First-order error analysis

First-order error analysis is a method for propagating uncertainty in the random parameters of a model into the model predictions using a fixed-form equation. This method is not a simulation like Monte Carlo but uses statistical theory to develop an equation that can easily be solved on a calculator. The method works well for linear models, but the accuracy of the method decreases as the model becomes more nonlinear. As a general rule, linear models that can be written down on a piece of paper work well with Ist-order error analysis. Complicated models that consist of a large number of pieced equations (like large exposure models) cannot be evaluated using Ist-order analysis. To use the technique, each partial differential equation of each random parameter with respect to the model must be solvable. 

In the equation, Yis the model output,/is the model, and (jCi. JCp) are random model parameters with standard error (5. 5p). The variance of model output is given by the Ist-order Taylor expansion  

As an example, in the following derivation the Ist-order error analysis equation for a simple model with both constants and random variables is found. The random terms ate X and Z, with constants a, b, and c. The model is 

Values for the variance of X and Z, including the correlation of X and Z, can easily be plugged into Equation (4.15) and solved. 

CHRONIC RISK CURVES FOR ATRAZINE IN TENNESSEE PONDS USING MONTE CARLO ANALYSIS 

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A Monte Carlo, Bayesian Monte Carlo, and First-Order Error Analysis... 

For example, n2 may be the number of counts obtained on a standard, while ni is the number of counts from the unknown. For similar compositions z is the ratio of the concentration of the element in the unknown to the known concentration in the standard. Using conventional, first-order, error analysis formulas [37,38] the relative standard deviation of the ratio z can be reported as... 

Forward Analysis In this type of analysis, we are interested in the propagation of initial perturbations Sxq along the flow of (1), i.e., in the growth of the perturbations 5x t xo) = (xo -h Sxq) — xq. The condition number K,(t) may be defined as the worst case error propagation factor (cf. textbook [4]), so that, in first order perturbation analysis and with a suitable norm j ... 

Figure 2. Histograms of Monte Carlo simulations for two synthetic analyses (Table 1) of a 330 ka sample. The lower precision analysis (A) has a distinctly asymmetric, non-Gaussian distribution of age errors and a misleading first-order error calculation. The higher precision analysis (B) yields a nearly symmetric, Gaussian age distribution with confidence limits almost identical those of the first-order error expansion.

The error analysis of this calculation procedure can be done using the equations in the previous section. It shows that the error made in using this scheme is of the order of 0(h + t). Thus, the scheme introduces an error term equivalent to a second-order partial differential term, which would add up to the RHS of Eq. 10.61, t.e., would decrease the apparent column efficiency. This procedure should not be used, unless very small values of the time increment t are selected. This, in turn, would make the computation time very long. In order to overcome this type of problem. Lax and Wendroff have suggested the addition to the axial dispersion term of an extra term, equivalent to the numerical dispersion term but of opposite sign [51]. This term compensates the first-order error contribution. In linear chromatography, the new finite difference equation, or Lax-Wendroff scheme, can be written as follows ... 

The benefit of (7.40) over the Euler-Maruyama scheme is that we can get second order accuracy in averages with only a single evaluation of the force at each timestep. However, our analysis presented here is only for the infinite time case. For averages computed in finite time, work in [224] has shown that (7.40) does indeed give a first order error, albeit one that vanishes exponentially fast in time as the simulation progresses. Hence for suitably long simulations, the observed error will be 0(h ). 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

If concentrations are known to —1-2 percent, a minimum of 10-fold excess over the stoichiometric concentration is required to evaluate k to within a few percent. The origins of error have been discussed.14,15 If the rate law is v = fc[A][B], with [B]o = 10[AJo, [B1 decreases during the run to 0.90[A]o. The data analysis provides k (the pseudo-first-order rate constant). To obtain k, one divides k by [B]av- If data were collected over the complete course of the reaction,... 

The first analysis is one with AS-level precision, the second with TIMS-level precision. The first order 2a error for the resulting 331 ka age is 96 ka, but examination of the distribution of a Monte Carlo simulation (Fig. 2) shows that the actual age distribution is strongly asymmetric, with 95% confidence limits of 158/-79 ka. For either younger ages or more-precise analyses, however, the first-order age errors are more than adequate, as shown by the Monte Carlo results for the same data, but with TIMS-level precision (Fig. 2B). 

They were also typical when the regression model chosen was first order. Mean-level bandwidths greater than 20-30% are probably indicative that errors have been made in the analysis process that should not be tolerated. In this case techniques would be carefully scrutinized to find errors, outliers, or changing chromatographic conditions. These should be remedied and the analysis repeated whenever possible. Certain manipulation can be done to reduce the bandwidth values. For example, they would be... 

Figure 2 is a good representation of almost all the 112 runs made with 1-octanol. There was curvature on only a few runs, undoubtedly caused by experimental error since they could not be reproduced. This feature was checked carefully after mathematical analysis indicated reasons to expect curvature. The conditions of reactions and values of k0 have been tabulated (Tables I and II). Since k0 depends upon 1-octanol and TMAE, it is called the pseudo-first-order rate constant. 

Thus, from a parabolic fit to the REDOR evolution data, the second moment can be evaluated. As mentioned in Section 1, this analysis has to be restricted to the initial part of the evolution curves AS/Sq <0.3, as exemplified in Figure 2. However, the first order approximation entails a systematic imderestimation of M2, as shovm by Bertmer and Eckert. Numerous variations of the original REDOR pulse sequence have been established to adapt the technique to specific needs. To accoimt for pulse imperfections and other experimental errors, Chan and Eckert introduced compensated REDOR. In this approach, an /-channel 7r-pulse in the centre of the pulse sequence cancels the reintroduction of the 7-S dipolar couplings hence the echo amplitudes are solely attenuated by the... 

Since the simple first-order treatment is generally adequate for ESR spectra of organic radicals in solution, their ESR spectra are easy to analyze, provided the lines do not overlap. Frequently, however, the lines are not all resolved and the analysis is difficult and subject to error. 

Calculate the percent error in 512 when a first-order analysis is applied to an AB spectrum when v08n/Jn has each of the following values20 (a) 2 (b) 4 (c) 7 (d) 10. 

For example, the output of a glass electrode (in mV) plotted against the antilog of activity of hydrogen ion yields a linear pH scale. It is the simplest form of performing analysis. This simplicity comes with a price, however. If the sample is contaminated by an unknown impurity, or if the response function 91 changes for whatever reason, an undetectable error accrues. Therefore, the first-order analysis relies on the invariability of the experimental conditions. 

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