Random number generation methods

Randomization At least 24 animals should be on pretest. Randomization to treatment groups is best done using a computerized blocking by body weight method or a random number generation method. 

The third (and now most common) method is to use a random number generator that is built into a calculator or computer program. Procedures for generating these are generally documented in user manuals. 

The universal algorithm of MC methods was provided early after computers came into use by Metropolis et al. (1953). The name MC stems from a random number generator in the method, similar to that used in casinos. 

Two classical methods are available in THINK to perform the conformational expansion of molecules systematic search and random search. When the systematic search option is used, the use of contacts check avoids high-energy conformations and reduces the overall processing time. The random method uses a random number generator to select the conformations from within the estimated total number of conformations. The implementation of the program does not prohibit identical conformations to be output resulting from symmetry. These conformations are used in the pharmacophore generation and site search modules. 

Synchrotron storage rings, for instance, are able to provide an extremely high flux of nearly monochromatic X-radiation on a small sample area. They could form the basis of XRF set-ups and enhance other microana-lytical methods to provide accurate determinations. In the future they could serve as a reference method for elemental trace analysis on the microscopical level (with the quality of the random number generator, a non-SI concept, as the prime source of error). 

The deterministic methods in Section 2.2.1 numerically solve the differential equations, thus determining the macroscopic composition of the system at a given time. On the contrary, the KMC method determines the time of change (reaction) of a species with the use of a random number generated with an appropriate distribution.89 The result is the system s composition, as well. 

Monte Carlo simulation can involve several methods for using a pseudo-random number generator to simulate random values from the probability distribution of each model input. The conceptually simplest method is the inverse cumulative distribution function (CDF) method, in which each pseudo-random number represents a percentile of the CDF of the model input. The corresponding numerical value of the model input, or fractile, is then sampled and entered into the model for one iteration of the model. For a given model iteration, one random number is sampled in a similar way for all probabilistic inputs to the model. For example, if there are 10 inputs with probability distributions, there will be one random sample drawn from each of the 10 and entered into the model, to produce one estimate of the model output of interest. This process is repeated perhaps hundreds or thousands of times to arrive at many estimates of the model output. These estimates are used to describe an empirical CDF of the model output. From the empirical CDF, any statistic of interest can be inferred, such as a particular fractile, the mean, the variance and so on. However, in practice, the inverse CDF method is just one of several methods used by Monte Carlo simulation software in order to generate samples from model inputs. Others include the composition and the function of random variable methods (e.g. Ang Tang, 1984). However, the details of the random number generation process are typically contained within the chosen Monte Carlo simulation software and thus are not usually chosen by the user. 

In an unmodified Monte Carlo method, simple random sampling is used to select each member of the 777-tuple set. Each of the input parameters for a model is represented by a probability density function that defines both the range of values that the input parameters can have and the probability that the parameters are within any subinterval of that range. In order to carry out a Monte Carlo sampling analysis, each input is represented by a cumulative distribution function (CDF) in which there is a one-to-one correspondence between a probability and values. A random number generator is used to select probability in the range of 0-1. This probability is then used to select a corresponding parameter value. 

In this method a random number generator is used to move and rotate molecules in a random fashion. If the system is held under specified conditions of temperature, volume and number of molecules, the probability of a particular arrangement of molecules is proportional to exp(-U/kT), where U is the total intermolecular energy of the assembly of molecules and k is the Boltzmann constant. Thus, within the MC scheme the movement of individual molecules is accepted or rejected in accordance with a probability determined by the Boltzmann distribution law. After the generation of a long sequence of moves, the results are averaged to give the equilibrium properties of the model system. 

A random sample means that every item in a population has an equal chance of being chosen. Simply choosing materials by eye does not satisfy this criterion. Each of the dose units should be assigned a number, starting at 1 and ending with the last number (i.e. the number of items in the sample). The materials to be chosen should then be picked by using either a computerized random-number generator or random-number tables. Whichever method is used, it should be documented. 

Monte Carlo simulation is a procedure for mimicking observations on a random variable that permits verification of results that would ordinarily require difficult mathematical calculations or extensive experimentation. The method normally uses computer programs called random number generators. A random number is a number selected from the interval (0,1) in such a way that the probabilities that the number comes from any two subintervals of equal length are equal. For example, the probability the number is in the subinterval (0.1, 0.3) is the same as the probability that the number is in the subinterval (0.5, 0.7). Thus, random numbers are observations on a random variable X having a uniform distribution on the interval (0,1). This means that the PDF of X is specified by... 

Often we are only interested in the equilibrium structure of a set of molecules. When the system is at high densities or includes a large number of conformations, other methods become computationally unfeasible, and one often uses Monte Carlo (MC) techniques. Monte Carlo methods use random number generators to integrate systems with very high degrees of freedom. These integrals are then used in theories of statistical mechanics to evaluate thermodynamic properties of materials. ... 

Niederreiter, H., Random Number Generation and Quasi-Monte Carlo Methods, Society for Industrial and Applied Mathematics, Philadelphia, 1992. 

The force fields can also be compared to a blank result, the mean absolute of all conformational energies to be predicted. This is the performance that would be expected by any random-number generator symmetrically centered around 0 kcal/ mol. Most force fields yield a performance substantially better than this random guess, but of the methods considered here, CVFF is not significantly different from the blank, and UFF is actually worse. 

The two major methods for the simulations are the Monte Carlo method (so named for its use of random number generation) and the molecular dynamics method. The Monte Carlo method, as applied to problems of chemistry, was first described by N. Metropolis and his co-workers at the Los Alamos... 

We use the effective mass approximation. The structure potential is approximated b y a c onsequence o f rectangular quantum barriers a nd wells. Their widths and potentials are randomly varied with the uniform distribution. The sequence of the parameters is calculated by a random number generator. Other parameters, e.g. effective masses of the carriers, are assumed equal in different layers. To simplify the transmission coefficient calculations, we approximate the electric field potential by a step function. The transmission coefficient is calculated using the transfer matrix method [9]. The 1-V curves of the MQW stmctures along the x-axis (growth direction) a re derived from the calculated transmission spectra... 

Another example is from the numerical study of phase transitions. Renormalization theory has proved accurate for the basic scaling properties of simple transitions. The attention of the research community is now shifting to corrections to scaling, and to more complex models. Very long simulations (also of the MCMC type) are done to investigate this effect, and it has been discovered that the random number generator can influence the results [3-6]. As computers become more powerful, and Monte Carlo methods become more commonly used and more central to scientific progress, the quality of the random number sequence becomes more important. 

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