Reweighting functions

Renormalization group 12 Reweighting functions 85 RISM theory 210, 218 Rosenbluth method, pruned-enriched (PERM) 15... 

A useful method of weighting is through the use of an iterative reweighted least squares algorithm. The first step in this process is to fit the data to an unweighted model. Table 11.7 shows a set of responses to a range of concentrations of an agonist in a functional assay. The data is fit to a three-parameter model of the form... 

Fig. 3.3. Typical results from a density-of-states simulation in which one generates the entropy for aliquid at fixed N and V (i.e., fixed density) (adapted from [29]). The dimensionless entropy. r/ In ( is shown as a function of potential energy U for the 110-particle Lennard-Jones fluid at p = 0.88. Given an input temperature, the entropy function can be reweighted to obtain canonical probabilities. The most probable potential energy U for a given temperature is related to the slope of this curve, d// /dU(U ) = l/k T, and this temperature-energy relationship is shown by the dotted line. Energy and temperature are expressed in Lennard-Jones units...

The histogram reweighting methodology for multicomponent systems [52-54] closely follows the one-component version described above. The probability distribution function for observing Ni particles of component 1 and No particles of component 2 with configurational energy in the vicinity of E for a GCMC simulation at imposed chemical potentials /. i and //,2, respectively, at inverse temperature ft in a box of volume V is... 

We can, therefore, let /cx be the subject of our calculations (which we approximate via an array in the computer). Post-simulation, we desire to examine the joint probability distribution p(N, U) at normal thermodynamic conditions. The reweighting ensemble which is appropriate to fluctuations in N and U is the grand-canonical ensemble consequently, we must specify a chemical potential and temperature to determine p. Assuming -7CX has converged upon the true function In f2ex, the state probabilities are given by... 

Note that there is a strong similarity to LDA (Section 5.2.1), because it can be shown that also for LDA the log-ratio of the posterior probabilities is modeled by a linear function of the x-variables. However, for LR, we make no assumption for the data distribution, and the parameters are estimated differently. The estimation of the coefficients b0, b, ..., bm is done by the maximum likelihood method which leads to an iteratively reweighted least squares (IRLS) algorithm (Hastie et al. 2001). 

It is obvious that LS regression yields values that are too high and cannot be used for interpretation of plant lead content. The introduction of the reweighted median-related values enables comparison of the results of the various LMS regressions with one another. For this purpose the regression coefficient a7/ of the function... 

Fig. 4.2. Time series of potential energy of the C-peptide system from the multi-canonical MD production run (a) and the average potential energy as a function of temperature (b). The latter was obtained from the trajectory of the multicanonical MD production run by the single-histogram reweighting techniques...

Fig. 4.6. The population ratios as functions of the inverse of temperature l/T at constant pressure of V = 0.1 MPa, which was obtained by the reweighting techniques from the results of the multibaric-multithermal MD simulation (a) those of WcrJWPu and WaK/W-pu, (b) that of Wa-p/Wpu, and (c) that of Wah/W-put...

Fig. 21. Ratio between the interface tension 7 and the simple expression for the strong segregation limit yssL in (54) as a function of inverse incompatibility. Symbols correspond to Monte Carlo results for the bond fluctuation model, the solid line shows the result of the SCF theory, and the dashed line presents first corrections to (54) calculated by Semenov. Also an estimate of the interface tension from the spectrum of capillary waves is shown to agree well with the results of the reweighting method. Adapted from Schmid and Muller [107]...

A correlated sampling method, known as reweighting [39,40] is much more efficient. One samples a set of configurations Rj (usually several thousand points at least) according to some distribution function, usually taken to be the square of the wavefunction for some initial trial function ] f T(-R q o) -Then, the variational energy (or the variance) for trial function nearby in parameter space can be calculated by using the same set of points ... 

We have studied the phase and micellization behavior of a series of model surfactant systems using Monte Carlo simulations on cubic lattices of coordination number z = 26. The phase behavior and thermodynamic properties were studied through the use of histogram reweighting methods, and the nanostructure formation was studied through examination ofthe behavior ofthe osmotic pressure as a function of composition and through analysis of configurations. 

The above reweighting technique [136] is analogous to the histogram Monte Carlo approach [141-143], but instead of determining the configurational density of states from the canonical potential energy distribution, g, effectively a density of minima, is obtained from the occupation probability of the different basins of attraction. A similar approach has been used to calculate the density of minima as a function of the potential energy for a bulk liquid [144,145]. 

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