Estimation errors discretization

There is an alternative - and for our purposes more powerful - way to estimate the discretization error, namely in terms of the Fourier expansion of a periodic -function. We write hf xk), see (A.l), as [24]... 

The discrete version of the equation for the estimate error covariance propagation is now... 

Likewise, we can initialize the notional-particle properties so that both fxO, 0 and 6s (x, t) are null.145 The estimation error in the initial conditions, (6.222), is then due only to discretization error ... 

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. 

Relative Free Energies Of Solvation in Water, in kcal/mole," Obtained by a QM/MM Discrete Molecular Solvent Method, Using The AMI Solute Hamiltonian. The Estimated Error Bars for The Calculated Values Are 0.5 Kcal/Mole... 

Once the possibility of estimator construction was established, as mentioned above, the estimator construction followed a straightforward application of the construction procedure given in [5], but with one tenuous modification the estimation sequence was delayed in one-step by replacing the discrete-instantaneous output estimation error... 

The discrepancy can be seen at various places along the eurves and is small in absolute terms. However, the difference is systematieally large in relative terms at concentrations approaching zero, a feature that would lead to potentially serious errors in the determination of low-concentration analytes. Analysts need to avoid this trap, by calibrating the analytical system over a much smaller range than that shown when near-zero concentrations are important. In reality, the quadratic curve would be estimated from discrete responses corresponding to a small number of calibration points and the ensuing random errors would be combined with these systematic discrepancies. 

To increase the accuracy of solving the problems of detection and separation of waveforms, a posteriori discrete optimization algorithms have been proposed and analyzed. High accuracy of the proposed algorithms has been proved by numerical experiments. Specifically, it has been shown that the root-mean-square deviation in the estimation of waveforms does not exceed 6 % and relative estimation errors of their arrival times are not worse than 0.1%. 

Hence, we use the trajectory that was obtained by numerical means to estimate the accuracy of the solution. Of course, the smaller the time step is, the smaller is the variance, and the probability distribution of errors becomes narrower and concentrates around zero. Note also that the Jacobian of transformation from e to must be such that log[J] is independent of X at the limit of e — 0. Similarly to the discussion on the Brownian particle we consider the Ito Calculus [10-12] by a specific choice of the discrete time... 

Unfortunately, this local error Cr cannot be calculated, since we do not know the exact solution to the QCMD equations. The clue to this problem is given by the introduction of an approximation to Let us consider another discrete evolution with an order q > p and define an error estimation via er t + z i) - z t). 

We replace the integral from Si to S2 by a sum over a regular grid. We do this by applying first a variable transformation (to be specified by some criteria) such that after this transformation an equidistant grid can be used. An estimate of the discretization error is possible by means of tricky and non-trivial application of analysis. Details on this are given in the appendix, which is a rather important part of this paper. 

Estimates for the discretization error are derived in the appendix. Unlike the estimates (2.6) these are not obtained as strict inequalities, but rather as leading terms of asymptotic expansions. For the integral (2.12a) with the integration limits —oo to oo the discretization error is (for large n and sufficiently small h, see appendix... 

The estimation of the discretization error is fortunately rather easy, relying on the results of appendix E (which contains the difficult part of the derivation). In fact the discretization error /a(r) given by (2.14) is simply proportional to 1/r. Hence... 

There is one difficulty insofar as (3.8) is only an estimate of the absolnte valne of the discretization error. It cannot be exclnded that (depending on how the limit xi —oo, X2 oo is performed, see appendix D) c and a have opposite sign. In this case the minimum absolute error may vanish, while (3109a) is still valid. 

The discretization error Cd for finite integration limits yi and y2 contains in addition to (D.8) two extra terms (under the sum) that contain incomplete Gamma functions. We don t need their explicit form for the estimation of the dominating part of the overall error. Of course, expanding these extra terms in powers of h would lead to the error estimation (A.4), that holds for extremely small h (and sufficiently small /) which is rather irrelevant in the present context. 

This estimate of the discretization error ought to be known in numerical mathematics. Usually it is easier to derive formulas like this than too look them up in the literature. 

But the main advantage of the SNR concept in modern analytical chemistry is the fact that the signal function is recorded continuously and, therefore, a large number of both background and signal values is available. As shown in Fig. 7.9, the principles of the evaluation of discrete and continuous measurement values are somewhat different. The basic measure for the estimation of the limit of detection is the confidence interval of the blank. It can be calculated from Eq. (7.52). For n = 10 measurements of both blank and signal values and a risk of error of a = 0.05 one obtains a critical signal-to-noise ratio (S/N)c = fo.95,9 = 1.83 and a = 0.01 (S/N)c = t0.99,9 = 2.82. The common value (S/N)c = 3 corresponds to a risk of error a = 0.05... 0.02 in case of a small number of measurements (n = 2... 5). When n > 6, a... 

Equation (4.33) requires the computation of time derivatives. In molecular dynamics discretization errors due to the finite time step dt are of order Oidt2). Therefore we would like to estimate the time derivative in (4.33) with the same accuracy. 

Selected entries from Methods in Enzymology [vol, page(s)] Generation, 240, 122-123 confidence limits, 240, 129-130 discrete variance profile, 240, 124-126, 128-129, 131-133, 146, 149 error response, 240, 125-126, 149-150 Monte Carlo validation, 240, 139, 141, 146, 148-149 parameter estimation, 240, 126-129 radioimmunoassay, 240, 122-123, 125-127, 131-139 standard errors of mean, 240, 135 unknown sample evaluation, 240, 130-131 zero concentration response, 240, 138, 150. 

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