Finite sampling

So basic is the notion of a statistical estimate of a physical parameter that statisticians use Greek letters for the parameters and Latin letters for the estimates. For many purposes, one uses the variance, which for the sample is s and for the entire populations is cr. The variance s of a finite sample is an unbiased estimate of cr, whereas the standard deviation 5- is not an unbiased estimate of cr. 

Having established that a finite volume of sample causes peak dispersion and that it is highly desirable to limit that dispersion to a level that does not impair the performance of the column, the maximum sample volume that can be tolerated can be evaluated by employing the principle of the summation of variances. Let a volume (Vi) be injected onto a column. This sample volume (Vi) will be dispersed on the front of the column in the form of a rectangular distribution. The eluted peak will have an overall variance that consists of that produced by the column and other parts of the mobile phase conduit system plus that due to the dispersion from the finite sample volume. For convenience, the dispersion contributed by parts of the mobile phase system, other than the column (except for that from the finite sample volume), will be considered negligible. In most well-designed chromatographic systems, this will be true, particularly for well-packed GC and LC columns. However, for open tubular columns in GC, and possibly microbore columns in LC, where peak volumes can be extremely small, this may not necessarily be true, and other extra-column dispersion sources may need to be taken into account. It is now possible to apply the principle of the summation of variances to the effect of sample volume. 

Given limits to the time resolution with which wave profiles can be detected and the existence of rate-dependent phenomena, finite sample thicknesses are required. To maintain a state of uniaxial strain, measurements must be completed before unloading waves arrive from lateral surfaces. Accordingly, larger loading diameters permit the use of thicker samples, and smaller loading diameters require the use of measurement devices with short time resolution. 

In the case of finite sample size in analytical practice, the quantiles of Student s f-distribution are used as realistic limits. 

Finding the best estimate of the free energy difference between two canonical ensembles on the same configurational space, for which finite samples are available, is a nontrivial problem. Charles Bennett [11] addressed this problem by developing the acceptance ratio estimator, which corresponds to the minimum statistical... 

Wood, R. H. Muhlbauer, W. C. F. Thompson, P. T., Systematic errors in free energy perturbation calculations due to a finite sample of configuration space. Sample-size hysteresis, J. Phys. Chem. 1991, 95, 6670-6675... 

Zuckerman, D. M. Woolf, T. B., Overcoming finite-sampling errors in fast-switching free-energy estimates. Extrapolative analysis of a molecular system, Chem. Phys. Lett. 2002, 351, 445 153... 

Note that the variance does not depend on the true value x, and the mean estimator x has the least variance. The finite sampling bias is the difference between the estimate x and the true value x, and represents the finite sampling systematic part of the generalized error... 

In this chapter we use the terms precision and accuracy in relation to the finite sampling variance and bias, respectively. Also, we describe the overall quality of an estimator - the mean square error - by the term reliability. Note the difference between our terminology and that in some statistics literature where accuracy is used to describe the overall quality (i.e., the reliability in this chapter). The decomposition of the error into the variance and bias allows us to use different approaches for studying the behavior of each term. 

In the equations above, the mean square error, the sample variance, and the finite sampling bias are all explicitly written as functions of the sample size N. Both the variance and bias diminish as /V — oc (infinite sampling). However, the variance... 

The sample size in a real simulation is always finite, and usually relatively small. Thus, understanding the error behavior in the finite-size sampling region is critical for free energy calculations based on molecular simulation. Despite the importance of finite sampling bias, it has received little attention from the community of molecular simulators. Consequently, we would like to emphasize the importance of finite sampling bias (accuracy) in this chapter. 

In principle, the forward and reverse calculations should produce identical free energy estimates. However in real simulations these estimates usually differ. Furthermore, as we explore finite sampling errors, we will see that the reliability (the error in each direction) of these estimates also differs [24, 26, 38]. 

Thus, the cause of the finite sampling error in free energy estimates from FEP or NEW simulations is the poor sampling of the important low-x tail of the / distribution and the high-x tail of the g distribution, for a forward and a reverse calculation, respectively. Sampling of these important tails corresponds to the sampling of the... 

Perhaps the most challenging part of analyzing free energy errors in FEP or NEW calculations is the characterization of finite sampling systematic error (bias). The perturbation distributions / and g enable us to carry out the analysis of both the finite sampling systematic error (bias) and the statistical error (variance). 

As discussed in Sect. 6.1, the bias due to finite sampling is usually the dominant error in free energy calculations using FEP or NEW. In extreme cases, the simulation result can be precise (small variance) but inaccurate (large bias) [24, 32], In contrast to precision, assessing the systematic part (accuracy) of finite sampling error in FEP or NEW calculations is less straightforward, since these errors may be due to choices of boundary conditions or potential functions that limit the results systematically. 

In this model, the finite sampling systematic error is due to the missed sampling of the important region x < Xf. The free energy estimate given by the model is... 

By subtracting (6.18) from (6.19), we obtain the finite sampling systematic error in... 

These bounds originate from the systematic errors (biases) due to the finite sampling in free energy simulations and they differ from other inequalities such as those based on mathematical statements or the second law of thermodynamics. The bounds become tighter with more sampling. It can be shown that, statistically, in a forward calculation AA(M) < AA(N) for sample sizes M and N and M > N. In a reverse calculation, AA(M) > AA(N). In addition, one can show that the inequality (6.27) presents a tighter bound than that of the second law of thermodynamics... 

In practice it is helpful to know the order of magnitude of the sample size N needed to reach a reasonably accurate free energy. The inaccuracy model described above presents an effective way to relate the sample size N and the finite sampling error through perturbation distribution functions. Alternatively, one can develop a heuristic that does not involve distribution functions and is determined by exploring the common behavior of free energy calculations for different systems [25]. Although only FEP calculations are considered in this section, the analysis extends to NEW calculations. 

The variance characterizes the spread of AA if an infinite number of independent simulations are carried out, each with a finite sample of size N. In practice, usually only one estimate (or a small number of repeats) of free energy differences are taken, and the variance in free energy must be estimated. One way to compute the variance is to use the error propagation formula (for a forward calculation)... 

Generally, a NEW calculation has finite sampling and is usually far from the nearequilibrium region. Thus, the perturbation distributions are not Gaussian, and the conclusion of (6.42) or (6.44) is no longer valid. However, (6.36) is still useful. 

Now consider the finite sampling systematic error. As discussed in Sect. 6.4.1, the fractional bias error in free energy is related to both the sample size and entropy difference 5e N exp(-AS/kB). With intermediates defined so that the entropy difference for each substage is the same (i.e., AS/n), the sampling length Ni required to reach a prescribed level of accuracy is the same for all stages, and satisfies... 

Finally, the material presented in this chapter paves the way for further improvements in free energy calculations. Two promising directions for future studies are improving the methods for sampling phase space so that is satisfies the most effective overlap and/or subset relationships and developing better techniques for averaging samples and extrapolating from finite-sampled sets. 

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