Mean-field estimation errors

As with all statistical methods, the mean-field estimate will have statistical error due to the finite sample size (X ), and deterministic errors due to the finite grid size (S ) and feedback of error in the coefficients of the SDEs Ui,p). Since error control is an important consideration in transported PDF simulations, we will now consider a simple example to illustrate the tradeoffs that must be made to minimize statistical error and bias. The example that we will use corresponds to (6.198), where the exact solution141 to the SDEs has the form ... 

The error e in this expression must be the true error (i.e., all the systematic and statistical errors). Uncontrolled approximations cannot be allowed otherwise, the complexity problem is not well posed. Chemistry is unique in that, first, there is a well-tested, virtually exact theory (the Schrddinger equation), and second, the mean-field estimates of chemical energies are often surprisingly accurate. Unfortunately, very accurate estimates are required to provide input to real-world chemistry, since much of the interesting chemistry takes place at room temperature. Currently, the level of chemical accuracy is considered to be =1 kcal/... 

We have seen that Lagrangian PDF methods allow us to express our closures in terms of SDEs for notional particles. Nevertheless, as discussed in detail in Chapter 7, these SDEs must be simulated numerically and are non-linear and coupled to the mean fields through the model coefficients. The numerical methods used to simulate the SDEs are statistical in nature (i.e., Monte-Carlo simulations). The results will thus be subject to statistical error, the magnitude of which depends on the sample size, and deterministic error or bias (Xu and Pope 1999). The purpose of this section is to present a brief introduction to the problem of particle-field estimation. A more detailed description of the statistical error and bias associated with particular simulation codes is presented in Chapter 7. 

Ideally, one would like to choose Np and M large enough that e is dominated by statistical error (X ), which can then be reduced through the use of multiple independent simulations. In any case, for fixed Np and M, the relative magnitudes of the errors will depend on the method used to estimate the mean fields from the notional-particle data. We will explore this in detail below after introducing the so-called empirical PDF. 

FIGURE 11.5 Safflower seed losses at harvest and predicted persistence in soil from field studies. The table at the top is the mean number of safflower seeds lost after harvest in commercial fields. The table in the middle is the results of the regression analysis of seed viability rate of decline over time at the EUerslie site established in 2002 with means and standard errors for intercept (a), rate of seed viability decline (b), and frequency of seeds in spring and fall for 2 follow years. Note that no viable seeds were recovered from the burial studies after 2 years. The figure at the bottom provides estimates of the percentage and number of seeds per unit area that would remain viable if 1,000 seeds m" were lost in the fall following safflower production, based on the rate of decline of safflower seeds in the artificial seed bank (burial) studies. 

In order to test observational errors nsing a fnll sample of unblended spectral lines, the Monte-Carlo method with a generator of normally distributed numbers was used. For N = 2545 measurements of magnetic fields on four yellow supergiants Aqr, a Aqr, e Gem, e Peg), including weak unblended spectral lines, the relation between mean the Monte-Carlo simulated standard error and the mean experimental standard error was estimated as = 1.033<(t>. Further, weak spectral lines for which z ro - rj < 0.2 were eliminated to strengthen the data uniformity. For A= 1844 measurements = 0.968<(t>. The discrepancy is 3.3 % in the first case and 3.2 % in the second case both appear to be very small. 

Fig. 1. Mean abundances for background giants in the fields of the globulars studied va distance om the Galactic center Re = ( + 2.046 ) in degrees. The error bars are the combination of the A(J K) error and the estimated error in the metaUidty of the globulars ( - 0.2 dex). For comparison, the solid line is the fit to the metallidties derived by Terndrup (1988) om optical photometry along minor fields.

The LER-HEP method (NUREG/ CR-3519) is a means of analyzing field data to estimate HEPs. It considers available data on specific human errors in similar to those being considered in the risk as.sessment. The application of the method in NUREG/CR-3519 is to the Licensee Event Report (LER) file so it is called the LER-HEP method. For each error analyzed by this method, an error rate is derived by dividing the number of similar errors by the estimated number of... 

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