Traditional Monte Research Articles

A fully Bayesian approach to sparse reduced-rank multivariate regression

In the context of high-dimensional multivariate linear regression, sparse reduced-rank regression (SRRR) provides a way to handle both variable selection and low-rank estimation problems. Although there has been extensive research on SRRR, statistical inference procedures that deal with the uncertainty due to variable selection and rank reduction are still limited. To fill this research gap, we develop a fully Bayesian approach to SRRR. A major difficulty that occurs in a fully Bayesian framework is that the dimension of parameter space varies with the selected variables and the reduced-rank. Due to the varying-dimensional problems, traditional Markov chain Monte Carlo (MCMC) methods such as Gibbs sampler and Metropolis-Hastings algorithm are inapplicable in our Bayesian framework. To address this issue, we propose a new posterior computation procedure based on the Laplace approximation within the collapsed Gibbs sampler. A key feature of our fully Bayesian method is that the model uncertainty is automatically integrated out by the proposed MCMC computation. The proposed method is examined via simulation study and real data analysis.

Mini-Batch Metropolis–Hastings With Reversible SGLD Proposal

Traditional Markov chain Monte Carlo (MCMC) algorithms are computationally intensive and do not scale well to large data. In particular, the Metropolis–Hastings (MH) algorithm requires passing over the entire dataset to evaluate the likelihood ratio in each iteration. We propose a general framework for performing MH-MCMC using mini-batches of the whole dataset and show that this gives rise to approximately a tempered stationary distribution. We prove that the algorithm preserves the modes of the original target distribution and derive an error bound on the approximation with mild assumptions on the likelihood. To further extend the utility of the algorithm to high-dimensional settings, we construct a proposal with forward and reverse moves using stochastic gradient and show that the construction leads to reasonable acceptance probabilities. We demonstrate the performance of our algorithm in both low dimensional models and high dimensional neural network applications. Particularly in the latter case, compared to popular optimization methods, our method is more robust to the choice of learning rate and improves testing accuracy. Supplementary materials for this article are available online.

An efficient Monte Carlo simulation method for thermal radiation transport

Thermal radiation transfer in material and the interaction between radiative photon and material are important research projects of the inertial confinement fusion, and Monte Carlo method is one of the important researching methods. Based on the implicit integral-differential transport equation, traditional implicit Monte Carlo method can accurately simulate the thermal radiation transport in material. However, the implicit Monte Carlo method would take quite a long computational time when the opacity is increased because scattering events are dominant in particles’ histories, thus reducing the simulation efficiency. In this paper, based on discrete diffusion Monte Carlo method, i.e. a radiation transport code, the discrete diffusion Monte Carlo radiation transport simulation, is developed. The code increases the efficiency of thermal radiation simulations in a high opacity range, but it can yield insufficiently accurate results in a low opacity range. There exist low opacity material and high opacity material in the inertial confinement fusion. In this work, investigated are several numerical techniques that can improve the utility and accuracy of discrete diffusion Monte Carlo for grey thermal radiation simulation. First, the discrete diffusion Monte Carlo method and implicit Monte Carlo method are combined. Second, a new method of treating the interface between the diffusive region and the transport region is proposed. Finally, a hybrid radiative transfer program is developed. In order to verify the hybrid radiative transfer program and the new interface method, a series of numerical experiments for typical thermal radiation transport problem is conducted. In these problems, materials with different opacities are tested. Then the simulation efficiencies and curves of temperature, obtained by the two Monte Carlo methods, are analyzed. According to the simulation results, the program can not only accurately simulate the radiation transport in material with high opacity, but also remarkably increase the simulation efficiency. This is because many implicit Monte Carlo steps are substituted by one diffusive step and the details in the diffusive step are ignored. Also, the propagation of thermal radiation depicted with figures and tables are consistent with the radiation transport theory. In addition, the results from the hybrid Monte Carlo method reach the same accuracy as that from the implicit Monte Carlo method, and the simulation efficiency is remarkably increased.

Dynamic Probabilistic Hazard Mapping in the Long Valley Volcanic Region CA: Integrating Vent Opening Maps and Statistical Surrogates of Physical Models of Pyroclastic Density Currents

AbstractIdeally, probabilistic hazard assessments combine available knowledge about physical mechanisms of the hazard, data on past hazards, and any precursor information. Systematically assessing the probability of rare, yet catastrophic hazards adds a layer of difficulty due to limited observation data. Via computer models, one can exercise potentially dangerous scenarios that may not have happened in the past but are probabilistically consistent with the aleatoric nature of previous volcanic behavior in the record. Traditional Monte Carlo‐based methods to calculate such hazard probabilities suffer from two issues: they are computationally expensive, and they are static. In light of new information, newly available data, signs of unrest, and new probabilistic analysis describing uncertainty about scenarios the Monte Carlo calculation would need to be redone under the same computational constraints. Here we present an alternative approach utilizing statistical emulators that provide an efficient way to overcome the computational bottleneck of typical Monte Carlo approaches. Moreover, this approach is independent of an aleatoric scenario model and yet can be applied rapidly to any scenario model making it dynamic. We present and apply this emulator‐based approach to create multiple probabilistic hazard maps for inundation of pyroclastic density currents in the Long Valley Volcanic Region. Further, we illustrate how this approach enables an exploration of the impact of epistemic uncertainties on these probabilistic hazard forecasts. Particularly, we focus on the uncertainty of vent opening models and how that uncertainty both aleatoric and epistemic impacts the resulting probabilistic hazard maps of pyroclastic density current inundation.

Certainty of Lifestyle: Contrasting a Simulation Over a Fixed Period Versus Multiple Period Models

The purpose of this paper is to demonstrate the differences between the commonly used Monte Carlo simulations that assume a static withdrawal based on a fixed mortality period and initial capital, versus the use of Monte Carlo simulations that re-estimate withdrawal amounts utilizing period life tables for longevity, in addition to the variable remaining capital of each simulation, both for each individual year within the model. Such models may be called Conditionally Adjusted Sequential Retirement Periods models (CASRP). The model approach demonstrates that, although a fixed period simulation has a failure rate at some future age, the multiple period model doesn’t run out of funds unless catastrophically spent. This paper explains why funds are available at superannuated ages and demonstrates methods that remove the fear of failure (running out of money before life) to give more control and insight to retirees over the future of their funds. The authors find there is a difference between a single fixed period Monte Carlo simulation, and a model made up of a series of connected simulations.The model approach results in a series of solutions at each age, rather than a series of computations that result in a single solution from a traditional fixed period Monte Carlo simulation.

Non-parametric Sampling Approximation via Voronoi Tessellations

In this article we propose a novel non-parametric sampling approach to estimate posterior distributions from parameters of interest. Starting from an initial sample over the parameter space, this method makes use of this initial information to form a geometrical structure known as Voronoi tessellation over the whole parameter space. This rough approximation to the posterior distribution provides a way to generate new points from the posterior distribution without any additional costly model evaluations. By using a traditional Markov Chain Monte Carlo (MCMC) over the non-parametric tessellation, the initial approximate distribution is refined sequentially. We applied this method to a couple of climate models to show that this hybrid scheme successfully approximates the posterior distribution of the model parameters.

An Adaptive Multi-Element Probabilistic Collocation Method for Statistical EMC/EMI Characterization

An adaptive multi-element probabilistic collocation (ME-PC) method for quantifying uncertainties in electromagnetic compatibility and interference phenomena involving electrically large, multi-scale, and complex platforms is presented. The method permits the efficient and accurate statistical characterization of observables (i.e., quantities of interest such as coupled voltages) that potentially vary rapidly and/or are discontinuous in the random variables (i.e., parameters that characterize uncertainty in a system's geometry, configuration, or excitation). The method achieves its efficiency and accuracy by recursively and adaptively dividing the domain of the random variables into subdomains using as a guide the decay rate of relative error in a polynomial chaos expansion of the observables. While constructing local polynomial expansions on each subdomain, a fast integral-equation-based deterministic field-cable-circuit simulator is used to compute the observable values at the collocation/integration points determined by the adaptive ME-PC scheme. The adaptive ME-PC scheme requires far fewer (computationally costly) deterministic simulations than traditional polynomial chaos collocation and Monte Carlo methods for computing averages, standard deviations, and probability density functions of rapidly varying observables. The efficiency and accuracy of the method are demonstrated via its applications to the statistical characterization of voltages in shielded/unshielded microwave amplifiers and magnetic fields induced on car tire pressure sensors.

Enhanced Statistical Blockade Approaches for Fast Robustness Estimation and Compensation of Nano-CMOS Circuits

The challenges to design engineers have been complicated due to the introduction of nanoscale process variation into the design phase. One of the ways to analyze the circuit behaviors under process variation is to determine the rare events that may be originated due to such process variation. A method called Statistical Blockade (SB) had been investigated to estimate the rare events statistics especially for high-replication circuits [1]. An enhanced statistical blockade method is proposed in this paper which is shown to be much faster compared to the traditional exhaustive Monte Carlo simulation. In SB, the classification threshold determination is quite important for different tail regions which is related to the number of rare events simulation. This paper presents the values of classification threshold tc for different tail regions of typical circuits and the training samples size n required for corresponding tc. It offers both fastest speed of simulation and highest accuracy for the proposed Statistical Blockade method. It is also proven that the obtained tc can be used for all the technology corners. The enhanced statistical blockade method thus performs fast estimate the robustness for different designs. In the proposed method, the tail part of the whole distribution is used in estimation; thereby saving time. It shows 7.3× faster than traditional evaluation methods. Furthermore, for the design which is proved to be robust even in worst-case, the optimal body bias voltage is applied to improve the performance and power while reducing the variability with Adaptive Body Bias (ABB) technique. Index Terms Statistical Blockade Method, Robustness Circuits, Nanoscale CMOS, Arithmetic Circuits, Monte Carlo, Rare Event

Simulations of pressure-driven flows through channels and pipes with unified flow solver

The aim of this paper is to demonstrate the benefits of direct methods of solving kinetic equations and adaptive kinetic-fluid solvers for vacuum science and technology. We consider pressure-driven flows through short channels for a wide range of gas rarefaction degrees. Our Unified Flow Solver combines Adaptive Mesh Refinement (AMR) with automatic selection of kinetic and fluid solvers in different parts of computational domain. The discrete velocity method is used for direct numerical solution of Boltzmann and model kinetic equations. The advantages of adaptive hybrid method are demonstrated for compressible flows at large pressure gradients. For small pressure drops, direct solutions of the Boltzmann equation provide accurate solutions of rarefied flows not achievable by the traditional direct simulation Monte Carlo methods.

A Green's function Monte Carlo algorithm for the Helmholtz equation subject to Neumann and mixed boundary conditions: Validation with an 1D benchmark problem

In this paper, we present the application of our recently developed Green's function Monte Carlo algorithm to the solution of the one-dimensional Helmholtz equation subject to Neumann and mixed boundary conditions problems. The traditional Green's function Monte Carlo approach for the solution of partial differential equations subjected to Neumann and mixed boundary conditions involves “reflecting boundaries” resulting in relatively large computational times. Our algorithm, motivated by the work of K. K. Sabelfeld is philosophically different in that there is no requirement for reflection at these boundaries. The underlying feature of this algorithm is the elimination of the use of reflecting boundaries through the use of novel Green's functions that mimic the boundary conditions of the problem of interest. In the past, we have applied it to the solution of the one-dimensional Laplace equation and the modified Helmholtz equation. In this work, we apply it to the solution of the Helmholtz equation. In the case of the Helmholtz equation, unlike the Laplace equation and modified Helmholtz equation, the algorithm is constrained to quarter-wavelength length scales, a constraint that is the result of resonance in the Green's function for the Helmholtz equation. This constraint is also present in the case of the Helmholtz equation subjected to Dirichlet conditions and is not specific to Neumann and mixed boundary conditions. However, within this constraint, excellent agreement has been obtained between an analytical solution and numerical results.

Multivariate Normal Slice Sampling

By introducing auxiliary variables, the traditional Markov chain Monte Carlo method can be improved in certain cases by implementing a “slice sampler.” In the current literature, this sampling technique is used to sample from multivariate distributions with both single and multiple auxiliary variables. When the latter is employed, it generally updates one component at a time.In this article, we propose two variations of a new multivariate normal slice sampling method that uses multiple auxiliary variables to perform multivariate updating. These methods are flexible enough to allow for truncation to a rectangular region and/or exclusion of any n-dimensional hyper-quadrant. We present results of our methods and existing state-of-the-art slice samplers by comparing efficiency and accuracy. We find that we can generate approximately iid samples at a rate that is more efficient than other methods that update all dimensions at once. Supplemental materials are available online.

Estimating the Variability of Estimated Variance Components for Generalizability Theory

Generalizability theory is widely applied in psychological and educational measurement.The variability of estimated variance component,which is constrained by sampling,is the "Achilles heel" of generalizability theory. Therefore,estimating the variability of estimated variance components needs to be further explored. In previous literature,some problems remain to be settled:first,the previous studies failed to compare the variability of estimated variance components among different methods simultaneously:traditional,bootstrap,jackknife and Markov Chain Monte Carlo (MCMC); second,some studies only focused on such variability of estimated variance components as the standard error,while neglected other variability such as confidence interval; last but not least,MCMC method which can be used in generalizability theory hasn't gained sufficient exploration. There are different methods to estimate the variability of variance components:standard error and confidence interval,including traditional,bootstrap,jackknife and MCMC. Based on these four methods,the study adopts Monte Carlo data simulation technique to compare the variability of estimated variance components for normal distribution data. For traditional method,ANOVA is used to estimate the variance components and their standard errors. Satterthwaite and TBGJL are used to estimate the confidence intervals. For bootstrap method,twelve bootstrap strategies are adopted,but only three strategies are considered in jackknife method and only two strategies,i.e.,informative and non-informative priors,for MCMC method. Some criteria are set to compare the four methods. The bias is cared about when variance components and their standard errors are estimated. The smaller the absolute bias is,the more reliable the result is. The criterion of confidence intervals is "80% interval coverage". If the "80% interval coverage" is more closed to 0.80,the confidence interval is more reliable. The simulation is implemented in R statistical programming environment. To link R program with WinBUGS,R2WinBUGS and Coda package are adopted. And the simulation results are as follows. First,it is more accurate to use traditional,jackknife,adjusted bootstrap and MCMC method with informative priors to estimate variance components. But unadjusted bootstrap and MCMC method with non-informative priors are not adequate. Second,traditional and MCMC method with informative priors are accurate to estimate standard errors of three variance components,while jackknife method is not. Bootstrap method needs to adopt a "divide-and-conquer" strategy to obtain good estimated standard errors:the most accurate estimation of standard error for person is consistently provided by adjusted boot-p or boot-ir; adjusted boot-pi is the clear winner in estimating standard error for item; adjusted boot-ir,boot-pir or boot-pr give good estimate of standard error for person and item. Finally,using traditional and MCMC method to estimate confidence intervals is suitable because their respective interval coverages are close to 0.80. But jackknife method is not accurate in estimating confidence intervals. There is no great difference between Bootstrap-PC and Bootstrap-BCa method that are used to estimate confidence intervals of three variance components. Bootstrap method should apply the "divide-and-conquer" strategy to get desirable estimated confidence intervals as following:adjusted boot-p for person; adjusted boot-pi for item; adjusted boot-ir,boot-pr or boot-p for person and item. This study shows that jackknife method is not accurate to estimate the variability of estimated variance components. If the "divide-and-conquer" strategy is not used in bootstrap method,it is advisable to use traditional method or MCMC method with informative priors.

Reverse Monte Carlo analysis of small-angle scattering data on colloids and nanoparticles

Applications of Reverse Monte Carlo-type simulations are summarized for understanding the inter-particle structure of colloids and nanoparticles. Results are shown for simple spherical polymer latex, for micelles and for a catalyst containing nanoparticles. Different methodological extensions of the traditional Reverse Monte Carlo simulations are developed in order to get reliable interpretation of small-angle scattering data and they are compared: the traditional Reverse Monte Carlo technique, a method for polydisperse spheres, a combination of direct minimization and Reverse Monte Carlo procedures, and a method for particles with arbitrary shape and without analytical particle form factor.

The Effect of Method of Constructing the Mass Distribution on Single Stellar Populations

We use two methods of constructing the initial mass distribution, the traditional way and Monte Carlo simulation, to obtain integrated U - B, B - V, V-R and V-I colours and absorption-line indices defined by the Lick Observatory image dissector scanner (referred to as Lick/IDS), for instantaneous burst solar-metallicity single stellar populations with ages in the range 1-15 Gyr. We find that the evolutionary curves of all colours obtained by the traditional method are smoother than those by Monte Carlo simulation, that the U - B and B - V colours obtained by the two methods agree with one another, while the V - R and V - I colours by the traditional method are bluer than those by Monte Carlo simulation. A comparison of the Lick/IDS absorption-line indices shows that the variations in all the indices by the traditional method are smoother than that for the Monte Carlo simulation, and that all the indices except for TiO1 and TiO2 are consistent with those for the Monte Carlo simulation.

USING AN OPTIONS APPROACH TO EVALUATE KOREAN INFORMATION TECHNOLOGY INFRASTRUCTURE

This article presents a practical case application of the real options framework that addresses a multi-stage investment decision in Korean information technology infrastructure. Traditional net present value and Monte Carlo simulation are initially performed to ascertain a range of project values. The investment scenario is then viewed from a growth and compound options perspective in order to explicitly account for the project's irreversible capital expenditures, managerial flexibility, and uncertainty. Appropriate sensitivity analyses and discussion for each real options framework provide guidelines for improved decision-making. In general, it is shown that the real options approach identifies value overlooked by discounted cash flow techniques.

Analysis of PBPK models for risk characterization.

Adoption of a Bayesian framework for risk characterization permits the seamless integration of different kinds of information available in order to choose and parameterize risk models. It also becomes easy to disentangle uncertainty from variability, through hierarchical statistical modeling. Appropriate numerical techniques can be found, for example, in the recently developed arsenal of Markov chain, Monte Carlo simulations. The developments in this area can actually be viewed as extensions of the traditional or standard Monte Carlo methods for uncertainty analysis. Following a brief review of the techniques, examples of Bayesian analyses of physiologically-based pharmacokinetic models are presented for tetrachloroethylene and dichloromethane. The discussion touches on some open problems and perspectives for the proposed methods.

Hybrid MC/RISM technique for simulation of polymer solutions: Monte Carlo+ RISM integral equations

A new (hybrid) method is reported for modelling complex macromolecular systems. The approach combines the traditional atomistic Monte Carlo (MC) computer simulation of flexible polymer chains with the numerical solution of the site-site Ornstein-Zernike-like (RISM) integral equations. The method is used for calculating properties of a linear polymer in dilute solution. Since the condensed-phase environment of a flexible macromolecule affects the equilibrium configuration probability distribution of the macromolecule, the site-site intramolecular correlation function and the intramolecular potential field are treated in a self-consistent manner. Briefly, the MC method is applied to generate the configurations of a single chain molecule. Using the coordinates of chain beads, the averaged intrapolymer correlation function is obtained. Then, solving the coupled RISM equations for a given density of solvent particles, we find the polymer-solvent correlation functions. This yields the medium-induced intrapolymer potential and the corresponding effective intramolecular energies, which are used in the standard Metropolis MC procedure. The structural properties of the polymer chain are computed by averaging over the statistically representative set of configurations. As a result of many such iterations, the intramolecular structure is determined self-consistently. Using the hybrid MC/RISM method, extensive studies have been made of static properties of flexible polymer chains surrounded by LJ particles with purely repulsive interactions between the particles and chain beads. Also, direct molecular dynamics simulations have been carried out and have demonstrated that the hybrid MC/RISM approach gives a quite accurate prediction for condensed-phase effects.

Massively parallel Monte Carlo simulations of images and analytical data for Electron Microscopy

Monte Carlo electron trajectory simulations have been adapted to run on massively parallel supercomputers. An nCUBE2 parallel supercomputer with 1024 processors has been used in these studies. The advantage of the parallel architecture is the great increase in computational speed and the fact that few changes in the standard serial Monte Carlo algorithms are required. The temporal performance of the massively parallel Monte Carlo electron trajectory simulation run on 1024 nodes has been compared with Monte Carlo codes run on other types of supercomputers (CRAY-YMP). It was found to be as much as 100 times faster than the CRAY-YMP and over 2000 times faster than a VAX 785. This increase in computational speed allows the exploration of problems, in particular those involving small probability events, which are not normally amenable to solution by traditional serial Monte Carlo simulations due tothe time intensive nature of the calculations. For example, the calculation of 1,000,000 electrons at 100 kV through a thin foil takes about 6 seconds on the nCUBE.