Random number sampling method

In general, Monte Carlo methods refer to any procedures which involve sampling from random numbers. These methods are used in simulations of natural phenomena, simulation of experimental apparatus, and numerical analysis. An important feature is the simple structure of the computational algorithm. The method was developed by von Neuman, Ulam, and Metroplois during World War II to study the difiiision of neutrons in fissionable materials (ie, atomic bomb design)- Let us consider atom diffusion and demonstrate the principle of the Monte Carlo method. A two-dimensional square grid (Fig. 3.20A) represents interstitial sites in a sofid. 

A randomly collected sample makes no assumptions about the target population, making it the least biased approach to sampling. On the other hand, random sampling requires more time and expense than other sampling methods since a greater number of samples are needed to characterize the target population. 

Monte Carlo search methods are stochastic techniques based on the use of random numbers and probability statistics to sample conformational space. The name Monte Carlo was originally coined by Metropolis and Ulam [4] during the Manhattan Project of World War II because of the similarity of this simulation technique to games of chance. Today a variety of Monte Carlo (MC) simulation methods are routinely used in diverse fields such as atmospheric studies, nuclear physics, traffic flow, and, of course, biochemistry and biophysics. In this section we focus on the application of the Monte Carlo method for... 

As an illustration of the biased sampling method in application to the problems of polymer chain adsorption on a hard wall we shall recall here briefly the procedure used on a diamond lattice [35]. Starting the chain at the origin, the first bond is fixed at the plane and all the following bonds are determined at random apart from the non-reversal condition. Suppose, after a certain number i of steps, that the (/+l)st monomer reaches the plane at z = 0 again. With = 4 on the diamond lattice one has the probability p = /3 for each new possible choice of a bond. Thus in... 

A reliable procedure for determination of molecular parameters number, weight and z-averages of the molecular weight (Mj, i = n, w and z respectively) for polyethylenes, PE, by means of Size Exclusion Chromatography, SEC, has been developed. The Waters Sci. Ltd. GPC/LC Model 150C was used at 135 C with trichlorobenzene, TCB, as a solvent. The standard samples as well as commercial stabilized and not stabilized PE-resins were evaluated. The effects of sampling, method of solution preparation, addition of antioxidant(s), thermal and shear degradation were studied. The adopted procedure allows reproducible determination of and M , with a random error of 4% and M2, with 9%, within 2 to 72 hrs from the initial moment of preparation of solutions. 

For quantitative evaluation a set of 6 serial dilutions of the original sample that should have an accurately known concentration of -2.0%. The concentration can be adjusted to give the optimum odor data. Each sample is diluted by a factor of 3 from the previous sample (i.e., 1 ml sample diluted with 2 ml of solvent) and then given a random number using a double blind labeling method. The samples are sniffed in random order. 

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). 

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. 

Figure 5,2 Examples of the laboratory systematic errors when samples are collected in numeric and random number order, (a) Shows error due to analytical drift and (b) carry over contamination during sample preparation after the preparation of a high sample. The random numbering of samples makes it easier to identify systematic laboratory errors. If the samples are collected and analysed in sequential order the results look like naturally occurring anomalies while the random numbering method tends to distinguish such errors as isolated highs.

Often, calibration of natural products and materials is a desirable goal. In these kinds of assays, it is usually not feasible to control the composition of calibration and validation standards. Some well-known examples include the determination of protein, starch, and moisture in whole-wheat kernels and the determination of gasoline octane number by NIR spectroscopy. In cases such as these, sets of randomly selected samples must be obtained and analyzed by reference methods. 

Recently, Gillespie (2001) introduced an approximate approach, termed the r-leap method, for solving stochastic models. The main idea is the same as in the WP-KMC method. One selects a time increment r that is larger than the microscopic KMC time increment, and multiple molecular bundles of fast events occur. However, one now samples how many times each reaction will be executed from a Poisson rather than a uniform random number distribution. Prototype examples indicate that the r-leap method provides comparable noise with the microscopic KMC when the leap condition is satisfied, i.e., the time increments are such that the populations do not change significantly between time steps. 

Figure 11.7 A schematic illustration of the Monte Carlo simulation method for computing the stochastic trajectories of a chemical reaction system following the CME. Two random numbers, r and r2, are sampled from a uniform distribution to simulate each stochastic step r determines when to move, and r2 determines where to move. For a given state of a master equation graph shown in the upper panel, there are four outward reactions, labeled 1-4, each with their corresponding rate constants q, (i = 1, 2, , 4). The upper and lower panels illustrate, respectively, the calculation of the random time T associated with a stochastic move, and the probability pm of moving to state m.

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. 

Each of the above aspects have been studied in great detail and a number of mathematical models proposed for evaluation stndies, bnt there is cnrrently no universally adopted method (apart from the United Nations recommendations). However, the latter themselves present their own problems - how are random numbers assigned to individnal doses in a batch of thonsands so that the samples can be chosen trnly randomly ... 

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