Dominant Errors

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. 

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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. 

As discussed in Section 6.8, the estimation errors can be categorized as statistical, bias, and discretization. In a well designed MC simulation, the statistical error will be controlling. In contrast, in FV methods the dominant error is usually discretization. 

The dominant error term is third order in At. The initial wavefunction (Qx,Qy,t) at t = 0 is normally the lowest energy eigenfunction of the initial state of the spectroscopic transition. The value of the wavefunction at incremental time intervals At is calculated by using Eq. (7) for each point on the (Qx,Qy) grid. The autocorrelation function is then calculated at each time interval and the resulting < (t> is Fourier transformed according to Eq. (2) to give the emission spectrum. 

Just n — 1 derivatives are needed on the right-hand side, and the dominant error term is indicated. The resulting system can be cast in vector/matrix form... 

Minimizing errors - determine early in your study what the dominant errors are likely to be and concentrate your time and effort on reducing these. 

As Table 3 illustrates, both the PYc and HNCc approximations give the correct zeroth-order contribution C(r x) max- The leading-order and dominant error in the HNCc approximation arises from the nonzero coefficient which it assigns to nmax-i One way to correct the approximation would be to take this contribution back out. For the PYc approximation, the leading-order contribution to the error arises from an incorrect nmax-2 coefficient. In fact, to this order in the Ree-Hoover series, PYc gives precisely... 

For trace analysis, counting rates are low and peak-to-background ratios are generally poor. This means that the dominant error in trace analysis is normally due to counting statistics. Since peak-to-background ratios are low, it becomes important to measure and subtract the background contribution accurately. 

With SA on, satellite clock dithering was the dominant error and was common to all users. It is also assumed that over the few hundred kilometers or less where the signal can be received, that errors such as ephemeris and ionospheric and tropospheric delay are correlated between base station and user receiver as well and can be considerably reduced. The algorithms to predict these delays are disabled in both the base station and user receivers. 

However equations (3) and (4) cannot be regarded as anything but helpful guides, since in cases where highly reliable systems are sought, common-mode failures will dominate errors in maintenance, errors in design, environmental conditions etc. (Reference 8) software design errors (Reference 9). 

Risk management culture, where an engineering view of human error causation is dominant. Errors and accidents are analysed in terms of mismatches between the operator and his environment. Remedial actions typically include design changes and provisions of procedural support. 

Ideally the aim of every evaluation effort is to present a best or recommended value plus a quantitative statement of its uncertainty. If the dominant errors are truly random, a standard deviation or 95% confidence interval can be quoted, which gives the user a sound basis for deciding the implication of this uncertainty for a given application. However, this ideal situation rarely applies instead, the most significant errors... 

Since RHF limit properties will differ (possibly substantially) from exact wavefunction values or experiment, do the deviations from the RHF limit shown in Table 1 really matter The answer is yes for two reasons. First, such deviations signal the inability to satisfy the HF equations for even the simplest approximate wavefunction, and they will not become easier to satisfy in the multiconfiguration case. Second, the RHF configuration usually has substantial weight in the exact wavefunction, so errors committed at the RHF level will propagate with comparable magnitude to higher levels and cannot be removed except by fortuitous error cancellation. For some properties (e.g., quadrupole moments, spin densities) deviations from the RHF limit easily can be the dominant error in BSE calculations. 

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