Topic

# Monte Carlo method

About: Monte Carlo method is a research topic. Over the lifetime, 95966 publications have been published within this topic receiving 2181896 citations. The topic is also known as: MC method & Monte Carlo experiments.

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TL;DR: In this article, a modified Monte Carlo integration over configuration space is used to investigate the properties of a two-dimensional rigid-sphere system with a set of interacting individual molecules, and the results are compared to free volume equations of state and a four-term virial coefficient expansion.

Abstract: A general method, suitable for fast computing machines, for investigating such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two‐dimensional rigid‐sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four‐term virial coefficient expansion.

35,161 citations

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TL;DR: A generalization of the sampling method introduced by Metropolis et al. as mentioned in this paper is presented along with an exposition of the relevant theory, techniques of application and methods and difficulties of assessing the error in Monte Carlo estimates.

Abstract: SUMMARY A generalization of the sampling method introduced by Metropolis et al. (1953) is presented along with an exposition of the relevant theory, techniques of application and methods and difficulties of assessing the error in Monte Carlo estimates. Examples of the methods, including the generation of random orthogonal matrices and potential applications of the methods to numerical problems arising in statistics, are discussed. For numerical problems in a large number of dimensions, Monte Carlo methods are often more efficient than conventional numerical methods. However, implementation of the Monte Carlo methods requires sampling from high dimensional probability distributions and this may be very difficult and expensive in analysis and computer time. General methods for sampling from, or estimating expectations with respect to, such distributions are as follows. (i) If possible, factorize the distribution into the product of one-dimensional conditional distributions from which samples may be obtained. (ii) Use importance sampling, which may also be used for variance reduction. That is, in order to evaluate the integral J = X) p(x)dx = Ev(f), where p(x) is a probability density function, instead of obtaining independent samples XI, ..., Xv from p(x) and using the estimate J, = Zf(xi)/N, we instead obtain the sample from a distribution with density q(x) and use the estimate J2 = Y{f(xj)p(x1)}/{q(xj)N}. This may be advantageous if it is easier to sample from q(x) thanp(x), but it is a difficult method to use in a large number of dimensions, since the values of the weights w(xi) = p(x1)/q(xj) for reasonable values of N may all be extremely small, or a few may be extremely large. In estimating the probability of an event A, however, these difficulties may not be as serious since the only values of w(x) which are important are those for which x -A. Since the methods proposed by Trotter & Tukey (1956) for the estimation of conditional expectations require the use of importance sampling, the same difficulties may be encountered in their use. (iii) Use a simulation technique; that is, if it is difficult to sample directly from p(x) or if p(x) is unknown, sample from some distribution q(y) and obtain the sample x values as some function of the corresponding y values. If we want samples from the conditional dis

14,965 citations

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01 Jun 1969

TL;DR: In this paper, Monte Carlo techniques are used to fit dependent and independent variables least squares fit to a polynomial least-squares fit to an arbitrary function fitting composite peaks direct application of the maximum likelihood.

Abstract: Uncertainties in measurements probability distributions error analysis estimates of means and errors Monte Carlo techniques dependent and independent variables least-squares fit to a polynomial least-squares fit to an arbitrary function fitting composite peaks direct application of the maximum likelihood. Appendices: numerical methods matrices graphs and tables histograms and graphs computer routines in Pascal.

12,737 citations

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TL;DR: Both optimal and suboptimal Bayesian algorithms for nonlinear/non-Gaussian tracking problems, with a focus on particle filters are reviewed.

Abstract: Increasingly, for many application areas, it is becoming important to include elements of nonlinearity and non-Gaussianity in order to model accurately the underlying dynamics of a physical system. Moreover, it is typically crucial to process data on-line as it arrives, both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics. In this paper, we review both optimal and suboptimal Bayesian algorithms for nonlinear/non-Gaussian tracking problems, with a focus on particle filters. Particle filters are sequential Monte Carlo methods based on point mass (or "particle") representations of probability densities, which can be applied to any state-space model and which generalize the traditional Kalman filtering methods. Several variants of the particle filter such as SIR, ASIR, and RPF are introduced within a generic framework of the sequential importance sampling (SIS) algorithm. These are discussed and compared with the standard EKF through an illustrative example.

11,409 citations

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TL;DR: A new reproducibility index is developed and studied that is simple to use and possesses desirable properties and the statistical properties of this estimate can be satisfactorily evaluated using an inverse hyperbolic tangent transformation.

Abstract: A new reproducibility index is developed and studied. This index is the correlation between the two readings that fall on the 45 degree line through the origin. It is simple to use and possesses desirable properties. The statistical properties of this estimate can be satisfactorily evaluated using an inverse hyperbolic tangent transformation. A Monte Carlo experiment with 5,000 runs was performed to confirm the estimate's validity. An application using actual data is given.

6,916 citations