Showing papers on "Monte Carlo method published in 1973"

Robust Regression: Asymptotics, Conjectures and Monte Carlo

Abstract: Maximum likelihood type robust estimates of regression are defined and their asymptotic properties are investigated both theoretically and empirically. Perhaps the most important new feature is that the number $p$ of parameters is allowed to increase with the number $n$ of observations. The initial terms of a formal power series expansion (essentially in powers of $p/n$) show an excellent agreement with Monte Carlo results, in most cases down to 4 observations per parameter.

Matched Sampling for Causal Effects: The Use of Matched Sampling and Regression Adjustment to Remove Bias in Observational Studies

Abstract: The ability of matched sampling and linear regression adjustment to reduce the bias of an estimate of the treatment eff ect in two sample observational studies is investigated for a simple matching method and five simple estimates. Monte Carlo results are given for moderately linear exponential response surfaces and analytic results are presented for quadratic response surfaces. The conclusions are (1) in general both matched sampling and regression adjustment can be expected to reduce bias, (2) in some cases when the variance of the matching variable differs in the two populations both matching and regression adjustment can increase bias, (3) when the variance of the matching variable is the same in the two populations and the distributions of the matching variable are symmetric the usual covariance adjusted estimate based on random samples is almost unbiased, and (4) the combination of regression adjustment in matched samples generally produces the least biased estimate.

Statistical Mechanics of Dense Ionized Matter. I. Equilibrium Properties of the Classical One-Component Plasma

Abstract: The equilibrium properties of a classical one-component plasma, in a uniform background of opposite charge, are computed for systems of various sizes by the Monte Carlo method of Metropolis et al. Following the work of Brush, Sahlin, and Teller, the periodicity of the system is accounted for by replacing the long-range Coulomb potential by an effective Ewald sum. Thermodynamic properties are computed over the whole density range of the fluid phase of the system, and their $N$ dependence is carefully investigated. A semiempirical equation of state is proposed from which all thermodynamic properties can be easily derived. Quantum corrections to these properties are calculated to first order in the Wigner expansion. Radial distribution functions, direct-correlation functions, and structure factors at various densities are tabulated. It is shown that at all densities, the direct-correlation function tends rapidly towards its Debye-H\"uckel form, in contrast to the radial distribution function. The behavior of the structure factor at small wave vectors is also shown to be in good agreement with the Debye-H\"uckel predictions at all densities.

Theory and Monte Carlo simulation of physical clusters in the imperfect vapor

Abstract: A formal physical cluster theory for an imperfect gas, valid for an arbitrary definition of a ``physical cluster,'' is described The role of the definition of the physical cluster is stressed For a particular definition of the physical cluster, which may be appropriate in nucleation theory, the Helmholtz free energy of 13‐, 43‐, 60‐, 70‐, 80‐, 87‐, and 100‐atom argon clusters is calculated in the classical limit for temperatures ranging from absolute zero to 100°K using Monte Carlo techniques It is found that a cluster's free energy is almost independent of its definition provided that the definition is reasonable and the temperature is sufficiently low The results are compared with the predictions using the harmonic approximation

On non‐RRKM unimolecular kinetics: Molecules in general, and CH3NC in particular

Abstract: Monte Carlo rate constants for model CH3NC isomerization, determined at 200, 100, and 70 kcal/mole, disagree with theoretical predictions. Also, three different approximate methods of generating initial conditions at 200 kcal lead to divergent results. The molecule does not appear to us to obey the random lifetime assumption of conventional unimolecular rate theory at any of these energies. A discussion is given of the systematics of this kind of effect, and comments are made on the relationship between our results and those obtained in the laboratory.

Perturbation theory for mixtures of simple liquids

Abstract: The perturbation approach developed by Weeks, Chandler, and Andersen (WCA) and by Verlet and Weis (VW) for pure systems is here generalized to the case of mixtures. We study binary mixtures of molecules interacting with the 12–6 Lennard-Jones potential, for which Monte Carlo simulations are available for comparison. The work is divided into two parts: The first part presents results of Monte Carlo calculations on mixtures of hard spheres of 864 and 1000 particles. The radial distribution functions generated are used to test the VW representation for the correlation functions of hard-sphere mixtures. This representation is found to work satisfactorily within the expected error limits. The second part deals with the two-step perturbation procedure for calculating the thermodynamic quantities of the Lennard-Jones system. The Lennard-Jones potential is divided into a reference potential, which is strictly repulsive, and an attractive part. The system of the reference potential is represented by a system of ha...

Investigations of a nonrandom numerical method for multidimensional integration

Abstract: A numerical integration technique based upon the use of nonrandom number sequences is examined with test integrations of a simple, analytical function. A comparison of the nonrandom technique with the familiar Monte Carlo method shows that the error of the new method decreases faster as more points are used in the calculation. Moreover, the new method needs fewer points to calculate an integral to an accuracy of 10% than does the Monte Carlo method.

Dynamic properties of the Monte Carlo method in statistical mechanics

Abstract: By means of the Monte Carlo sampling technique the equilibrium thermodynamics of fluids and magnets can be calculated numerically. We show that the questions of convergence and accuracy of this method can be understood in terms of the dynamics of the appropriate stochastic model. Also, we discuss to what extent various choices of transition probabilities lead to different dynamic properties of the system. As examples of applications, we consider Ising and Heisenberg spin systems. The numerical results about the dynamic correlation functions are compared to simple approximations taken from the theory of the kinetic Ising model.

A Monte-Carlo study

Abstract: The theory of the estimation methods discussed in Chapter II actually involves no difficulties at all. In fact, the matrix of observations of the explanatory variables X is of full rank and in such a case the least squares method is applied; though if matrix X is not of full rank it may be better to use the best linear P-unbiased estimators.

Integral equation solutions for the classical electron gas

Abstract: The hypernetted chain and Percus‐Yevick approximations are used to compute the radial distribution functions and the average potential energy for a classical electron gas. The results are compared with Monte Carlo calculations. The Percus‐Yevick equation gives poor results above Γ = 1.0; Γ = e2/kTa, where a is the ion‐sphere radius. The hypernetted chain equation was solved for 0.05 ≤ Γ ≤ 50. The results agree qualitatively with the Monte Carlo calculations everywhere except at very small Γ, where they agree with the Debye‐Huckel approximation. Definite short‐range order is predicted for Γ>3.0, but the size of the peak in the radial distribution function is underestimated. The error in the peak size is about 9% at Γ = 50. The average potential energy is in error by less than 2% for 1 ≤ Γ ≤ 50.

Monte Carlo trajectories: Dynamics of the reaction F +D2 on a semiempirical valence-bond potential energy surface

Abstract: A semiempirical valence‐bond calculation was carried out for the potential energy surface of H2F treating explicitly seven valence electrons (2pF51sHa1sHb). The integrals were evaluated from diatomic potential energy curves using the Cashion‐Herschbach method. Using this potential energy surface, the chemical reaction F+D2→FD+D was studied by Monte Carlo calculations of quasiclassical trajectories. Cross‐section calculations were carried out for initial conditions in these ranges: relative translational energy, 1.56–19.3 kcal/mole; rotational angular momentum (in units of ℏ), 0–5; vibrational quantum number, 0–1. In addition, distributions of internal energy and scattering angles for the molecular product were calculated. The results were compared with those of previous theoretical studies and the comparison indicates that subtle features (not well understood) of the potential surface may be important for obtaining correct results. The results were also compared with molecular beam, chemical laser, and in...

Maximum-Likelihood Estimation of the Number-Flux Distribution of Radio Sources in the Presence of Noise and Confusion

Abstract: The application of the method of maximum likelinood to the determination of the slope of the number- flux-density relationship is extended to include the presence of experimental errors in the flux-density measurements. It is shown that these experimental errors have a significant effect on the number counts at higher natios of flux density to error than is often recognized. The case of noise-limited fiux-density measurements is treated in some detail, and it is found that, provided the lower limit of a survey is chosen to be at least five times the rms noise, the enhancement in the source density as a function of fiux density can be readily calculated. For the case of significart confusion errors in the flux-density measurements the importance of a Monte Carlo approach is emphasized. Several methods that have been used previously are discussed and a number of shortcomings noted. (auth)

Monte-Carlo investigations of the motion of gaseous ions in electrostatic fields

Abstract: The motion of gaseous ions in arbitrarily strong electrostatic fields has been studied by Monte-Carlo simulation techniques, assuming various forms of the ion-neutral interaction law. Approximate expressions for the mobility and the mean square velocity suggested by Wannier (1953) were in most cases found to predict the correct values within 20%. The lateral diffusion coefficient to mobility ratio Dperpendicular to / mu was in all cases studied found to be remarkably accurately connected to the mean square velocity normal to the electric field through a generalized Nernst-Townsend relation. The longitudinal diffusion coefficient D/sub /// does not, on the other hand, seem to be simply related in any way to the mean square random velocity in the field direction, except near thermal energies. For hard sphere interaction and non-vanishing fields the lateral diffusion coefficient is always larger than the longitudinal one.

Computational investigations of low-discrepancy point-sets.

Abstract: The quasi-Monte Carlo method of integration offers an attractive solution to the problem of evaluating integrals in a large number of dimensions; however, the associated error bounds are difficult to obtain theoretically. Since these bounds are associated with the L2 discrepancy of the set of points used in the integration. Numerical calculations of the L2 discrepancy for several types of quasi-Monte Carlo formulae are presented.

The rapid estimation and control of precision by duplicate determinations

Abstract: Studies on computer-simulated models have provided several new methods of estimating, studying or controlling analytical precision in real systems. The methods are based upon precision estimators derived from the difference between duplicate analyses, and take into account variations in the precision of a determination with the concentration of the substance being determined. The methods have been checked by applying them to simulated samples of many duplicate analyses drawn by Monte Carlo techniques from specified populations, that is, in effect, from analytical systems with known precision characteristics. Some examples show the application of the methods in practice.

Thermalization and transport of photoelectrons - A comparison of theoretical approaches.

Abstract: Three methods of studying ionospheric photoelectron thermalization, in which transport effects are evaluated in different ways, are outlined and compared. One method uses a diffusion equation formulation, the second proceeds from a two-stream approximation to the general particle transfer equation, and the third employs a Monte Carlo technique to simulate electron paths. The methods are compared by applying them to identically prepared models. The methods are in close agreement for altitudes in the region of local energy loss, but higher altitudes reveal a discrepancy of as much as a factor of 2 in the net flux of photoelectrons. The discrepancy is removed by appropriately modifying the computational techniques.

On Using the Box‐Müller Transformation with Multiplicative Congruential Pseudo‐Random Number Generators

Abstract: SUMMARY In a recent Monte Carlo study, a most unsatisfactory sampling distribution was obtained when supposedly standard techniques were employed to simulate a simple random sample from the standard normal distribution A selection of observed and expected frequencies (the latter rounded to the nearest integer) in the tails of the distribution for a sample of size 1,000,000 is given in Table 1 It hardly needs a chi-square test to indicate that the observed frequencies are not following the expected pattern! Note particularly that all the 1,000,000 observations are restricted to the range (-333 36), and that the sampling distribution has marked local maxima approximately at the points - 33, 3 0 and 3-6 This paper investigates such phenomena

A Monte Carlo simulation of the omega phase transformation

Abstract: A model of the β to ω phase transformation in Ti, Zr and Hf alloys is proposed whereby the three-dimensional displacive transition is decomposed into one-dimensional and two-dimensional components. The one-dimensional component is postulated to be a linear displacive defect along 〈111〉 closepacked rows of the b.c.c. structure, the two-dimensional component then consists of the cooperative ordering of neighbouring parallel linear defects in the (111) planes normal to the linear component. That the two-dimensional cooperative ordering of displaced row segments can lead to a sharp phase transition has been demonstrated by a two-dimensional Monte Carlo simulation involving three types of symbols with nearest-neighbour interactions. Computer simulation exhibits characteristic two-domain ω structures which, along with the linear defect structure, appear to explain, at least qualitatively, observed electron and neutron diffraction patterns. It is tentatively suggested that the postulated linear defects ...

A Monte Carlo Swindle for Estimators of Location

Abstract: Savings in time and money can be achieved through the use of a Monte Carlo swindle in place of unaided experimental sampling. A description is provided of a swindle currently in use which often achieves efficiencies of several orders of magnitude in terms of the number of samples needed to achieve a given precision. Data are provided to show some efficiencies actually observed.

A Monte Carlo calculation of thermodynamic properties for the liquid NaCl+KCl mixture

Abstract: The mixing process NaCl+KCl →(Na, K)Cl in the liquid state at T=1083 K and zero pressure is simulated by means of Monte Carlo calculations. A cubic box with periodic boundary conditions containing 216 ions interacting according to a pair potential of the Huggins-Mayer form served as a model of the system. Thermodynamic properties are calculated for the mixture and the pure salt models and compared with experiments. Radial and angular distribution functions are obtained. A perturbation theory of the heats of mixing of molten salts and the ‘random mixture approximation’ are commented on in the light of the Monte Carlo results.

New Approach to Cosmic-Ray Diffusion Theory

Abstract: We have investigated a new approach to deriving a diffusion equation for charged particles in a static, random magnetic field. Our method incorporates essential effects of the magnetic fluctuations in the lowest order particle orbits. Significant corrections to the usual quasilinear diffusion coefficient for cosmic rays with pitch angles near 90\ifmmode^\circ\else\textdegree\fi{} are a consequence. Monte Carlo results bear out the validity of our theory.

Some Estimators for a Random Coefficient Regression Model

Abstract: The justification of applying a random coefficient regression model in econometric work has been discussed by numerous econometricians. Hildreth and Houck have derived a set of consistent estimators for such a model. Alternatives to these are developed but analytic attempts to ascertain the small sampling properties of these alternative estimators have not been very successful so far. A Monte Carlo experiment is made and the relative performance of these estimators is described.

Beam-pointing errors of planar-phased arrays

Abstract: The beam-pointing error of an (M + 1) \times (N + 1) planar phased array is analyzed by the use of both analytical and Monte Carlo techniques. Simple formulas for the rms pointing error are obtained for uniform planar arrays with both uniform and Gaussian uncorrelated phase-error distributions and for any arbitrary scan angle.

Radial Distribution Function of Liquid Sodium

Abstract: The radial distribution function and static structure factor of liquid sodium at temperatures of 100 and 200\ifmmode^\circ\else\textdegree\fi{}C have been calculated by a Monte Carlo method using a realistic ion-ion potential. The calculated structure factor is in excellent agreement with recent x-ray-diffraction measurements.

Monte Carlo calculations of reaction rates and energy distributions among reaction products. IV. F+HF(ν) → HF(ν')+F and F+DF(ν) → DF(ν')+F

Abstract: Rate constants are calculated for the vibrational relaxation HF(ν) and DF(ν) molecules by F atoms with ν = 1, 2, 3, and 6. Three‐dimensional classical trajectories of the collision dynamics of these reactions were calculated by means of a modified London‐Eyring‐Polanyi‐Sato (LEPS) potential energy surface used to calculate rate constants for the reactions between H atoms and F2 molecules, and D atoms and F2 molecules. The Monte Carlo procedure is used to start each collision trajectory. By means of this calculation, it is predicted for the reactant HF molecule in the ν = 2, J = 8 state that 13.3% of the mean fraction of available energy will become rotation in the product HF, 80.7% will become vibrational energy in the product HF, and 6% will become relative translational energy of the products. As the vibrational energy of the reactant HF molecule increases the mean fraction of available energy that becomes rotational energy increases slightly. For example, if the reactant molecule HF is in the ν = 6, J ...