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Monte Carlo correlation coefficient

A Monte Carlo simulation is fast to perform on a computer, and the presentation of the results is attractive. However, one cannot guarantee that the outcome of a Monte Carlo simulation run twice with the same input variables will yield exactly the same output, making the result less auditable. The more simulation runs performed, the less of a problem this becomes. The simulation as described does not indicate which of the input variables the result is most sensitive to, but one of the routines in Crystal Ball and Risk does allow a sensitivity analysis to be performed as the simulation is run.This is done by calculating the correlation coefficient of each input variable with the outcome (for example between area and UR). The higher the coefficient, the stronger the dependence between the input variable and the outcome. [Pg.167]

Debye and Hiickel s theory of ionic atmospheres was the first to present an account of the activity of ions in solution. Mayer showed that a virial coefficient approach relating back to the treatment of the properties of real gases could be used to extend the range of the successful treatment of the excess properties of solutions from 10 to 1 mol dm". Monte Carlo and molecular dynamics are two computational techniques for calculating many properties of liquids or solutions. There is one more approach, which is likely to be the last. Thus, as shown later, if one knows the correlation functions for the species in a solution, one can calculate its properties. Now, correlation functions can be obtained in two ways that complement each other. On the one hand, neutron diffraction measurements allow their experimental determination. On the other, Monte Carlo and molecular dynamics approaches can be used to compute them. This gives a pathway purely to calculate the properties of ionic solutions. [Pg.324]

The limit in front of the ratio means that the time t has to be much longer than the longest relaxation time of the chain. The resulting diffusion coefficients obtained by Monte Carlo simulation of the Evans-Edwards model of entangled polymers are presented in Fig. 9.33(a). The diffusion coefficient decreases with the number of monomers in the chain. Another quantity that can be extracted from the Monte Carlo simulations of the Evans-Edwards model is the relaxation time of the chain. It can be defined as the characteristic decay time of the time correlation function of the end-to-end vector R[t)R 0)) exp( t/Trep). Figure 9.33(b) presents the results of such simulations. [Pg.399]

The concept of squared distances has important functional consequences on how the value of the correlation coefficient reacts to various specific arrangements of data. The significance of correlation is based on the assumption that the distribution of the residual values (i.e., the deviations from the regression line) for the dependent variable y follows the normal distribution and that the variability of the residual values is the same for all values of the independent variable. However, Monte Carlo studies have shown that meeting these assumptions closely is not crucial if the sample size is very large. Serious biases are unlikely if the sample size is 50 or more normality can be assumed if the sample size exceeds 100. [Pg.86]

In some respects, this approach is very attractive since, if the spherical harmonic expansions of the correlation functions are sufficiently rapidly convergent, the approximate solution of the Ornstein-Zernike equation for a molecular fluid can be placed upon essentially the same footing as that for a simple atomic fluid. The question of convergence of the spherical harmonic expansions turns out to be the key issue in determining the efficacy of the approach, so it is worthwhile to review briefly the available evidence on this question. Most of the work on this problem has concerned the spherical harmonic expansion of (1,2) for linear molecules. This work was pioneered by Streett and Tildesley, who showed how it was possible to write the spherical harmonic expansion coefficients as ensemble averages obtainable from a Monte Carlo or molecular dynamics simulation via... [Pg.475]

Molecular dynamics generates configurations of the system that are connected in time and so an MD simulation can be used to calculate time-dependent properties. This is a major advantage of molecular dynamics over the Monte Carlo method. Time-dependent properties are often calculated as time correlation coefficients. [Pg.374]

The wavelet coefficients Wjnn ci) in nearby pixels are correlated and their statistical behaviour is complex (Grebenev et al., 1995). To set the correspondence between the computed S/N ratio of the coefficients and actual levels of significance we performed Monte-Carlo simulations. The probability that one of the sources, detected at the 4a S/N threshold in the 164x172 pixel wavelet image with scale a = 5, is spurious was found to be only 20%. [Pg.158]

Figure 12.5 contains the graphical representation of the Monte Carlo optimization. This approach is based on calculations of the correlation weights which give maximum correlation coefficient between experimental and predicted endpoint. [Pg.361]

Fig. 12.5 The general scheme of the Monte Carlo optimization used as the basis of caleulation of optimal descriptors. The row Correlation weight contains graphical images of various features (extracted from graph or SMILES) characterized by positive values of the correlation weights (they are indicated by white color) or by negative values of correlation weights (those are indieated by black color). Blocked (rare) features have correlation weights which are fixed to be equal to zero (indicated by grey b ). The R(X,Y) is correlation coefficient between descriptor and endpoint... Fig. 12.5 The general scheme of the Monte Carlo optimization used as the basis of caleulation of optimal descriptors. The row Correlation weight contains graphical images of various features (extracted from graph or SMILES) characterized by positive values of the correlation weights (they are indicated by white color) or by negative values of correlation weights (those are indieated by black color). Blocked (rare) features have correlation weights which are fixed to be equal to zero (indicated by grey b ). The R(X,Y) is correlation coefficient between descriptor and endpoint...
This can be done readily in a spreadsheet. First, a Monte Carlo simulation is employed to generate normal data for x and y respectively based on their fi and a. After the data for z are generated based on z = a x + b y, /x and can be calculated for the set of data z via equations (21.2) and (21.3). In this way, the calculation is simpler as there is no need to calculate correlation coefficient R. Otherwise, equations (21.5) and (21.6) are used to calculate and with R estimated based on correlation of x and y. [Pg.452]

The Monte Carlo method, which is an alternative to molecular dynamics, consists of the generation of a sequence of molecular configurations in such a way that there is a Boltzmann distribution in the potential energies of the selected configurations. Then, the average of a molecular property over all configurations gives the appropriate value for comparisons with experiment. This method can only be u.sed to calculate equilibrium properties of a system, while equilibrium, transport (e.g., diffusion coefficients), time dependent (e.g., time correlation functions), and spectroscopic (e.g., infrared and Raman lineshapes) properties are accessible to calculation with the molecular dynamics method. [Pg.1018]


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