distribution, a Bayesian estimator for the parameter of AR model is estimated. Because the order of AR model is considered a parameter, this Bayesian estimator cannot be explicitly calculated. To resolve this problem, a method of reversible jump Markov Chain Monte Carlo (MCMC) is adopted. A result is a estimation of the parameter AR model can be simultaneously calculated.

Keywords: autoregressive (AR) model, exponential white Noise, bayesian, reversible jump Markov Chain Monte Carlo (MCMC)

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Authors: Autcha Araveeporn

Abstract:

This paper is to compare the parameter estimation of the mean in normal distribution by Maximum Likelihood (ML), Bayes, and Markov Chain Monte Carlo (MCMC) methods. The ML estimator is estimated by the average of data, the Bayes method is considered from the prior distribution to estimate Bayes estimator, and MCMC estimator is approximated by Gibbs sampling from posterior distribution. These methods are also to estimate a parameter then the hypothesis testing is used to check a robustness of the estimators. Data are simulated from normal distribution with the true parameter of mean 2, and variance 4, 9, and 16 when the sample sizes is set as 10, 20, 30, and 50. From the results, it can be seen that the estimation of MLE, and MCMC are perceivably different from the true parameter when the sample size is 10 and 20 with variance 16. Furthermore, the Bayes estimator is estimated from the prior distribution when mean is 1, and variance is 12 which showed the significant difference in mean with variance 9 at the sample size 10 and 20.

Keywords: Bayes method, Markov chain Monte Carlo method, maximum likelihood method, normal distribution

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Authors: Yuzhi Cai

Abstract:

This talk presents a Bayesian approach to quantile autoregressive (QAR) time series model estimation and forecasting. We establish that the joint posterior distribution of the model parameters and future values is well defined. The associated MCMC algorithm for parameter estimation and forecasting converges to the posterior distribution quickly. We also present a combining forecasts technique to produce more accurate out-of-sample forecasts by using a weighted sequence of fitted QAR models. A moving window method to check the quality of the estimated conditional quantiles is developed. We verify our methodology using simulation studies and then apply it to currency exchange rate data. An application of the method to the USD to GBP daily currency exchange rates will also be discussed. The results obtained show that an unequally weighted combining method performs better than other forecasting methodology.

Keywords: combining forecasts, MCMC, quantile modelling, quantile forecasting, predictive density functions

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Authors: Jeremie Coullon, Robert J. Webber

Abstract:

We introduce a Markov chain Monte Carlo (MCMC) sam-pler for infinite-dimensional inverse problems. Our sam-pler is based on the affine invariant ensemble sampler, which uses interacting walkers to adapt to the covariance structure of the target distribution. We extend this ensem-ble sampler for the first time to infinite-dimensional func-tion spaces, yielding a highly efficient gradient-free MCMC algorithm. Because our ensemble sampler does not require gradients or posterior covariance estimates, it is simple to implement and broadly applicable. In many Bayes-ian inverse problems, Markov chain Monte Carlo (MCMC) meth-ods are needed to approximate distributions on infinite-dimensional function spaces, for example, in groundwater flow, medical imaging, and traffic flow. Yet designing efficient MCMC methods for function spaces has proved challenging. Recent gradi-ent-based MCMC methods preconditioned MCMC methods, and SMC methods have improved the computational efficiency of functional random walk. However, these samplers require gradi-ents or posterior covariance estimates that may be challenging to obtain. Calculating gradients is difficult or impossible in many high-dimensional inverse problems involving a numerical integra-tor with a black-box code base. Additionally, accurately estimating posterior covariances can require a lengthy pilot run or adaptation period. These concerns raise the question: is there a functional sampler that outperforms functional random walk without requir-ing gradients or posterior covariance estimates? To address this question, we consider a gradient-free sampler that avoids explicit covariance estimation yet adapts naturally to the covariance struc-ture of the sampled distribution. This sampler works by consider-ing an ensemble of walkers and interpolating and extrapolating between walkers to make a proposal. This is called the affine in-variant ensemble sampler (AIES), which is easy to tune, easy to parallelize, and efficient at sampling spaces of moderate dimen-sionality (less than 20). The main contribution of this work is to propose a functional ensemble sampler (FES) that combines func-tional random walk and AIES. To apply this sampler, we first cal-culate the Karhunen–Loeve (KL) expansion for the Bayesian prior distribution, assumed to be Gaussian and trace-class. Then, we use AIES to sample the posterior distribution on the low-wavenumber KL components and use the functional random walk to sample the posterior distribution on the high-wavenumber KL components. Alternating between AIES and functional random walk updates, we obtain our functional ensemble sampler that is efficient and easy to use without requiring detailed knowledge of the target dis-tribution. In past work, several authors have proposed splitting the Bayesian posterior into low-wavenumber and high-wavenumber components and then applying enhanced sampling to the low-wavenumber components. Yet compared to these other samplers, FES is unique in its simplicity and broad applicability. FES does not require any derivatives, and the need for derivative-free sam-plers has previously been emphasized. FES also eliminates the requirement for posterior covariance estimates. Lastly, FES is more efficient than other gradient-free samplers in our tests. In two nu-merical examples, we apply FES to challenging inverse problems that involve estimating a functional parameter and one or more scalar parameters. We compare the performance of functional random walk, FES, and an alternative derivative-free sampler that explicitly estimates the posterior covariance matrix. We conclude that FES is the fastest available gradient-free sampler for these challenging and multimodal test problems.

Keywords: Bayesian inverse problems, Markov chain Monte Carlo, infinite-dimensional inverse problems, dimensionality reduction

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Authors: Endeshaw Assefa Derso, Maria Gabriella Campolo, Angela Alibrandi

Abstract:

Objectives:Early childhood malnutrition can have long-term and irreversible effects on a child's health and development. This study uses the Bayesian method with spatial variation to investigate the flexible trends of metrical covariates and to identify communities at high risk of injury. Methods: Cross-sectional data on underweight are collected from the 2016 Ethiopian Demographic and Health Survey (EDHS). The Bayesian geo-additive model is performed. Appropriate prior distributions were provided for scall parameters in the models, and the inference is entirely Bayesian, using Monte Carlo Markov chain (MCMC) stimulation. Results: The results show that metrical covariates like child age, maternal body mass index (BMI), and maternal age affect a child's underweight non-linearly. Lower and higher maternal BMI seem to have a significant impact on the child’s high underweight. There was also a significant spatial heterogeneity, and based on IDW interpolation of predictive values, the western, central, and eastern parts of the country are hotspot areas. Conclusion: Socio-demographic and community- based programs development should be considered compressively in Ethiopian policy to combat childhood underweight malnutrition.

Keywords: bayesX, Ethiopia, malnutrition, MCMC, semi-parametric bayesian analysis, spatial distribution, P- splines

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Authors: P. Byrnes, F. A. DiazDelaO

Abstract:

The Support Vector Machine (SVM) has become widely recognised as one of the leading algorithms in machine learning for both regression and binary classification. It expresses predictions in terms of a linear combination of kernel functions, referred to as support vectors. Despite its popularity amongst practitioners, SVM has some limitations, with the most significant being the generation of point prediction as opposed to predictive distributions. Stemming from this issue, a probabilistic model namely, Probabilistic Classification Vector Machines (PCVM), has been proposed which respects the original functional form of SVM whilst also providing a predictive distribution. As physical system designs become more complex, an increasing number of classification tasks involving industrial applications consist of more than two classes. Consequently, this research proposes a framework which allows for the extension of PCVM to a multi class setting. Additionally, the original PCVM framework relies on the use of type II maximum likelihood to provide estimates for both the kernel hyperparameters and model evidence. In a high dimensional multi class setting, however, this approach has been shown to be ineffective due to bad scaling as the number of classes increases. Accordingly, we propose the application of Markov Chain Monte Carlo (MCMC) based methods to provide a posterior distribution over both parameters and hyperparameters. The proposed framework will be validated against current multi class classifiers through synthetic and real life implementations.

Keywords: probabilistic classification vector machines, multi class classification, MCMC, support vector machines

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Authors: Ofosuhene O. Apenteng, Noor Azina Ismail

Abstract:

The human immunodeficiency virus (HIV) that causes acquired immune deficiency syndrome (AIDS) remains a global cause of morbidity and mortality. It has caused panic since its emergence. Relationships between migration and HIV/AIDS have become complex. In the absence of prospectively designed studies, dynamic mathematical models that take into account the migration movement which will give very useful information. We have explored the utility of mathematical models in understanding transmission dynamics of HIV and AIDS and in assessing the magnitude of how migration has impact on the disease. The model was calibrated to HIV and AIDS incidence data from Malaysia Ministry of Health from the period of 1986 to 2011 using Bayesian analysis with combination of Markov chain Monte Carlo method (MCMC) approach to estimate the model parameters. From the estimated parameters, the estimated basic reproduction number was 22.5812. The rate at which the susceptible individual moved to HIV compartment has the highest sensitivity value which is more significant as compared to the remaining parameters. Thus, the disease becomes unstable. This is a big concern and not good indicator from the public health point of view since the aim is to stabilize the epidemic at the disease-free equilibrium. However, these results suggest that the government as a policy maker should make further efforts to curb illegal activities performed by migrants. It is shown that our models reflect considerably the dynamic behavior of the HIV/AIDS epidemic in Malaysia and eventually could be used strategically for other countries.

Keywords: epidemic model, reproduction number, HIV, MCMC, parameter estimation

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Authors: S. Bouhouche, R. Drai, J. Bast

Abstract:

This paper is concerned with a method for uncertainty evaluation of steel sample content using X-Ray Fluorescence method. The considered method of analysis is a comparative technique based on the X-Ray Fluorescence; the calibration step assumes the adequate chemical composition of metallic analyzed sample. It is proposed in this work a new combined approach using the Kalman Filter and Markov Chain Monte Carlo (MCMC) for uncertainty estimation of steel content analysis. The Kalman filter algorithm is extended to the model identification of the chemical analysis process using the main factors affecting the analysis results; in this case, the estimated states are reduced to the model parameters. The MCMC is a stochastic method that computes the statistical properties of the considered states such as the probability distribution function (PDF) according to the initial state and the target distribution using Monte Carlo simulation algorithm. Conventional approach is based on the linear correlation, the uncertainty budget is established for steel Mn(wt%), Cr(wt%), Ni(wt%) and Mo(wt%) content respectively. A comparative study between the conventional procedure and the proposed method is given. This kind of approaches is applied for constructing an accurate computing procedure of uncertainty measurement.

Keywords: Kalman filter, Markov chain Monte Carlo, x-ray fluorescence calibration and testing, steel content measurement, uncertainty measurement

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Authors: Niroshan K. Walgama Wellalage, Tieling Zhang, Richard Dwight

Abstract:

State-based Markov deterioration models (SMDM) sometimes fail to find accurate transition probability matrix (TPM) values, and hence lead to invalid future condition prediction or incorrect average deterioration rates mainly due to drawbacks of existing nonlinear optimization-based algorithms and/or subjective function types used for regression analysis. Furthermore, a set of separate functions for each condition state with age cannot be directly derived by using Markov model for a given bridge element group, which however is of interest to industrial partners. This paper presents a new approach for generating Homogeneous SMDM model output, namely, the Modified Weibull approach, which consists of a set of appropriate functions to describe the percentage condition prediction of bridge elements in each state. These functions are combined with Bayesian approach and Metropolis Hasting Algorithm (MHA) based Markov Chain Monte Carlo (MCMC) simulation technique for quantifying the uncertainty in model parameter estimates. In this study, factors contributing to rail bridge deterioration were identified. The inspection data for 1,000 Australian railway bridges over 15 years were reviewed and filtered accordingly based on the real operational experience. Network level deterioration model for a typical bridge element group was developed using the proposed Modified Weibull approach. The condition state predictions obtained from this method were validated using statistical hypothesis tests with a test data set. Results show that the proposed model is able to not only predict the conditions in network-level accurately but also capture the model uncertainties with given confidence interval.

Keywords: bridge deterioration modelling, modified weibull approach, MCMC, metropolis-hasting algorithm, bayesian approach, Markov deterioration models

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Authors: Muhammad Arif Khan, Haroon Idrees, Imran Aziz, Sidra Mushtaq

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The purpose of this study is to investigate the current level of knowledge mapping skills of the public sector employees in Pakistan. National Institute of Management is one of the premiere public sector training organization for mid-career public sector employees in Pakistan. This study is conducted on participants of fourteen weeks long training course called Mid-Career Management Course (MCMC) which is mandatory for public sector employees in order to ascertain how to enhance their knowledge mapping skills. Methodology: Researcher used both qualitative and quantitative approach to conduct this study. Primary data about current level of participants’ understanding of knowledge mapping was collected through structured questionnaire. Later on, Participant Observation method was used where researchers acted as part of the group to gathered data from the trainees during their performance in training activities and tasks. Findings: Respondents of the study were examined for skills and abilities to organizing ideas, helping groups to develop conceptual framework, identifying critical knowledge areas of an organization, study large networks and identifying the knowledge flow using nodes and vertices, visualizing information, represent organizational structure etc. Overall, the responses varied in different skills depending on the performance and presentations. However, generally all participants have demonstrated average level of using both the IT and Non-IT K-mapping tools and techniques during simulation exercises, analysis paper de-briefing, case study reports, post visit presentation, course review, current issue presentation, syndicate meetings, and daily synopsis. Research Limitations: This study is conducted on a small-scale population of 67 public sector employees nominated by federal government to undergo 14 weeks extensive training program called MCMC (Mid-Career Management Course) at National Institute of Management, Peshawar, Pakistan. Results, however, reflects only a specific class of public sector employees i.e. working in grade 18 and having more than 5 years of work. Practical Implications: Research findings are useful for trainers, training agencies, government functionaries, and organizations working for capacity building of public sector employees.

Keywords: knowledge management, km in public sector, knowledge management and professional development, knowledge management in training, knowledge mapping

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Authors: Yuzhi Cai

Abstract:

In this paper, we discuss a Bayesian approach to quantile autoregressive (QAR) time series model estimation and forecasting. Together with a combining forecasts technique, we then predict USD to GBP currency exchange rates. Combined forecasts contain all the information captured by the fitted QAR models at different quantile levels and are therefore better than those obtained from individual models. Our results show that an unequally weighted combining method performs better than other forecasting methodology. We found that a median AR model can perform well in point forecasting when the predictive density functions are symmetric. However, in practice, using the median AR model alone may involve the loss of information about the data captured by other QAR models. We recommend that combined forecasts should be used whenever possible.

Keywords: combining forecasts, MCMC, predictive density functions, quantile forecasting, quantile modelling

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Authors: Muhammad Farooq, Ahtasham Gul

Abstract:

To meet the desired standard, it is important to monitor and analyze different engineering processes to get desired output. The bivariate distributions got a lot of attention in recent years to describe the randomness of natural as well as artificial mechanisms. In this article, a bivariate model is constructed using two independent models developed by the nesting approach to study the effect of each component on reliability for better understanding. Further, the Bayes analysis of system availability is studied by considering prior parametric variations in the failure time and repair time distributions. Basic statistical characteristics of marginal distribution, like mean median and quantile function, are discussed. We use inverse Gamma prior to study its frequentist properties by conducting Monte Carlo Markov Chain (MCMC) sampling scheme.

Keywords: reliability, system availability Weibull, inverse Lomax, Monte Carlo Markov Chain, Bayesian

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Authors: Watcharin Sangma, Onsiri Chanmuang, Pitsanu Tongkhow

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A forecasting model for steel demand uncertainty in Thailand is proposed. It consists of trend, autocorrelation, and outliers in a hierarchical Bayesian frame work. The proposed model uses a cumulative Weibull distribution function, latent first-order autocorrelation, and binary selection, to account for trend, time-varying autocorrelation, and outliers, respectively. The Gibbs sampling Markov Chain Monte Carlo (MCMC) is used for parameter estimation. The proposed model is applied to steel demand index data in Thailand. The root mean square error (RMSE), mean absolute percentage error (MAPE), and mean absolute error (MAE) criteria are used for model comparison. The study reveals that the proposed model is more appropriate than the exponential smoothing method.

Keywords: forecasting model, steel demand uncertainty, hierarchical Bayesian framework, exponential smoothing method

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Authors: Panudet Saengseedam, Nanthachai Kantanantha

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This paper proposes a linear mixed model (LMM) with spatial effects to forecast rice and cassava yields in Thailand at the same time. A multivariate conditional autoregressive (MCAR) model is assumed to present the spatial effects. A Bayesian method is used for parameter estimation via Gibbs sampling Markov Chain Monte Carlo (MCMC). The model is applied to the rice and cassava yields monthly data which have been extracted from the Office of Agricultural Economics, Ministry of Agriculture and Cooperatives of Thailand. The results show that the proposed model has better performance in most provinces in both fitting part and validation part compared to the simple exponential smoothing and conditional auto regressive models (CAR) from our previous study.

Keywords: Bayesian method, linear mixed model, multivariate conditional autoregressive model, spatial time series

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Authors: Krisada Lekdee, Sunee Sammatat, Nittaya Boonsit

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This paper proposes a GLMM with spatial and temporal effects for malaria data in Thailand. A Bayesian method is used for parameter estimation via Gibbs sampling MCMC. A conditional autoregressive (CAR) model is assumed to present the spatial effects. The temporal correlation is presented through the covariance matrix of the random effects. The malaria quarterly data have been extracted from the Bureau of Epidemiology, Ministry of Public Health of Thailand. The factors considered are rainfall and temperature. The result shows that rainfall and temperature are positively related to the malaria morbidity rate. The posterior means of the estimated morbidity rates are used to construct the malaria maps. The top 5 highest morbidity rates (per 100,000 population) are in Trat (Q3, 111.70), Chiang Mai (Q3, 104.70), Narathiwat (Q4, 97.69), Chiang Mai (Q2, 88.51), and Chanthaburi (Q3, 86.82). According to the DIC criterion, the proposed model has a better performance than the GLMM with spatial effects but without temporal terms.

Keywords: Bayesian method, generalized linear mixed model (GLMM), malaria, spatial effects, temporal correlation

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Authors: Betty Johanna Garzon Rozo, Jonathan Crook, Fernando Moreira

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Operational risk losses are heavy tailed and are likely to be asymmetric and extremely dependent among business lines/event types. We propose a new methodology to assess, in a multivariate way, the asymmetry and extreme dependence between severity distributions, and to calculate the capital for Operational Risk. This methodology simultaneously uses (i) several parametric distributions and an alternative mix distribution (the Lognormal for the body of losses and the Generalized Pareto Distribution for the tail) via extreme value theory using SAS®, (ii) the multivariate skew t-copula applied for the first time for operational losses and (iii) Bayesian theory to estimate new n-dimensional skew t-copula models via Markov chain Monte Carlo (MCMC) simulation. This paper analyses a newly operational loss data set, SAS Global Operational Risk Data [SAS OpRisk], to model operational risk at international financial institutions. All the severity models are constructed in SAS® 9.2. We implement the procedure PROC SEVERITY and PROC NLMIXED. This paper focuses in describing this implementation.

Keywords: operational risk, loss distribution approach, extreme value theory, copulas

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Authors: Jefferson Hernandez, Juan Padilla

Abstract:

Estimation of price elasticity of demand is a valuable tool for the task of price settling. Given its relevance, it is an active field for microeconomic and statistical research. Price elasticity in the industry of oil and gas, in particular for fuels sold in gas stations, has shown to be a challenging topic given the market and state restrictions, and underlying correlations structures between the types of fuels sold by the same gas station. This paper explores the Lotka-Volterra model for the problem for price elasticity estimation in the context of fuels; in addition, it is introduced multivariate random effects with the purpose of dealing with errors, e.g., measurement or missing data errors. In order to model the underlying correlation structures, the Inverse-Wishart, Hierarchical Half-t and LKJ distributions are studied. Here, the Bayesian paradigm through Markov Chain Monte Carlo (MCMC) algorithms for model estimation is considered. Simulation studies covering a wide range of situations were performed in order to evaluate parameter recovery for the proposed models and algorithms. Results revealed that the proposed algorithms recovered quite well all model parameters. Also, a real data set analysis was performed in order to illustrate the proposed approach.

Keywords: price elasticity, volume, correlation structures, Bayesian models

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Authors: Ofosuhene O. Apenteng, Noor Azina Ismail

Abstract:

Over the last several years, concern has developed over how to minimize the spread of HIV/AIDS epidemic in many countries. AIDS epidemic has tremendously stimulated the development of mathematical models of infectious diseases. The transmission dynamics of HIV infection that eventually developed AIDS has taken a pivotal role of much on building mathematical models. From the initial HIV and AIDS models introduced in the 80s, various improvements have been taken into account as how to model HIV/AIDS frameworks. In this paper, we present the impact of migration on the spread of HIV/AIDS. Epidemic model is considered by a system of nonlinear differential equations to supplement the statistical method approach. The model is calibrated using HIV incidence data from Malaysia between 1986 and 2011. Bayesian inference based on Markov Chain Monte Carlo is used to validate the model by fitting it to the data and to estimate the unknown parameters for the model. The results suggest that the migrants stay for a long time contributes to the spread of HIV. The model also indicates that susceptible individual becomes infected and moved to HIV compartment at a rate that is more significant than the removal rate from HIV compartment to AIDS compartment. The disease-free steady state is unstable since the basic reproduction number is 1.627309. This is a big concern and not a good indicator from the public heath point of view since the aim is to stabilize the epidemic at the disease equilibrium.

Keywords: epidemic model, HIV, MCMC, parameter estimation

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Authors: Xiang Jia, Zhijun Cheng

Abstract:

The residual lifetime of a product is the operation time between the current time and the time point when the failure happens. The residual lifetime estimation is rather important in reliability analysis. To predict the residual lifetime, it is necessary to assume or verify a particular distribution that the lifetime of the product follows. And the two-parameter Weibull distribution is frequently adopted to describe the lifetime in reliability engineering. Due to the time constraint and cost reduction, a life testing experiment is usually terminated before all the units have failed. Then the censored data is usually collected. In addition, other information could also be obtained for reliability analysis. The expert judgements are considered as it is common that the experts could present some useful information concerning the reliability. Therefore, the residual lifetime is estimated for Weibull distribution by fusing the censored data and expert judgements in this paper. First, the closed-forms concerning the point estimate and confidence interval for the residual lifetime under the Weibull distribution are both presented. Next, the expert judgements are regarded as the prior information and how to determine the prior distribution of Weibull parameters is developed. For completeness, the cases that there is only one, and there are more than two expert judgements are both focused on. Further, the posterior distribution of Weibull parameters is derived. Considering that it is difficult to derive the posterior distribution of residual lifetime, a sample-based method is proposed to generate the posterior samples of Weibull parameters based on the Monte Carlo Markov Chain (MCMC) method. And these samples are used to obtain the Bayes estimation and credible interval for the residual lifetime. Finally, an illustrative example is discussed to show the application. It demonstrates that the proposed method is rather simple, satisfactory, and robust.

Keywords: expert judgements, information fusion, residual lifetime, Weibull distribution

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Authors: Ren Moses, Emmanuel Kidando, Eren Ozguven, Yassir Abdelrazig

Abstract:

Although older drivers (defined as those of age 65 and above) are less involved with speeding, alcohol use as well as night driving, they are more vulnerable to severe crashes. The major contributing factors for severe crashes include frailty and medical complications. Several studies have evaluated the contributing factors on severity of crashes. However, few studies have established the impact of travel time reliability (TTR) on road safety. In particular, the impact of TTR on senior adults who face several challenges including hearing difficulties, decreasing of the processing skills and cognitive problems in driving is not well established. Therefore, this study focuses on determining possible impacts of TTR on the traffic safety with focus on elderly drivers. Historical travel speed data from freeway links in the study area were used to calculate travel time and the associated TTR metrics that is, planning time index, the buffer index, the standard deviation of the travel time and the probability of congestion. Four-year information on crashes occurring on these freeway links was acquired. The binary logit model estimated using the Markov Chain Monte Carlo (MCMC) sampling technique was used to evaluate variables that could be influencing elderly crash severity. Preliminary results of the analysis suggest that TTR is statistically significant in affecting the severity of a crash involving an elderly driver. The result suggests that one unit increase in the probability of congestion reduces the likelihood of the elderly severe crash by nearly 22%. These findings will enhance the understanding of TTR and its impact on the elderly crash severity.

Keywords: highway safety, travel time reliability, elderly drivers, traffic modeling

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Authors: Nikolaos Papadimas, Joanna Bennett, Amir Sakhaei, Timothy Dodwell

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Composite modeling of consolidation processes is playing an important role in the process and part design by indicating the formation of possible unwanted prior to expensive experimental iterative trial and development programs. Composite materials in their uncured state display complex constitutive behavior, which has received much academic interest, and this with different models proposed. Errors from modeling and statistical which arise from this fitting will propagate through any simulation in which the material model is used. A general hyperelastic polynomial representation was proposed, which can be readily implemented in various nonlinear finite element packages. In our case, FEniCS was chosen. The coefficients are assumed uncertain, and therefore the distribution of parameters learned using Markov Chain Monte Carlo (MCMC) methods. In engineering, the approach often followed is to select a single set of model parameters, which on average, best fits a set of experiments. There are good statistical reasons why this is not a rigorous approach to take. To overcome these challenges, A hierarchical Bayesian framework was proposed in which population distribution of model parameters is inferred from an ensemble of experiments tests. The resulting sampled distribution of hyperparameters is approximated using Maximum Entropy methods so that the distribution of samples can be readily sampled when embedded within a stochastic finite element simulation. The methodology is validated and demonstrated on a set of consolidation experiments of AS4/8852 with various stacking sequences. The resulting distributions are then applied to stochastic finite element simulations of the consolidation of curved parts, leading to a distribution of possible model outputs. With this, the paper, as far as the authors are aware, represents the first stochastic finite element implementation in composite process modelling.

Keywords: data-driven , material consolidation, stochastic finite elements, surrogate models

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Authors: Thomas Wetere

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Background: The coronavirus disease 2019 (COVID-19) pandemic (COVID-19) virus infection is a severe infectious disease with the highly transmissible variant, which become the global public health treat now. It has taken the life of more than 4 million people so far. What makes the disease the worst of all is no specific effective treatment available, its dynamics is not much researched and understood. Methodology: To end the global COVID-19 pandemic, implementation of multiple population-wide strategies, including vaccination, environmental factors, Government action, testing, and contact tracing, is required. In this article, a new mathematical model incorporating both temperature and government action to study the dynamics of the COVID-19 pandemic has been developed and comprehensively analysed. The model considers eight stages of infection: susceptible (S), infected Asymptomatic and Undetected(IAU ), infected Asymptomatic and detected(IAD), infected symptomatic and Undetected(ISU ), infected Symptomatic and detected(ISD), Hospitalized or threatened(H), Recovered(R) and Died(D). Results: The existence as well as non-negativity of the solution to the model is also verified, and the basic reproduction number is calculated. Besides, stability conditions are also checked, and finally, simulation results are compared with real data. The results demonstrates that effective government action will need to be combined with vaccination to end the ongoing COVID-19 pandemic. Conclusion: Vaccination and Government action are highly the crucial measures to control the COVID-19 pandemic. Besides, as the cost of vaccination might be high, we recommend an optimal control to reduce the cost and number of infected individuals. Moreover, in order to prevent COVID-19 pandemic, through the analysis of the model, the government must strictly manage the policy on COVID-19 and carry it out. This, in turn, helps for health campaigning and raising health literacy which plays a role to control the quick spread of the disease. We finally strongly believe that our study will play its own role in the current effort of controlling the pandemic.

Keywords: modeling, COVID-19, MCMC, stability

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Authors: José Luis Gallizo, Pilar Gargallo, Ramon Saladrigues, Manuel Salvador

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This study offers an exploratory statistical analysis of the persistence of annual profits across a sample of firms from different European Union (EU) countries. To this end, a hierarchical Bayesian dynamic model has been used which enables the annual behaviour of those profits to be broken down into a permanent structural and a transitory component, while also distinguishing between general effects affecting the industry as a whole to which each firm belongs and specific effects affecting each firm in particular. This breakdown enables the relative importance of those fundamental components to be more accurately evaluated by country and sector. Furthermore, Bayesian approach allows for testing different hypotheses about the homogeneity of the behaviour of the above components with respect to the sector and the country where the firm develops its activity. The data analysed come from a sample of 23,293 firms in EU countries selected from the AMADEUS data-base. The period analysed ran from 1999 to 2007 and 21 sectors were analysed, chosen in such a way that there was a sufficiently large number of firms in each country sector combination for the industry effects to be estimated accurately enough for meaningful comparisons to be made by sector and country. The analysis has been conducted by sector and by country from a Bayesian perspective, thus making the study more flexible and realistic since the estimates obtained do not depend on asymptotic results. In general terms, the study finds that, although the industry effects are significant, more important are the firm specific effects. That importance varies depending on the sector or the country in which the firm carries out its activity. The influence of firm effects accounts for around 81% of total variation and display a significantly lower degree of persistence, with adjustment speeds oscillating around 34%. However, this pattern is not homogeneous but depends on the sector and country analysed. Industry effects depends also on sector and country analysed have a more marginal importance, being significantly more persistent, with adjustment speeds oscillating around 7-8% with this degree of persistence being very similar for most of sectors and countries analysed.

Keywords: dynamic models, Bayesian inference, MCMC, abnormal returns, persistence of profits, return on assets

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Authors: Sabyasachi Mukhopadhyay, Joseph Ogutu, Gundula Bartzke, Hans-Peter Piepho

Abstract:

Rainfall is a critical component of climate governing vegetation growth and production, forage availability and quality for herbivores. However, reliable rainfall measurements are not always available, making it necessary to predict rainfall values for particular locations through time. Predicting rainfall in space and time can be a complex and challenging task, especially where the rain gauge network is sparse and measurements are not recorded consistently for all rain gauges, leading to many missing values. Here, we develop a flexible Bayesian model for predicting rainfall in space and time and apply it to Narok County, situated in southwestern Kenya, using data collected at 23 rain gauges from 1965 to 2015. Narok County encompasses the Maasai Mara ecosystem, the northern-most section of the Mara-Serengeti ecosystem, famous for its diverse and abundant large mammal populations and spectacular migration of enormous herds of wildebeest, zebra and Thomson's gazelle. The model incorporates geographical and meteorological predictor variables, including elevation, distance to Lake Victoria and minimum temperature. We assess the efficiency of the model by comparing it empirically with the established Gaussian process, Kriging, simple linear and Bayesian linear models. We use the model to predict total monthly rainfall and its standard error for all 5 * 5 km grid cells in Narok County. Using the Monte Carlo integration method, we estimate seasonal and annual rainfall and their standard errors for 29 sub-regions in Narok. Finally, we use the predicted rainfall to predict large herbivore biomass in the Maasai Mara ecosystem on a 5 * 5 km grid for both the wet and dry seasons. We show that herbivore biomass increases with rainfall in both seasons. The model can handle data from a sparse network of observations with many missing values and performs at least as well as or better than four established and widely used models, on the Narok data set. The model produces rainfall predictions consistent with expectation and in good agreement with the blended station and satellite rainfall values. The predictions are precise enough for most practical purposes. The model is very general and applicable to other variables besides rainfall.

Keywords: non-stationary covariance function, gaussian process, ungulate biomass, MCMC, maasai mara ecosystem

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Authors: M. A. C. S. Sampath Fernando, James M. Curran, Renate Meyer

Abstract:

Model assessment, in the Bayesian context, involves evaluation of the goodness-of-fit and the comparison of several alternative candidate models for predictive accuracy and improvements. In posterior predictive checks, the data simulated under the fitted model is compared with the actual data. Predictive model accuracy is estimated using information criteria such as the Akaike information criterion (AIC), the Bayesian information criterion (BIC), the Deviance information criterion (DIC), and the Watanabe-Akaike information criterion (WAIC). The goal of an information criterion is to obtain an unbiased measure of out-of-sample prediction error. Since posterior checks use the data twice; once for model estimation and once for testing, a bias correction which penalises the model complexity is incorporated in these criteria. Cross-validation (CV) is another method used for examining out-of-sample prediction accuracy. Leave-one-out cross-validation (LOO-CV) is the most computationally expensive variant among the other CV methods, as it fits as many models as the number of observations. Importance sampling (IS), truncated importance sampling (TIS) and Pareto-smoothed importance sampling (PSIS) are generally used as approximations to the exact LOO-CV and utilise the existing MCMC results avoiding expensive computational issues. The reciprocals of the predictive densities calculated over posterior draws for each observation are treated as the raw importance weights. These are in turn used to calculate the approximate LOO-CV of the observation as a weighted average of posterior densities. In IS-LOO, the raw weights are directly used. In contrast, the larger weights are replaced by their modified truncated weights in calculating TIS-LOO and PSIS-LOO. Although, information criteria and LOO-CV are unable to reflect the goodness-of-fit in absolute sense, the differences can be used to measure the relative performance of the models of interest. However, the use of these measures is only valid under specific circumstances. This study has developed 11 models using normal, log-normal, gamma, and student’s t distributions to improve the PCR stutter prediction with forensic data. These models are comprised of four with profile-wide variances, four with locus specific variances, and three which are two-component mixture models. The mean stutter ratio in each model is modeled as a locus specific simple linear regression against a feature of the alleles under study known as the longest uninterrupted sequence (LUS). The use of AIC, BIC, DIC, and WAIC in model comparison has some practical limitations. Even though, IS-LOO, TIS-LOO, and PSIS-LOO are considered to be approximations of the exact LOO-CV, the study observed some drastic deviations in the results. However, there are some interesting relationships among the logarithms of pointwise predictive densities (lppd) calculated under WAIC and the LOO approximation methods. The estimated overall lppd is a relative measure that reflects the overall goodness-of-fit of the model. Parallel log-likelihood profiles for the models conditional on equal posterior variances in lppds were observed. This study illustrates the limitations of the information criteria in practical model comparison problems. In addition, the relationships among LOO-CV approximation methods and WAIC with their limitations are discussed. Finally, useful recommendations that may help in practical model comparisons with these methods are provided.

Keywords: cross-validation, importance sampling, information criteria, predictive accuracy

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