Markov Chain Monte Carlo Procedure Research Articles

Bayesian estimation in generalized linear models for longitudinal data with hyperspherical coordinates

Under the framework of generalized linear models (GLM), the generalized estimating equation (GEE) method is typically applied for longitudinal data analysis. However, there are a series of problems due to the misspecification of the within-subject correlation structure, especially in Bayesian estimation. To handle these difficulties, in this paper, we construct a class of generalized estimating equations for longitudinal data with hyperspherical coordinates (HPC) and propose a Bayesian approach established through empirical likelihood (EL). Additionally, an efficient Markov chain Monte Carlo (MCMC) procedure is developed for the required computation of the posterior distribution. As proved by the simulation studies and an application to a real longitudinal data set, our method not only performs better than traditional empirical likelihood estimation and Bayesian estimation with partial autocorrelations (PAC) but also is suitable for non-Gaussian data.

Hospital Volume of Elective Abdominal Aortic Aneurysm Repair as a Predictor of Mortality After Ruptured Abdominal Aortic Aneurysm Repair

Registry data suggest that centralising abdominal aortic aneurysm (AAA) surgery decreases the mortality rate after AAA repair. However, the impact of higher elective volumes on ruptured AAA (rAAA) repair associated mortality rates remains uncertain. This study aimed to examine associations between intact AAA (iAAA) repair volume and post-operative rAAA death. Using data from official national registries between 2015 - 2019, all iAAA and rAAA repairs were separately analysed across 10 public hospitals. The following were assessed: 30 day and 12 month mortality rate following open surgical repair (OSR) and endovascular aneurysm repair (EVAR). Associations between the 5 year hospital iAAA repair volumes (organised into tertiles) and rAAA associated mortality rate were analysed, regardless of treatment modality. Receiver operating characteristic (ROC) curves were generated to identify iAAA volume thresholds for decreasing the rAAA mortality rate. Subanalysis by treatment type was conducted. Threshold analysis was repeated with the Markov chain Monte Carlo (MCMC) procedure to confirm the findings. A total of 1 599 iAAAs (80.2% EVAR, 19.8% OSR) and 196 rAAAs (66.3% EVAR, 33.7% OSR) repairs were analysed. The median and interquartile range of the volume/hospital/year for all iAAA repairs were 39.2 (31.2, 47.4). The top volume iAAA tertile exhibited lower rAAA associated 30 day (odds ratio [OR] 0.374; p= .007) and 12 month (OR 0.264; p < .001) mortality rates. The ROC analysis revealed a threshold of 40 iAAA repairs/hospital/year (EVAR+ OSR) for a reduced rAAA mortality rate. Middle volume hospitals for open iAAA repair had reduced 30 day (OR 0.267; p= .033) and 12 month (OR 0.223; p= .020) mortality rates, with a threshold of five OSR procedures/year. The MCMC procedure found similar thresholds. No significant association was found between elective EVAR volumes and ruptured EVAR mortality. Higher iAAA repair volumes correlated with a lower rAAA mortality rate, particularly for OSR. The recommended iAAA repair threshold is 40 procedures/year and five procedures/year for OSR. These findings support high elective volumes for improving the rAAA mortality rate, especially for OSR.

Statistical inference of the Birnbaum-Saunders model using adaptive progressively hybrid censored data and its applications

&lt;abstract&gt;&lt;p&gt;Lately, the Birnbaum-Saunders distribution has gained a lot of attention, mainly due to its different density shapes and the non-monotonicity property of its failure rates. This work considered some estimation issues for the Birnbaum-Saunders distribution using adaptive progressive Type-Ⅱ hybrid censoring. Point and interval estimations were performed employing both conventional and Bayesian methodologies. In addition to estimating the model parameters, we obtained point and interval estimates for the reliability and hazard rate functions. We looked at the method of maximum likelihood as a classical approach, and its asymptotic traits were employed to obtain approximate confidence ranges. From a Bayesian point of perspective, we considered the squared error loss function to obtain the point estimates of the various parameters. The Bayes and highest posterior density credible intervals were additionally determined. For the complex form of the posterior distribution, Bayes estimates and credible intervals were computed by sampling from the posterior distribution through the Markov chain Monte Carlo procedure. For assessing the performance of all of these estimators, a Monte Carlo simulation was employed. Several statistical standards were applied to check the effectiveness of various estimates for multiple levels of censoring with small, moderate, and large sample sizes. Finally, two scenarios for applications were given in order to highlight the usefulness of the supplied approaches.&lt;/p&gt;&lt;/abstract&gt;

Bayesian backcalculation of pavement properties using parallel transitional Markov chain Monte Carlo

AbstractThis paper presents a novel Bayesian method for backcalculation of pavement dynamic modulus, stiffness, thickness, and damping using falling weight deflectometer (FWD) data. The backcalculation procedure yields estimates and uncertainties for each pavement property of interest. As a by‐product of the Bayesian procedure, information about measurement error is recovered. The Bayesian method is tested on simulated FWD backcalculations and compared with a state‐of‐the‐art trust‐region optimization algorithm, and it achieves estimation errors that are nearly an order of magnitude lower than the trust‐region solver. Confidence intervals are computed from thousands of simulated backcalculations and are shown to quantify uncertainty in estimated pavement properties. To cope with the computational expense of backcalculation, a fully parallel transitional Markov chain Monte Carlo procedure is developed. The fully parallel algorithm scales well to computation with many processor cores, and it yields up to a 50% reduction in computation time when compared to existing parallel implementations.

Improved Drought Characteristics in the Pearl River Basin Based on Reconstructed GRACE Solution with Enhanced Temporal Resolution

As global warming intensifies, the damage caused by drought cannot be disregarded. Traditional drought monitoring is often carried out with monthly resolution, which fails to monitor the sub-monthly climatic event. The GRACE-based drought severity index (DSI) is a drought index based on terrestrial water storage anomalies (TWSA) observed by the gravity recovery and climate experiment (GRACE) satellite. DSI has the ability to monitor drought effectively, and it is in good consistency with other drought monitoring methods. However, the temporal resolution of DSI is limited by that of GRACE observations, so it is necessary to obtain TWSA with a higher temporal resolution to calculate DSI. We use a statistical method to reconstruct the TWSA, which adopts precipitation and temperature to obtain TWSA on a daily resolution. This statistical method needs to be combined with the time series decomposition method, and then the parameters are simulated by the Markov chain Monte Carlo (MCMC) procedure. In this study, we use this TWSA reconstruction method to obtain high-quality TWSA at daily time resolution. The correlation coefficient between CSR–TWSA and the reconstructed TWSA is 0.97, the Nash–Sutcliffe efficiency is 0.93, and the root mean square error is 16.57. The quality of the reconstructed daily TWSA is evaluated, and the DSI on a daily resolution is calculated to analyze the drought phenomenon in the Pearl River basin (PRB). The results show that the TWSA reconstructed by this method has high consistency with other daily publicly available TWSA products and TWSA provided by the Center for Space Research (CSR), which proves the feasibility of this method. The correlation between DSI based on reconstructed daily TWSA, SPI, and SPEI is greater than 0.65, which is feasible for drought monitoring. From 2003 to 2021, the DSI recorded six drought events in the PRB, and the recorded drought is more consistent with SPI-6 and SPEI-6. There was a drought event from 27 May 2011 to 12 October 2011, and this drought event had the lowest DSI minimum (minimum DSI = −1.76) recorded among the six drought events. The drought monitored by the DSI is in line with government announcements. This study provides a method to analyze drought events at a higher temporal resolution, and this method is also applicable in other areas.

Regression in extremes using the four-parameter Kappa distribution

The four-parameter Kappa distribution (4KD) is known to be flexible and has been used for extreme value analysis, especially in hydrological sciences. This distribution has the great advantage of encompassing the two main distributions of extreme values used in the literature: the Generalized Pareto Distribution (GPD) and the Generalized Extreme Value (GEV) distribution. However, the analyses made in previous works for 4KD in the context of extreme values are based only on models with static parameters, not allowing the capture of covariate effects such as seasonality and location variables in a specific dataset. This study proposes that the inference about the 4KD can be made under the Bayesian paradigm by inserting a regression structure on the parameters to assess possible effects of covariates under the model. This proposal allows us to interpret the parameters and predict extreme events when there are covariates. The estimation of the parameters is performed using Markov Chain Monte Carlo (MCMC) procedures. The model was applied with both approaches in extremes: maximum and exceedance data. In this analysis, the proposed model was compared with models that use a regression structure in the usual extreme values distribution (GEV and GPD), and it presented better results in fit measures.

Two super-Earths at the edge of the habitable zone of the nearby M dwarf TOI-2095

The main scientific goal of TESS is to find planets smaller than Neptune around stars that are bright enough to allow for further characterization studies. Given our current instrumentation and detection biases, M dwarfs are prime targets in the search for small planets that are in (or near) the habitable zone of their host star. In this work, we use photometric observations and CARMENES radial velocity (RV) measurements to validate a pair of transiting planet candidates found by TESS. The data were fitted simultaneously, using a Bayesian Markov chain Monte Carlo (MCMC) procedure and taking into account the stellar variability present in the photometric and spectroscopic time series. We confirm the planetary origin of the two transiting candidates orbiting around TOI-2095 (LSPM J1902+7525). The star is a nearby M dwarf (d = 41.90 ± 0.03 pc, Teff = 3759 ± 87 K, V = 12.6 mag), with a stellar mass and radius of M* = 0.44 ± 0.02 M⊙ and R* = 0.44 ± 0.02 R⊙, respectively. The planetary system is composed of two transiting planets: TOI-2095b, with an orbital period of Pb = 17.66484 ± (7 × 10−5) days, and TOI-2095c, with Pc = 28.17232 ± (14 × 10−5) days. Both planets have similar sizes with Rb = 1.25 ± 0.07 R⊕ and Rc = 1.33 ± 0.08 R⊕ for planet b and planet c, respectively. Although we did not detect the induced RV variations of any planet with significance, our CARMENES data allow us to set stringent upper limits on the masses of these objects. We find Mb &lt; 4.1 M⊕ for the inner and Mc &lt; 7.4 M⊕ for the outer planet (95% confidence level). These two planets present equilibrium temperatures in the range of 300–350 K and are close to the inner edge of the habitable zone of their star.

A sensitivity analysis framework for SMA-based self-centering systems considering uncertainty in hysteretic material parameters

Shape memory alloys (SMAs) are widely utilized as the core component to develop self-centering (SC) structures owing to their excellent super-elasticity. Seismic performance of SMA-based structures is significantly influenced by the hysteretic parameters of the exploited SMAs. Uncertainties in the hysteretic parameters of SMAs could cause variation in seismic performance however have not received sufficient attentions. This paper proposes a sensitivity analysis framework to realistically and efficiently quantify the sensitivity of SMA-based structures to hysteretic model parameters. The experimental data from SMA bars are analyzed through the Markov chain Monte Carlo (MCMC) procedure. The probabilistic distributions of SMA hysteretic parameters are then sampled using the linear moment theory. Polynomial chaos expansions (PCE) based surrogate model is then used to evaluate sensitivity of response to different parametric uncertainties through the Sobol indices. The proposed framework is then applied to a total of 120 single-degree-of-freedom (SDOF) systems to demonstrate its efficacy to generalize observations and findings. The results show that the proposed framework provided more effective and realistic sensitivity analysis of seismic responses of SMA-based SDOF systems to account for uncertainty in the hysteretic model parameters. The Sobol indices also indicate that the sensitivity of different seismic response quantities varies for different hysteretic model parameters of SMAs.

Stochastic modeling and statistical calibration with model error and scarce data

This paper introduces a procedure to assess the predictive accuracy of stochastic models subject to model error and sparse data. Model error is introduced as uncertainty on the coefficients of appropriate polynomial chaos expansions (PCE). The error associated with finite sample size allows us to conceive of these coefficients as statistics of the data that we describe as random variables whose influence on output quantities of interest is evaluated through the extended polynomial chaos expansion (EPCE). A Bayesian data assimilation scheme is introduced to update these expansions by considering the resulting nested chaos expansion as a hierarchical probabilistic model. Stochastic models of quantities of interest (QoI) are thus constructed and efficiently evaluated. The Metropolis–Hastings Markov chain Monte Carlo procedure is used to sample the posterior. Two illustrative analytical and numerical problems are used to demonstrate the proposed approach.

Flexible Bayesian quantile regression for nonlinear mixed effects models based on the generalized asymmetric Laplace distribution

We propose flexible Bayesian quantile regression for a class of parametric nonlinear mixed effects models for longitudinal data based on the generalized asymmetric Laplace distribution, which exhibits more flexibility in skewness, mode and tail behaviour than the frequently used asymmetric Laplace distribution in quantile regression. An efficient Markov chain Monte Carlo procedure based on the adaptive random walk Metropolis-within-Gibbs sampling algorithm is derived for posterior inference. We demonstrate through simulation studies and empirical analysis that the proposed method could provide more accurate parameter estimation and better model fit than the existing methods.

Statistical Analysis of Inverse Lindley Data Using Adaptive Type-II Progressively Hybrid Censoring with Applications

This paper deals with the statistical inference of the unknown parameter and some life parameters of inverse Lindley distribution under the assumption that the data are adaptive Type-II progressively censored. The maximum likelihood method is considered to acquire the point and interval estimates of the distribution parameter, reliability, and hazard rate functions. The approximate confidence intervals are also addressed. The delta method is taken into consideration to approximate the variances of the estimators of the reliability and hazard rate functions to get the required intervals. Based on the assumption of gamma prior, we further consider Bayesian estimation of the different parameters. The Bayes estimates are obtained by considering squared error and general entropy loss functions. The Bayes estimates and highest posterior density credible intervals are obtained by employing the Markov chain Monte Carlo procedure. An exhaustive numerical study is conducted to compare the offered estimates with regard to their root means squared error, relative absolute biases, confidence lengths, and coverage probabilities. To explain the suggested methods, two applications are investigated. The numerical findings show that the Bayes estimates perform better than those obtained based on the maximum likelihood method. The Bayesian estimations using the asymmetric loss function give more efficient estimates than the symmetric loss. Finally, the inverse Lindley distribution is recommended to be used as a suitable model to fit airborne communication transceiver and wooden toys data sets when compared with some competitive models including inverse Weibull, inverse gamma and alpha power inverted exponential.

Bayesian inversion of log-normal eikonal equations

We study the Bayesian inverse problem for inferring the log-normal slowness function of the eikonal equation, given noisy observation data on its solution at a set of spatial points. We contribute rigorous proofs on the existence and well-posedness of the problem. We then study approximation of the posterior probability measure by solving the truncated eikonal equation, which contains only a finite number of terms in the Karhunen–Loeve expansion of the slowness function, by the fast marching method (FMM). The error of this approximation in the Hellinger metric is deduced in terms of the truncation level of the slowness and the grid size in the FMM resolution. It is well known that the plain Markov chain Monte Carlo (MCMC) procedure for sampling the posterior probability is highly expensive. We develop and justify the convergence of a multilevel MCMC method. Using the heap sort procedure in solving the forward eikonal equation by the FMM, our multilevel MCMC method achieves a prescribed level of accuracy for approximating the posterior expectation of quantities of interest, requiring only an essentially optimal level of complexity. Numerical examples confirm the theoretical results.

Reliability Estimation under Normal Operating Conditions for Progressively Type-II XLindley Censored Data

This paper assumes constant-stress accelerated life tests when the lifespan of the test units follows the XLindley distribution. In addition to the maximum likelihood estimation, the Bayesian estimation of the model parameters is acquired based on progressively Type-II censored samples. The point and interval estimations of the model parameters and some reliability indices under normal operating conditions at mission time are derived using both estimation methods. Using the Markov chain Monte Carlo algorithm, the Bayes estimates are calculated using the squared error loss function. Simulating the performances of the different estimation methods is performed to illustrate the proposed methodology. As an example of how the proposed methods can be applied, we look at two real-life accelerated life test cases. According to the numerical outcomes and based on some criteria, including the root of the mean square error and interval length, we can conclude that the Bayesian estimation method based on the Markov chain Monte Carlo procedure performs better than the classical methods in evaluating the XLindley parameters and some of its reliability measures when a constant-stress accelerated life test is applied with progressively Type-II censoring.

Reliability Estimation of XLindley Constant-Stress Partially Accelerated Life Tests using Progressively Censored Samples

It often takes a lot of time to conduct life-testing studies on products or components. Units can be tested under more severe circumstances than usual, known as accelerated life tests, to reduce the testing period. This study’s goal is to look into certain estimation issues related to point and interval estimations for XLindley distribution under constant stress partially accelerated life tests with progressive Type-II censored samples. The maximum likelihood approach is utilized to acquire the point and interval estimates of the model parameters as well as the reliability function under normal use conditions. The Bayesian estimation method using the Monte Carlo Markov Chain procedure using the squared error loss function is also provided. Moreover, the Bayes credible intervals as well as the highest posterior density credible intervals of the different parameters are considered. To make comparisons between the proposed methods, a simulation study is conducted with various sample sizes and different censoring schemes. The usefulness of the suggested methodologies is then demonstrated by the analysis of two data sets. A summary of the major findings of the study can be found in the conclusion.

Testing independence of bivariate censored data using random walk on restricted permutation graph

In this paper, we propose a procedure to test the independence of bivariate censored data, which is generic and applicable to any censoring types in the literature. To test the hypothesis, we consider a rank-based statistic, Kendall’s tau statistic. The censored data defines a restricted permutation space of all possible ranks of the observations. We propose the statistic, the average of Kendall’s tau over the ranks in the restricted permutation space. To evaluate the statistic and its reference distribution, we develop a Markov chain Monte Carlo (MCMC) procedure to obtain uniform samples on the restricted permutation space and numerically approximate the null distribution of the averaged Kendall’s tau. We numerically compare the power of our procedure to existing state of the art procedures in the literature under various censoring types. We apply the procedure to three real data examples with different censoring types, and compare the results with those by existing methods.

A Bayesian Variable Selection Method for Spatial Autoregressive Quantile Models

In this paper, a Bayesian variable selection method for spatial autoregressive (SAR) quantile models is proposed on the basis of spike and slab prior for regression parameters. The SAR quantile models, which are more generalized than SAR models and quantile regression models, are specified by adopting the asymmetric Laplace distribution for the error term in the classical SAR models. The proposed approach could perform simultaneously robust parametric estimation and variable selection in the context of SAR quantile models. Bayesian statistical inferences are implemented by a detailed Markov chain Monte Carlo (MCMC) procedure that combines Gibbs samplers with a probability integral transformation (PIT) algorithm. In the end, empirical numerical examples including several simulation studies and a Boston housing price data analysis are employed to demonstrate the newly developed methodologies.

An efficient Bayesian modelling of extreme winds in the favour​ of energy generation in Pakistan

Daily and annual maximum wind speed quantiles can be estimated using extreme value theory for any metrological site of interest. These estimates are of vast importance for modelling and predicting maximum wind speed. This paper develops an efficient modelling paradigm of extreme winds by analysing daily and annual maximum wind speed data via frequentist and Bayesian methodologies. For this purpose, the generalized extreme value (GEV) model is used for yearly maxima, and the generalized Pareto distribution (GPD) is used for daily exceedance over a high threshold. In frequentist inference, the parameters of both models are estimated using the maximum likelihood and linear moments method. In contrast, the Bayesian Markov Chain Monte Carlo procedure with the Metropolis–Hasting algorithm is used. In addition, the informative priors for both models are constructed empirically using historical records of wind speed data from five other weather stations of Pakistan and one belonging to India. The results show that the Bayesian modelling provides apparent benefits in terms of improved accuracy in the estimation of the parameters as well as return levels of both distributions. Furthermore, the Bayesian analysis expresses that posterior inference might be affected by the choice of priors used to formulate the informative priors. Overall, based on assessment measures, the GPD fitted through Bayesian informative priors provides an efficient estimation strategy in terms of precision than other frameworks when uncertainty in parameters and return levels are taken into account. Our methodology can be implemented easily to other regions by considering the prior information from the border area stations of other countries (e.g., China, Afghanistan, India, and Iran). Moreover, the return level estimates of the GPD based on informative Bayesian priors are very beneficial in policymaking and wind energy generation engineering for the Thatta region of the country.

Cosmicflows-4

With Cosmicflows-4, distances are compiled for 55,877 galaxies gathered into 38,065 groups. Eight methodologies are employed, with the largest numbers coming from the correlations between the photometric and kinematic properties of spiral galaxies (TF) and elliptical galaxies (FP). Supernovae that arise from degenerate progenitors (type Ia SNe) are an important overlapping component. Smaller contributions come from distance estimates from the surface brightness fluctuations of elliptical galaxies and the luminosities and expansion rates of core-collapse supernovae (SNe II). Cepheid period–luminosity relation and tip of the red giant branch observations founded on local stellar parallax measurements along with the geometric maser distance to NGC 4258 provide the absolute scaling of distances. The assembly of galaxies into groups is an important feature of the study in facilitating overlaps between methodologies. Merging between multiple contributions within a methodology and between methodologies is carried out with Bayesian Markov chain Monte Carlo procedures. The final assembly of distances is compatible with a value of the Hubble constant of H 0 = 74.6 km s−1 Mpc−1 with the small statistical error of ±0.8 km s−1 Mpc−1 but a large potential systematic error of ∼3 km s−1 Mpc−1. Peculiar velocities can be inferred from the measured distances. The interpretation of the field of peculiar velocities is complex because of large errors on individual components and invites analyses beyond the scope of this study.

A Bayesian Method for Material Identification of Composite Plates via Dispersion Curves

Ultrasonic guided waves offer a convenient and practical approach to structural health monitoring and non-destructive evaluation. A key property of guided waves is the fully defined relationship between central frequency and propagation characteristics (phase velocity, group velocity and wavenumber)-which is described using dispersion curves. For many guided wave-based strategies, accurate dispersion curve information is invaluable, such as group velocity for localisation. From experimental observations of dispersion curves, a system identification procedure can be used to determine the governing material properties. As well as returning an estimated value, it is useful to determine the distribution of these properties based on measured data. A method of simulating samples from these distributions is to use the iterative Markov-Chain Monte Carlo (MCMC) procedure, which allows for freedom in the shape of the posterior. In this work, a scanning-laser Doppler vibrometer is used to record the propagation of Lamb waves in a unidirectional-glass-fibre composite plate, and dispersion curve data for various propagation angles are extracted. Using these measured dispersion curve data, the MCMC sampling procedure is performed to provide a Bayesian approach to determining the dispersion curve information for an arbitrary plate. The distribution of the material properties at each angle is discussed, including the inferred confidence in the predicted parameters. The percentage errors of the estimated values for the parameters were 10-15 points larger when using the most likely estimates, as opposed to calculating from the posterior distributions, highlighting the advantages of using a probabilistic approach.

Solar data uncertainty impacts on MCMC methods for r-process nucleosynthesis

In recent work, we developed a Markov Chain Monte Carlo (MCMC) procedure to predict the ground state masses capable of forming the observed Solar r-process rare-earth abundance peak. By applying this method to nucleosynthesis calculations which make use of distinct astrophysical conditions and comparing our results to the latest precision mass measurements, we are able to shed light on the conditions/masses capable of producing a rare-earth peak which matches Solar data. Here we examine how our mass predictions change when using a few different sets of r-process Solar abundance residuals that have been reported in the literature. We explore how the differing error estimates of these Solar evaluations propagate through the Markov Chain Monte Carlo to our mass predictions. We find that Solar data which reports the rare-earth peak to have its highest abundance at mass number A = 162 can require distinctly different mass predictions from data with the peak centered at A = 164. Nevertheless, we find that two important general conclusions from past work, regarding the inconsistency of ‘cold’ astrophysical outflows with current mass measurements and the need for local stability at N = 104 in ‘hot’ scenarios, remain robust in the face of differing Solar data evaluations. Additionally, we show that the masses our procedure finds capable of producing a peak at A &amp;lt; 164 are not in line with the latest precision mass measurements.