Showing papers on "Monte Carlo method published in 1998"

Model Selection and Inference: A Practical Information-Theoretic Approach

Abstract: Information theory and log-likelihood models - a basis for model selection and inference practical use of the information theoretic approach model selection uncertainty with examples Monte Carlo insights and extended examples statistical theory.

Consistent Covariance Matrix Estimation with Spatially Dependent Panel Data

Abstract: Many panel data sets encountered in macroeconomics, international economics, regional science, and finance are characterized by cross-sectional or “spatial” dependence. Standard techniques that fail to account for this dependence will result in inconsistently estimated standard errors. In this paper we present conditions under which a simple extension of common nonparametric covariance matrix estimation techniques yields standard error estimates that are robust to very general forms of spatial and temporal dependence as the time dimension becomes large. We illustrate the relevance of this approach using Monte Carlo simulations and a number of empirical examples.

Monte Carlo and quasi-Monte Carlo methods

Abstract: Monte Carlo is one of the most versatile and widely used numerical methods. Its convergence rate, O(N−1/2), is independent of dimension, which shows Monte Carlo to be very robust but also slow. This article presents an introduction to Monte Carlo methods for integration problems, including convergence theory, sampling methods and variance reduction techniques. Accelerated convergence for Monte Carlo quadrature is attained using quasi-random (also called low-discrepancy) sequences, which are a deterministic alternative to random or pseudo-random sequences. The points in a quasi-random sequence are correlated to provide greater uniformity. The resulting quadrature method, called quasi-Monte Carlo, has a convergence rate of approximately O((logN)kN−1). For quasi-Monte Carlo, both theoretical error estimates and practical limitations are presented. Although the emphasis in this article is on integration, Monte Carlo simulation of rarefied gas dynamics is also discussed. In the limit of small mean free path (that is, the fluid dynamic limit), Monte Carlo loses its effectiveness because the collisional distance is much less than the fluid dynamic length scale. Computational examples are presented throughout the text to illustrate the theory. A number of open problems are described.

Statistical data analysis

Abstract: Preface Notation 1. Fundamental Concepts 2. Examples of Probability Functions 3. The Monte Carlo Method 4. Statistical Tests 5. General Concepts of Parameter Estimation 6. The Method of Maximum Likelihood 7. The Method of Least Squares 8. The Method of Moments 9. Statistical Errors, Confidence Intervals and Limits 10. Characteristic Functions and Related Examples 11. Unfolding Bibliography Index

Simulating normalizing constants: from importance sampling to bridge sampling to path sampling

Abstract: Computing (ratios of) normalizing constants of probability models is a fundamental computational problem for many statistical and scientific studies. Monte Carlo simulation is an effective technique, es- pecially with complex and high-dimensional models. This paper aims to bring to the attention of general statistical audiences of some effective methods originating from theoretical physics and at the same time to ex- plore these methods from a more statistical perspective, through estab- lishing theoretical connections and illustrating their uses with statistical problems. We show that the acceptance ratio method and thermodynamic integration are natural generalizations of importance sampling, which is most familiar to statistical audiences. The former generalizes importance sampling through the use of a single "bridge" density and is thus a case of bridge sampling in the sense of Meng and Wong. Thermodynamic integration, which is also known in the numerical analysis literature as Ogata's method for high-dimensional integration, corresponds to the use of infinitely many and continuously connected bridges (and thus a "path"). Our path sampling formulation offers more flexibility and thus potential efficiency to thermodynamic integration, and the search of op- timal paths turns out to have close connections with the Jeffreys prior density and the Rao and Hellinger distances between two densities. We provide an informative theoretical example as well as two empirical ex- amples (involving 17- to 70-dimensional integrations) to illustrate the potential and implementation of path sampling. We also discuss some open problems.

Small-sample Confidence Intervals for Impulse Response Functions

Abstract: Bias-corrected bootstrap confidence intervals explicitly account for the bias and skewness of the small-sample distribution of the impulse response estimator, while retaining asymptotic validity in stationary autoregressions. Monte Carlo simulations for a wide range of bivariate models show that in small samples bias-corrected bootstrap intervals tend to be more accurate than delta method intervals, standard bootstrap intervals, and Monte Carlo integration intervals. This conclusion holds for VAR models estimated in levels, as deviations from a linear time trend, and in first differences. It also holds for random walk processes and cointegrated processes estimated in levels. An empirical example shows that bias-corrected bootstrap intervals may imply economic interpretations of the data that are substantively different from standard methods.

Random number generation and Monte Carlo methods

Abstract: Simulating Random Numbers from a Uniform Distribution * Quality of Random Number Generation * Quasirandom Numbers * Transformations of Uniform Deviates: General Methods * Simulating Random Numbers from Specific Distributions * Generation of Random Samples, Permutations, and Stochastic Processes * Monte Carlo Methods * Software for Random Number Generation * Monte Carlo Studies in Statistics

Analysis of multivariate probit models

Abstract: SUMMARY This paper provides a practical simulation-based Bayesian and non-Bayesian analysis of correlated binary data using the multivariate probit model. The posterior distribution is simulated by Markov chain Monte Carlo methods and maximum likelihood estimates are obtained by a Monte Carlo version of the EM algorithm. A practical approach for the computation of Bayes factors from the simulation output is also developed. The methods are applied to a dataset with a bivariate binary response, to a four-year longitudinal dataset from the Six Cities study of the health effects of air pollution and to a sevenvariate binary response dataset on the labour supply of married women from the Panel Survey of Income Dynamics.

Markov chain Monte Carlo method and its application

Abstract: The Markov chain Monte Carlo (MCMC) method, as a computer-intensive statistical tool, has enjoyed an enormous upsurge in interest over the last few years. This paper provides a simple, comprehensive and tutorial review of some of the most common areas of research in this field. We begin by discussing how MCMC algorithms can be constructed from standard building-blocks to produce Markov chains with the desired stationary distribution. We also motivate and discuss more complex ideas that have been proposed in the literature, such as continuous time and dimension jumping methods. We discuss some implementational issues associated with MCMC methods. We take a look at the arguments for and against multiple replications, consider how long chains should be run for and how to determine suitable starting points. We also take a look at graphical models and how graphical approaches can be used to simplify MCMC implementation. Finally, we present a couple of examples, which we use as case-studies to highlight some of the points made earlier in the text. In particular, we use a simple changepoint model to illustrate how to tackle a typical Bayesian modelling problem via the MCMC method, before using mixture model problems to provide illustrations of good sampler output and of the implementation of a reversible jump MCMC algorithm.

The rendering equation

Abstract: We present an integral equation which generalizes a variety of known rendering algorithms. In the course of discussing a monte carlo solution we also present a new form of variance reduction, called Hierarchical sampling and give a number of elaborations shows that it may be an efficient new technique for a wide variety of monte carlo procedures. The resulting rendering algorithm extends the range of optical phenomena which can be effectively simulated.

Introduction to Monte Carlo methods

Abstract: This chapter describes a sequence of Monte Carlo methods: importance sampling, rejection sampling, the Metropolis method, and Gibbs sampling. For each method, we discuss whether the method is expected to be useful for high—dimensional problems such as arise in inference with graphical models. After the methods have been described, the terminology of Markov chain Monte Carlo methods is presented. The chapter concludes with a discussion of advanced methods, including methods for reducing random walk behaviour.

Data analysis of gravitational-wave signals from spinning neutron stars. I. The signal and its detection

Abstract: We present a theoretical background for the data analysis of the gravitational-wave signals from spinning neutron stars for Earth-based laser interferometric detectors. We introduce a detailed model of the signal including both the frequency and the amplitude modulations. We include the effects of the intrinsic frequency changes and the modulation of the frequency at the detector due to Earth's motion. We estimate the effects of the star's proper motion and of relativistic corrections. Moreover we consider a signal consisting of two components corresponding to a frequency $f$ and twice that frequency. From the maximum likelihood principle we derive the detection statistics for the signal and we calculate the probability density function of the statistics. We obtain the data analysis procedure to detect the signal and to estimate its parameters. We show that for optimal detection of the amplitude modulated signal we need four linear filters instead of one linear filter needed for a constant amplitude signal. Searching for the doubled frequency signal increases further the number of linear filters by a factor of 2. We indicate how the fast Fourier transform algorithm and resampling methods commonly proposed in the analysis of periodic signals can be used to calculate the detection statistics for our signal. We find that the probability density function of the detection statistics is determined by one parameter: the optimal signal-to-noise ratio. We study the signal-to-noise ratio by means of the Monte Carlo method for all long-arm interferometers that are currently under construction. We show how our analysis can be extended to perform a joint search for periodic signals by a network of detectors and we perform a Monte Carlo study of the signal-to-noise ratio for a network of detectors.

Filtered density function for large eddy simulation of turbulent reacting flows

Abstract: A methodology termed the “filtered density function” (FDF) is developed and implemented for large eddy simulation (LES) of chemically reacting turbulent flows. In this methodology, the effects of the unresolved scalar fluctuations are taken into account by considering the probability density function (PDF) of subgrid scale (SGS) scalar quantities. A transport equation is derived for the FDF in which the effect of chemical reactions appears in a closed form. The influences of scalar mixing and convection within the subgrid are modeled. The FDF transport equation is solved numerically via a Lagrangian Monte Carlo scheme in which the solutions of the equivalent stochastic differential equations (SDEs) are obtained. These solutions preserve the Ito-Gikhman nature of the SDEs. The consistency of the FDF approach, the convergence of its Monte Carlo solution and the performance of the closures employed in the FDF transport equation are assessed by comparisons with results obtained by direct numerical simulation ...

A new order parameter for tetrahedral configurations

Abstract: A new order parameter, S, is introduced to test for tetrahedral configurations. It is applied to analyse the results of three simulations: (1) molecular dynamics simulation of ice Ih (hexagonal ice) at 200 K, using the SPC/E water potential; (2) Monte Carlo simulation of aqueous solution of methane, using the OPLS methane potential and the TIP4P water potential; and (3) Monte Carlo simulation of Lennard-Jones spheres. The quasi-tetrahedral configurations of water molecules in different states can be adequately described using S.

On the simulation of vapor-liquid equilibria for alkanes

Abstract: A Monte Carlo simulation study has been conducted to assess the ability of recently proposed force fields to predict orthobaric densities, second virial coefficients, and P-V-T data for short and long alkanes. A new, modified force field is proposed that provides good agreement with experimental phase equilibrium and second virial coefficient data over wide ranges of temperature and chain length.

Temperature dependence of TIP3P, SPC, and TIP4P water from NPT Monte Carlo simulations: Seeking temperatures of maximum density

Abstract: Monte Carlo statistical mechanics simulations have been carried out with the TIP3P, SPC, and TIP4P models for liquid water at 13 temperatures from −50°C to 100°C at 1 atm. Long runs with 512 water molecules provided definitive results for densities. Although the TIP4P model yields a flat region in the density profile and a temperature of maximum density near −15°C, the SPC and TIP3P models show monotonically increasing density with decreasing temperature. Results for heats of vaporization, isothermal compressibilities, and coefficients of thermal expansion and their convergence characteristics are also reported. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1179–1186, 1998

Velocity distributions in homogeneous granular fluids: the free and the heated case

Abstract: Non-Gaussian properties (cumulants, high energy tails) of the single particle velocity distribution for homogeneous granular fluids of inelastic hard spheres or disks are studied, based on the Enskog-Boltzmann equation for the unforced and heated case. The latter is in a steady state. The non-Gaussian corrections have small effects on the cooling rate, and on the stationary temperature in the heated case, at all inelasticities. The velocity distribution in the heated steady state exhibits a high energy tail ˜exp(-A c3/2), where c is the velocity scaled by the thermal velocity and A˜ 1/ with e the inelasticity. The results are compared with molecular dynamics simulations, as well as direct Monte Carlo simulations of the Boltzmann equation.

Stochastic response surface methods (SRSMs) for uncertainty propagation: application to environmental and biological systems.

Abstract: Comprehensive uncertainty analyses of complex models of environmental and biological systems are essential but often not feasible due to the computational resources they require. "Traditional" methods, such as standard Monte Carlo and Latin Hypercube Sampling, for propagating uncertainty and developing probability densities of model outputs, may in fact require performing a prohibitive number of model simulations. An alternative is offered, for a wide range of problems, by the computationally efficient "Stochastic Response Surface Methods (SRSMs)" for uncertainty propagation. These methods extend the classical response surface methodology to systems with stochastic inputs and outputs. This is accomplished by approximating both inputs and outputs of the uncertain system through stochastic series of "well behaved" standard random variables; the series expansions of the outputs contain unknown coefficients which are calculated by a method that uses the results of a limited number of model simulations. Two case studies are presented here involving (a) a physiologically-based pharmacokinetic (PBPK) model for perchloroethylene (PERC) for humans, and (b) an atmospheric photochemical model, the Reactive Plume Model (RPM-IV). The results obtained agree closely with those of traditional Monte Carlo and Latin Hypercube Sampling methods, while significantly reducing the required number of model simulations.

Quantifying uncertainty in tracer‐based hydrograph separations

Abstract: A method is presented for quantifying the uncertainty in two- and three-component tracer-based hydrograph separations. The method relates the uncertainty in computed mixing fractions to both the tracer concentrations used to perform the hydrograph separation and the uncertainties in those concentrations. A two-component example and a three-component example illustrate the application of the method. The three-component example yields uncertainty results very similar to those from a previously published Monte Carlo analysis and requires less computation.

Critical point and phase behavior of the pure fluid and a Lennard-Jones mixture

Abstract: Monte Carlo simulations in the grand canonical ensemble were used to obtain liquid-vapor coexistence curves and critical points of the pure fluid and a binary mixture of Lennard-Jones particles. Critical parameters were obtained from mixed-field finite-size scaling analysis and subcritical coexistence data from histogram reweighting methods. The critical parameters of the untruncated Lennard-Jones potential were obtained as Tc*=1.3120±0.0007, ρc*=0.316±0.001 and pc*=0.1279±0.0006. Our results for the critical temperature and pressure are not in agreement with the recent study of Caillol [J. Chem. Phys. 109, 4885 (1998)] on a four-dimensional hypersphere. Mixture parameters were e1=2e2 and σ1=σ2, with Lorentz–Berthelot combining rules for the unlike-pair interactions. We determined the critical point at T*=1.0 and pressure-composition diagrams at three temperatures. Our results have much smaller statistical uncertainties relative to comparable Gibbs ensemble simulations.

The Monte Carlo Method in Radiative Heat Transfer

Abstract: The use of the Monte Carlo method in radiative heat transfer is reviewed. The review covers surface-surface, enclosure, and participating media problems. Discussio. is included of research on the fundamentals of the method and on applications to surface-surface interchange in enclosures, exchange between surfaces with roughness characteristics, determination of configuration factors, inverse design, transfer through packed beds and fiber layers, participating media, scattering, hybrid methods, spectrally dependent problems including media with line structure, effects of using parallel algorithms, practical applications, and extensions of the method. Conclusions are presented on needed future work and the place of Monte Carlo techniques in radiative heat transfer computations

Electron transport characteristics of GaN for high temperature device modeling

Abstract: Monte Carlo simulations of electron transport based upon an analytical representation of the lowest conduction bands of bulk, wurtzite phase GaN are used to develop a set of transport parameters for devices with electron conduction in GaN. Analytic expressions for spherical, nonparabolic conduction band valleys at the Γ, U, M, and K symmetry points of the Brillouin zone are matched to experimental effective mass data and to a pseudopotential band structure. The low-field electron drift mobility is calculated for temperatures in the range of 300–600 K and for ionized impurity concentrations between 1016 and 1018 cm−3. Compensation effects on the mobility are also examined. Electron drift velocities for fields up to 500 kV/cm are calculated for the above temperature range. To aid GaN device modeling, the drift mobility dependences on ambient temperature, donor concentration, and compensation ratio are expressed in analytic form with parameters determined from the Monte Carlo results. Analytic forms are also...

A Bayesian CART algorithm

Abstract: A stochastic search form of classification and regression tree (CART) analysis (Breiman et al., 1984) is proposed, motivated by a Bayesian model. An approximation to a probability distribution over the space of possible trees is explored using reversible jump Markov chain Monte Carlo methods (Green, 1995).

Noninvasive determination of the optical properties of two-layered turbid media

Abstract: Light propagation in two-layered turbid media having an infinitely thick second layer is investigated in the steady-state, frequency, and time domains. A solution of the diffusion approximation to the transport equation is derived by employing the extrapolated boundary condition. We compare the reflectance calculated from this solution with that computed with Monte Carlo simulations and show good agreement. To investigate if it is possible to determine the optical coefficients of the two layers and the thickness of the first layer, the solution of the diffusion equation is fitted to reflectance data obtained from both the diffusion equation and the Monte Carlo simulations. Although it is found that it is, in principle, possible to derive the optical coefficients of the two layers and the thickness of the first layer, we concentrate on the determination of the optical coefficients, knowing the thickness of the first layer. In the frequency domain, for example, it is shown that it is sufficient to make relative measurements of the phase and the steady-state reflectance at three distances from the illumination point to obtain useful estimates of the optical coefficients. Measurements of the absolute steady-state spatially resolved reflectance performed on two-layered solid phantoms confirm the theoretical results.