Metropolis-Hastings Research Articles

Statistical inference and data analysis for inverted Kumaraswamy distribution based on maximum ranked set sampling with unequal samples

A very useful modification to ranked set sampling (RSS) that allows a larger set size without significantly increasing ranking errors is the maximum ranked set sampling with unequal samples (MRSSU) approach. This article covers the parameter estimation of the inverted Kumaraswamy distribution using MRSSU and RSS designs. The maximum likelihood and Bayesian estimation techniques are considered. The regarded Bayesian estimation technique is determined in the case of non-informative and informative priors represented by Jeffreys and gamma priors, respectively. Squared error and minimum expected are the two loss functions that are employed. We presented a simulation study to evaluate the performance of the recommended estimations using root mean squared error and relative bias. The Bayes point estimates were computed using the Metropolis–Hastings algorithm. Additional conclusions have been made based on actual geological data regarding the intervals between Kiama Blowhole’s 64 consecutive eruptions. Based on the same number of measured units, the results of simulation and real data analysis showed that MRSSU estimators performed much better than their RSS counterparts.

A multi-objective approach for timber harvest scheduling to include management of at-risk species and spatial configuration objectives.

Sustainable forestry typically involves integration of several economic and ecological objectives which, at times, may not be compatible with one another. Multi-objective prioritization via harvest scheduling programs can be used to elucidate these relationships and explore solutions. One such program is a spatially explicit harvest scheduler that adopts the Metropolis-Hastings algorithm to iteratively find management solutions to achieve multiple objectives (Habplan). Although this program has been used to address forest management scheduling and simulation-based tasks, its utility is constrained by time-intensive data preparation and challenges with incorporating spatial configuration objectives. To address these shortcomings, we introduce an open-source software package, HabplanR, streamlines data preparation, sets parameters, visualizes results, and assesses spatial components of ecological objectives. We developed four example objectives to incorporate into a multi-objective management problem: habitat quality indices for three species "types" (open, closed, and intermediate-canopy-associated species), and harvested pine pulpwood (revenue). We demonstrate the utility of this package to find management schedules that can accommodate potentially conflicting habitat needs of species, while achieving economic targets. We produced 100 software runs and prioritized individual objectives to select four management schedules for further comparisons. We compared outcome differences of the four schedules, including a spatial comparison of two high performing schedules. The software package makes costs and benefits of different schedules explicit and allows for consideration of the spatial configuration of management outcomes in decision-making.

Partition function approach to non-Gaussian likelihoods: macrocanonical partitions and replicating Markov-chains

Monte-Carlo techniques are standard numerical tools for exploring non-Gaussian and multivariate likelihoods. Many variants of the original Metropolis-Hastings algorithm have been proposed to increase the sampling efficiency. Motivated by Ensemble Monte Carlo we allow the number of Markov chains to vary by exchanging particles with a reservoir, controlled by a parameter analogous to a chemical potential μ, which effectively establishes a random process that samples microstates from a macrocanonical instead of a canonical ensemble. In this paper, we develop the theory of macrocanonical sampling for statistical inference on the basis of Bayesian macrocanonical partition functions, thereby bringing to light the relations between information-theoretical quantities and thermodynamic properties. Furthermore, we propose an algorithm for macrocanonical sampling, 𝙰𝚟𝚊𝚕𝚊𝚗𝚌𝚑𝚎 𝚂𝚊𝚖𝚙𝚕𝚒𝚗𝚐, and apply it to various toy problems as well as the likelihood on the cosmological parameters Ωm and w on the basis of data from the supernova distance redshift relation.

Modeling evacuation activities amid compound hazards: Insights from hurricane Irma in Southeast Florida

Given the destructive nature of hurricanes in tropical regions, pre-disaster evacuation has emerged as a critical approach for hurricane preparedness. Nevertheless, the compounding effects of natural hazards and the outbreak of infectious diseases, such as Covid-19, significantly challenge hurricane evacuation management. To investigate emergency responses under compound hazards, this study develops an activity-based model to measure the evacuation behaviors of individuals, using Hurricane Irma as a case study. Four scenarios are designed, including a single hurricane hazard, Hurricane Irma compounded with a pandemic like Covid-19, Hurricane Irma compounded with flood damage to the transportation network, and a combination of all these hazards. The metropolis-hasting algorithm is utilized to generate a population with socioeconomic attributes, which is then allocated to census block groups covering Palm Beach, Broward, Miami-Dade, and Monroe Counties in Florida. Datasets from multiple sources are used to measure evacuation decisions, which are subsequently simulated using MATSim. The results highlight the potential impacts of compound hazards on transportation systems, including increased congestions in scenarios involving compounded hurricanes and floods, especially between 10 a.m. and 7p.m. Moreover, a higher proportion of socially vulnerable populations is observed in scenarios involving compounded hurricanes and pandemics, particularly in the Key West area. The developed model could be further applied to measure the indirect impacts of natural hazards on transportation systems.

Parallel multiple-chain DRAM probabilistic model for ultrasonic pulse velocity of concrete performance in the pipe of arch bridge

The existing models for predicting the compaction state of concrete inside steel tubes based on ultrasonic wave velocity have a major limitation in that they cannot comprehensively consider concrete compressive strength, instar, and various sources of uncertainties, both subjective and objective. To overcome this limitation, the study introduced a probability model for ultrasonic wave velocity in the concrete performance inside arch bridge steel tubes, based on the parallel multiple-chain Delayed Rejection Adaptive Metropolis (DRAM) algorithm. Initially, the study considers the influence of concrete compressive strength and instar, establishing a simplified deterministic model for predicting concrete compactness inside the tubes. Building on this, the study incorporates material stochastic uncertainty and cognitive uncertainty, deriving analytical expressions for a probabilistic prediction model of concrete compactness. Furthermore, using parallel processing alongside the DRAM algorithm, the study explores how the number of parallel chains and the order of acceptance probabilities influence the probabilistic model parameters. The study selected and updated the posterior distribution information of the probabilistic model parameters. Finally, the effectiveness of the model was validated using experimental data. As shown in the analysis, with better prediction accuracy and lower dispersion, the model not only selects the optimal Markov chain iteration path in a short time, but also corrects the parameters affecting the prediction accuracy of the existing model, and also provides a probabilistic method for correcting the confidence intervals of the existing model.

Modeling and Hybrid Inversion of Mineral Deposits Using the Dipping Dike Model with Finite Depth Extent

The dipping dike model has shown to be a useful approximation for mineral deposits. To make this model more realistic, we include the thickness, which yields the depth to the bottom, as an additional parameter. The magnetic anomaly is obtained by combining the anomalies of two infinite dikes, so that the resulting expression is simpler than the classical prismatic models with polygonal cross section. We employ a Metropolis-Hasting (MH) algorithm coupled with the Levenberg-Marquardt (LM) method to invert magnetic profiles assuming a model of multiple dike-like sources. We use a few iterations of the LM method to improve the candidate solutions at the end of each random walk generated by MH. The following parameters are obtained: depth to the top, thickness, half-width, horizontal location of the top center, geological dip, in addition to two effective parameters that depend on the intensity of magnetization and the directions of the induced and remanent fields. For synthetic anomalies, both noise-free and noisy magnetic data are considered, with examples presented for each scenario. These examples highlight the discrepancy between models with finite and infinite sources. They also illustrate the higher accuracy of the hybrid MH-LM method over the pure MH approach. Moreover, two field examples related to mineral exploration have been considered: the Pima copper mine, United States, where the relative differences between the parameters obtained by our algorithm and those known from drilling are not higher than 10%, and a magnetic profile over iron ore deposits located in Laje, northeast Brazil, where the inverted parameters were useful for detailing previous studies.

A new Bayesian method for estimation of value at risk and conditional value at risk

AbstractValue at Risk (VaR) and Conditional Value at Risk (CVaR) have become the most popular measures of market risk in Financial and Insurance fields. However, the estimation of both risk measures is challenging, because it requires the knowledge of the tail of the distribution. Therefore, Extreme Value Theory initially seemed to be one of the best tools for this kind of problems, because using peaks-over-threshold method, we can assume the tail data approximately follow a Generalized Pareto distribution (GPD). The main objection to its use is that it only employs observations over the threshold, which are usually scarce. With the aim of improving the inference process, we propose a new Bayesian method that computes estimates built with all the information available. Informative prior Bayesian (IPB) method employs the existing relations between the parameters of the loss distribution and the parameters of the GPD that models the tail data to define informative priors in order to perform Metropolis–Hastings algorithm. We show how to apply IPB when the distribution of the observations is Exponential, stable or Gamma, to make inference and predictions. .Afterwards, we perform a thorough simulation study to compare the accuracy and precision of the estimates computed by IPB and the most employed methods to estimate VaR and CVaR. Results show that IPB provides the most accurate, precise and least biased estimates, especially when there are very few tail data. Finally, data from two real examples are analysed to show the practical application of the method.

Stochastic relativistic advection diffusion equation from the Metropolis algorithm

Stochastic relativistic advection diffusion equation from the Metropolis algorithm

Analysis of recurrent event data with spatial random effects using a Bayesian approach.

Recurrent event data, which represent the occurrence of repeated incidences, are common in observational studies. Furthermore, collecting possible spatial correlations in health and environmental data is likely to provide more information for risk prediction. This article proposes a comprehensive proportional intensity model considering spatial random effects for recurrent event data using a Bayesian approach. The spatial information for areal data (where the spatial location is known up to a geographic unit such as a county) and georeferenced data (where the location is exactly observed) is examined. A traditional constant baseline intensity function, as well as a flexible piecewise constant baseline intensity function, are both under consideration. To estimate the parameters, a Markov chain Monte Carlo method with the Metropolis-Hastings algorithm and the adaptive Metropolis algorithm are applied. To assess the performance of model fitting, the deviance information criterion and log pseudo marginal likelihood are proposed. Overall, simulation studies demonstrate that the proposed model is significantly better than models that do not consider spatial effects if spatial correlations exist. Finally, our approach is implemented using a dataset related to the recurrence of cardiovascular diseases, which incorporates spatial information.

Markov chain Monte Carlo methods applied to the stochastic inversion of 1D viscoelastic parameters

Abstract The characterization of the subsurface profile properties that are susceptible to liquefaction during earthquakes is of great importance in accident prevention. Field data can be used in a stochastic inverse problem to estimate the material properties using Markov chain Monte Carlo Methods (McMC) that incorporates prior knowledge about the unknown parameters as well as the data available. Here, we numerically study the Random Walk (RW) and Differential Evolution (DE) variations of the Metropolis algorithm in the context of seismic modeling considering the viscoelastic wave equation. The algorithms are tested with data from the 1987 Superstition Hills earthquake recorded at the Wildlife Site. The results show the effectiveness and accuracy of these algorithms, with emphasis to DE which reached convergence earlier compared to RW.

Bayesian Inference for Estimating Heat Sources through Temperature Assimilation

Abstract This paper utilizes a Bayesian inference framework to address the two-dimensional steady-state heat conduction problem, focusing on the estimation of unknown distributed heat sources in a thermally-conducting medium with uniform conductivity. The goal is to infer the locations, strength, and shape of heaters by assimilating temperature data in Euclidean space, employing a Fourier series to represent each heater's shape. The Markov Chain Monte Carlo method, incorporating the random-walk Metropolis-Hasting algorithm and parallel tempering, is utilized for posterior distribution exploration in both unbounded and wall-bounded domains. It is found that multiple solutions arise in cases where the number of temperature sensors is less than the number of unknown states. Moreover, smaller heaters introduce greater uncertainty in estimated strength. To address the challenge of estimating the heater's strength and shape simultaneously due to their strong correlation, our method incorporates sharp priors on one to ensure accurate and feasible solutions of the other. The diffusive nature of heat conduction smooths out any deformations in the temperature contours, especially in the presence of multiple heaters positioned near each other, impacting convergence. In wall-bounded domains with Neumann boundary conditions, the inference of heater parameters tends to be more accurate than in unbounded domains.

Thermodynamic equilibrium of [formula omitted] Ising model on square lattice

We constructed a theoretical magnetic phase diagram in an external magnetic field T(P+), making it possible to determine the conditions for the existence of ferromagnets, antiferromagnets, and spin glass phases. The high-performance CUDA software package was used to the complete enumeration of all configurations of finite number spins in the ±J Ising model. We performed the rigorous numerical calculation of the partition function of N=8×8 systems of interacting spins with open boundary conditions. We used Monte Carlo methods like the Metropolis algorithm to calculate the critical temperatures for N=40×40 spins. The results of the Monte Carlo experiments are consistent with rigorous calculation data. The transition from the spin glass to the induced ferromagnetic state in an external field occurs without any critical change in the heat capacity. We used the ±J Ising model to calculate the instability line (AT—line) for the heat capacity of spin glass in the H−T diagram in an external magnetic field and the behavior of magnetic susceptibility in an external magnetic field. A rigorous calculation of the partition function allowed us to calculate all possible states and their thermodynamic probability. The calculation of the partition function meant that the model’s physics was obtained in an equilibrium state. The instability line was calculated for spin glass in the equilibrium state.

Model-independent cosmographic constraints from DESI 2024

Context. We explore model-independent constraints on the Universe kinematics up to the snap and jerk hierarchical terms, considering the latest baryon acoustic oscillation (BAO) release provided by the DESI collaboration. Aims. We intend to place novel and more stringent constraints on the cosmographic series, incorporating three combinations of data catalogs: the first made by BAO and observational cosmic chronometers, the second made by BAO and type Ia supernovae, and the last including all the cited data sets. Methods. Considering the latest BAO data provided by the DESI collaboration and tackling the rd parameter to span within the range [144,152] Mpc, with a fixed step of δrd = 2 Mpc, we employed Monte Carlo Markov chain analyses based on the Metropolis algorithm to fix novel bounds on the cosmographic series, fixing the deceleration, q0 , the jerk, j0 , and the snap, s0, parameters, up to the 2σ level. A comparison between the results of the Planck satellite with those obtained by the DESI collaboration is also reported. Results. Our findings showcase a significant departure in terms of j0 even at the 1σ confidence level, albeit compatible with the ACDM paradigm in regard to q0 and s0 at the 2σ level. Analogously, the h0 tension appears alleviated in the second hierarchy when including snap. Conclusions. Our method excludes models that significantly depart from the standard cosmological model. Particularly, direct comparisons with the ACDM and wCDM models and the Chevallier-Polarski-Linder parameterisation are explored, which definitively favour the wCDM scenario over other approaches, contradicting the findings of the original DESI collaboration.

Multivariate moment least-squares variance estimators for reversible Markov chains

Markov chain Monte Carlo (MCMC) is a commonly used method for approximating expectations with respect to probability distributions. Uncertainty assessment for MCMC estimators is essential in practical applications. Moreover, for multivariate functions of a Markov chain, it is important to estimate not only the auto-correlation for each component but also to estimate cross-correlations, in order to better assess sample quality, improve estimates of effective sample size, and use more effective stopping rules. Berg and Song (2023) introduced the moment least squares (momentLS) estimator, a shape-constrained estimator for the autocovariance sequence from a reversible Markov chain, for univariate functions of the Markov chain. Based on this sequence estimator, they proposed an estimator of the asymptotic variance of the sample mean from MCMC samples. In this study, we propose novel autocovariance sequence and asymptotic variance estimators for Markov chain functions with multiple components, based on the univariate momentLS estimators from Berg and Song (2023). We demonstrate strong consistency of the proposed auto(cross)-covariance sequence and asymptotic variance matrix estimators. We conduct empirical comparisons of our method with other state-of-the-art approaches on simulated and real-data examples, using popular samplers including the random-walk Metropolis sampler and the No-U-Turn sampler from STAN. Supplemental materials for this article are available online.

AN EFFICIENT ANALYTICAL METHOD FOR DETERMINING TRANSITION TEMPERATURES IN CORE‐SHELL SPIN CROSSOVER NANOPARTICLES DESCRIBED BY AN EXTENDED ISING‐LIKE MODEL

This contribution is dedicated to the analysis of the size and interaction parameters effects on the thermal properties of square‐shaped cross‐spin transition core‐shell nanoparticles. The present study is carried out by varying the shell size at fixed core size and vice versa. In this problem, the core and shell are active from the spin transition point of view, while having different properties in terms of ligand field (transition temperatures) and interactions. The resulting nanoparticle is described by an Ising‐type model including the three interaction parameters JCC, JSS and JCS coupling core‐core, shell‐shell and core‐shell sites, respectively. The thermodynamic investigations have been carried out in 3 different ways: (i) by Monte Carlo Entropic Sampling (MCES) for small size nanoparticles ([[EQUATION]]) (ii) by usual Monte Carlo Metropolis for higher dimensions (iii) by an analytical method deducing the equilibrium temperature of the system from the various local environments. We demonstrate that, as long as a clear plateau region occurs in the thermal‐dependence of the high‐spin fraction, an excellent agreement is obtained, allowing to predict analytically the evolution of the transition temperatures. The limitations of the method are discussed in the case of strong interference between the thermal dependences of the core and shell.

An Elite Wolf Pack Algorithm Based on the Probability Threshold for a Multi-UAV Cooperative Reconnaissance Mission

In the task assignment problem of multi-UAV collaborative reconnaissance, existing algorithms have issues with inadequate solution accuracy, specifically manifested as large spatial spans and knots of routes in the task execution of UAVs. To address the above challenges, this paper presents a multi-UAV task assignment model under complex conditions (MTAMCC). To efficiently solve this model, this paper proposes an elite wolf pack algorithm based on probability threshold (EWPA-PT). The EWPA-PT algorithm combines the wandering behavior in the traditional wolf pack algorithm with the genetic algorithm. It introduces an ordered permutation problem to calculate the adaptive wandering times of the detective wolves in a specific direction. During the calling phase of the algorithm, the fierce wolves in the wolf pack randomly learn the task assignment results of the head wolf. The sieging behavior introduces the Metropolis criterion from the simulated annealing algorithm to replace the distance threshold in traditional wolf pack algorithms with a probability threshold, which dynamically changes during the iteration process. The wolf pack updating mechanism leverages the task assignment experience of the elite group to reconstruct individual wolves, thereby improving the individual reconstruction’s efficiency. Experiments demonstrate that the EWPA-PT algorithm significantly improves solution accuracy compared to typical methods in recent years.

Estimation and Bayesian Prediction of the Generalized Pareto Distribution in the Context of a Progressive Type-II Censoring Scheme

The generalized Pareto distribution plays a significant role in reliability research. This study concentrates on the statistical inference of the generalized Pareto distribution utilizing progressively Type-II censored data. Estimations are performed using maximum likelihood estimation through the expectation–maximization approach. Confidence intervals are derived using the asymptotic confidence intervals. Bayesian estimations are conducted using the Tierney and Kadane method alongside the Metropolis–Hastings algorithm, and the highest posterior density credible interval estimation is accomplished. Furthermore, Bayesian predictive intervals and future sample estimations are explored. To illustrate these inference techniques, a simulation and practical example are presented for analysis.

Kinetics of Viral Shedding for Outbreak Surveillance of Emerging Infectious Diseases: Modeling Approach to SARS-CoV-2 Alpha and Omicron Infection.

Previous studies have highlighted the importance of viral shedding using cycle threshold (Ct) values obtained via reverse transcription polymerase chain reaction to understand the epidemic trajectories of SARS-CoV-2 infections. However, it is rare to elucidate the transition kinetics of Ct values from the asymptomatic or presymptomatic phase to the symptomatic phase before recovery using individual repeated Ct values. This study proposes a novel Ct-enshrined compartment model to provide a series of quantitative measures for delineating the full trajectories of the dynamics of viral load from infection until recovery. This Ct-enshrined compartment model was constructed by leveraging Ct-classified states within and between presymptomatic and symptomatic compartments before recovery or death among people with infections. A series of recovery indices were developed to assess the net kinetic movement of Ct-up toward and Ct-down off recovery. The model was applied to (1) a small-scale community-acquired Alpha variant outbreak under the "zero-COVID-19" policy without vaccines in May 2021 and (2) a large-scale community-acquired Omicron variant outbreak with high booster vaccination rates following the lifting of the "zero-COVID-19" policy in April 2022 in Taiwan. The model used Bayesian Markov chain Monte Carlo methods with the Metropolis-Hastings algorithm for parameter estimation. Sensitivity analyses were conducted by varying Ct cutoff values to assess the robustness of the model. The kinetic indicators revealed a marked difference in viral shedding dynamics between the Alpha and Omicron variants. The Alpha variant exhibited slower viral shedding and lower recovery rates, but the Omicron variant demonstrated swifter viral shedding and higher recovery rates. Specifically, the Alpha variant showed gradual Ct-up transitions and moderate recovery rates, yielding a presymptomatic recovery index slightly higher than 1 (1.10), whereas the Omicron variant had remarkable Ct-up transitions and significantly higher asymptomatic recovery rates, resulting in a presymptomatic recovery index much higher than 1 (152.5). Sensitivity analysis confirmed the robustness of the chosen Ct values of 18 and 25 across different recovery phases. Regarding the impact of vaccination, individuals without booster vaccination had a 19% higher presymptomatic incidence rate compared to those with booster vaccination. Breakthrough infections in boosted individuals initially showed similar Ct-up transition rates but higher rates in later stages compared to nonboosted individuals. Overall, booster vaccination improved recovery rates, particularly during the symptomatic phase, although recovery rates for persistent asymptomatic infection were similar regardless of vaccination status once the Ct level exceeded 25. The study provides new insights into dynamic Ct transitions, with the notable finding that Ct-up transitions toward recovery outpaced Ct-down and symptom-surfacing transitions during the presymptomatic phase. The Ct-up against Ct-down transition varies with variants and vaccination status. The proposed Ct-enshrined compartment model is useful for the surveillance of emerging infectious diseases in the future to prevent community-acquired outbreaks.

Simulation of Discovering Market Opportunities Using AI Methods

Due to the strong turbulence that exists in the business environment, the view is sometimes articulated that the time horizon of decisions is no longer a hallmark of strategic management. Opposing views argue that a turbulent environment changes the trajectory of reaching future objectives determined by strategic decisions. Such trajectories require the company to be agile, which is manifested by seizing opportunities. Opportunities arise because the environment is volatile. An example of market opportunity is the unsatisfied demand for products. Buyers are driven by different motives when choosing between products that are alternative to them. From the producer's point of view, the purchase is therefore random. As a consequence, the alignment of the offered product portfolio with the actual purchasing decisions of customers is undetermined. In this article, we address the problem of determining the product portfolio structure expected by customers, i.e. one that is consistent with their future purchasing decisions. We hypothesise that this problem can be represented by a model of flows between alternative resources. Problems of this type are solved using Markov Chain Monte Carlo (MCMC) methods. In order to discover the expected demand (opportunity), we simulated the flow model we developed using the Metropolis-Hastings algorithm, which is a special case of the MCMC method used in AI. Due to operating on numerical data, such simulations fall under quantitative research. In the article, we present a model of customer flows between alternative products and then the simulation result, which is the expected value of these flows (stationary point). We convert this value into the structure of the product portfolio using a model of customer purchasing decisions that we have developed. The obtained results confirm the effectiveness of our method and have significant practical value, especially for SMEs. It also enables the assessment of the company's product strategy based on real sales data. The application of our methodology is limited to discovering opportunities in situations where purchasing decisions are influenced by variable environmental factors, causing irregular purchases by individual customers. It is useful, for example, in discovering opportunities in markets such as Household Articles, Furniture, Vehicles, Equipment Repairs, Construction, and many others.

Monte Carlo simulations of glass-forming liquids beyond Metropolis.

Monte Carlo simulations are widely employed to measure the physical properties of glass-forming liquids in thermal equilibrium. Combined with local Monte Carlo moves, the Metropolis algorithm can also be used to simulate the relaxation dynamics, thus offering an efficient alternative to molecular dynamics. Monte Carlo simulations are, however, more versatile because carefully designed Monte Carlo algorithms can more efficiently sample the rugged free energy landscape characteristic of glassy systems. After a brief overview of Monte Carlo studies of glass-formers, we define and implement a series of Monte Carlo algorithms in a three-dimensional model of polydisperse hard spheres. We show that the standard local Metropolis algorithm is the slowest and that implementing collective moves or breaking detailed balance enhances the efficiency of the Monte Carlo sampling. We use time correlation functions to provide a microscopic interpretation of these observations. Seventy years after its invention, the Monte Carlo method remains the most efficient and versatile tool to compute low-temperature properties in supercooled liquids.