Biased sampling

As first step, it is important to analyze what happens if an additional potential V(s), which acts only on the CVs s, is added to the physical one U q). The resulting potential will be U q) + V s q)), so that the explored conformations will be distributed according to a biased canonical distribution 

If one tries to evaluate the free-energy landscape from such a biased distribution of conformations, one will end up in a different free energy F which is related to the original one by 

by supposing that the free-energy landscape is a priori known, at least approximately, it is possible to imagine performing a biased simulation, using V s) = —F s) as a bias where F(s) is our estimate for the free-energy landscape. The resulting simulation will explore the distribution. 

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The biased-sampling approach may be considerably generalized, to allow the construction of MC moves step-by-step, with each step depending on the success or failure of the last. Such a procedure is biased, but it is then possible to correct for the bias (by considering the possible reverse moves). The technique has dramatically speeded up polymer simulations, and is capable of wider application. 

There are a few variations on this procedure called importance sampling or biased sampling. These are designed to reduce the number of iterations required to obtain the given accuracy of results. They involve changes in the details of how steps 3 and 5 are performed. For more information, see the book by Allen and Tildesly cited in the end-of-chapter references. 

An improvement of this method—the so-called biased sampling [55] (or inversely restrieted sampling)—suggests to look ahead at least one step in order to overcome the attrition. Consider a SAW of i steps on a -coordination number lattice. To add the / + 1st step one first checks which of the = q — neighboring sites are empty. If k qQ > k>0) sites are empty one takes one of these with equal probability 1 /A if A = 0 the walk is terminated and one starts from the beginning. This reduces the attrition dramatically. Now each A-step walk has a probability PAr( i ) = Ylf=i so that dense configurations are clearly more probable. To compensate for this bias, each chain does not count as 1 in the sample but with a weight... 

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... 

Speleothem frequency distributions have provided a useful tool for broad comparisons, but they suffer from the problem of biased sampling strategies and low resolution at times of known abrupt change. The increased precision afforded by mass-spectrometric techniques will result in fewer studies using this approach to assess of growth frequency and, more often, records of continuous deposition and growth rate studies will be graphically illustrated. 

Transient currents in biased samples at room temperature. 

Thus far, we have been discussing the crystallization of a multichain system. However, under suitable conditions, crystallization can even occur in a single-chain system. Using a combination of biased sampling, multihistogram techniques, and parallel tempering [ 125], we can directly compute the... 

Oberhofer, H. Dellago, C. Geissler, P. L., Biased sampling of nonequilibrium trajectories. Can fast switching simulations outperform conventional free energy calculation methods, J. Phys. Chem. B 2005,109, 6902-6915... 

Surveillance is biased sampling directed at particular carcasses or products. Surveillance comes into play when the Program receives information from monitoring or other sources, e.g., from slaughter inspection, indicating that adulterating residues may be present. Product may be held until laboratory tests determine the appropriate regulatory action. 

R. Delgado-Buscaiohni, G. De Fabritiis, and P. V. Coveney, Determination of the chemical potential using energy-biased sampling. J. Chem. Phys. 123, 054105 (2005). 

Consider that we wanted to compare the height of adult males and females in the Netherlands. It would obviously be impractical to measure the heights of the whole population. We would take representative samples from both sexes and use these data to draw conclusions about the larger population. Population characteristics have to be inferred or estimated from measures taken from samples. If the sample is not truly representative of the population from which it is drawn, i.e. if it is a biased sample, then it is virtually impossible to make accurate predictions about the population. For example, choosing our sample of males from a basketball team would introduce considerable bias-basketball players are usually very tall The way to minimise bias of this type is to take random samples, where each member of the population under study has an equal chance of being selected. 

Lunar meteorites (see review by Korotev et al., 2003) are mostly brecciated samples of highlands crust (FAN) and regolith, although a few mare basalts are included in this collection. It is likely that the source craters for the meteorites are randomly distributed and thus include materials from the lunar farside. As we will see, these meteorites provide a better estimate of the crustal composition than do the geographically biased samples returned by spacecraft. 

Figure 13.6, which plots 120 versus Mg/(Mg+Fe) from Lunar Prospector gamma-ray spectrometer data in 5° bins, shows that the global data overlap major rock types in the Apollo collection. Such comparisons give confidence that major geologic units have not been overlooked by biased sampling. 

Fig. 4. Schematic representation of electrokinetic sample injection in a a volume defined injection scheme (unbiased loading) and b conventional CE (biased sample loading). The sample migrates through the injection volume in a way resembling a frontal electropherogram...

Indirect Methods Test particle method Grand canonical ensemble method Biased sampling methods Thermodynamic integration... 

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