Uncertainty propagation analysis

Mathematical model and uncertainty propagation analysis of a hypothetical measurement method in the context of the three-step model. 

Due to the fact that the measurands are stochastic variables an uncertainty propagation analysis was carried out. An uncertainty analysis can answer two questions (1) the expected accuracy (uncertainty) of the method, that is, the expected uncertainty with respect to the sought quantity (2) the most uncertain (sensitive) measurands. 

The main assumptions of the method are not explicitly declared. For example, Rogers does not discuss the problems with respect to the conflict between the transient behaviour of a batch process and the implicit steady-state assumption of the mass balance. However, the mathematical model behind the mass-balance method is quite clear. The work does not include any uncertainty propagation analysis of the mass-balance method. 

Keywords uncertainty propagation analysis powder mixtures dosage forms drug compounding. 

Abstract Every analytical result should be expressed with some indication of its quality. The uncertainty as defined by Eurachem ( parameter associated with the result of a measurement that characterises the dispersion of the values that could reasonably be attributed to the,. .., quantity subjected to measurement ) is a good tool to accomplish this goal in quantitative analysis. Eurachem has produced a guide to the estimation of the uncertainty attached to an analytical result. Indeed, the estimation of the total uncertainty by using uncertainty propagation laws is com-ponents-dependent. The estimation of some of those components is based on subjective criteria. The identification of the uncertainty sources and of their importance,... 

Together with the customer it was decided to perform the analysis according to option 2. The samples were analysed in one analytical run. The result is shown in Table 3. The result of the estimation compares well with the found results. The found uncertainty is smaller than the estimation. Assessment of the results of the validation of the manufacturing formula becomes easier from the customers point of view because the customer is able to deceide if a variation of his result is related to his process or to the uncertainty of the analytical method. Additionally, the influence of the individual contributions to uncertainty becomes smaller because of the uncertainty propagation. Therefore, the difference between the estimated and found uncertainty becomes smaller with an increasing number of parameters that influence uncertainty. 

Type B evaluation Method of evaluation of uncertainty by means other than the statistical analysis of series of observations (ISO 1995). uncertainty propagation See propagation of uncertainty. 

A comparison between different uncertainty propagation methods in multivariate analysis... 

Baraldi, P. Zio, E. (2008) A combined Monte Carlo and possibilistic approach to uncertainty propagation in event tree analysis. Risl nal 28 (5) 1309-1325. 

Estimation of model error bars and sensitivity analyses are based rai the same principle. AU rate coefficients (or other model parameters) of a system are randomly varied within a certain range. The chemical evolution is then computed for each set of rate coefficients. For a network containing 4,000 reactions, the model is typically run 2,000 times with different sets of rate coefficients. The distribution of the rate coefficients can be either log-normal or log-uniform (see Fig. 4.5). The first choice implies that the mean value ko is a preferred value. This is usually the case for rate coefficients, which are measured with an uncertainty defined by statistical errors. The factor Fq, which defines the range of variation, can be a fixed factor for aU reactiOTis for a sensitivity analysis or specific to each reaction for an uncertainty propagation study. Use of the same Fq for all reactions, in the case of a sensitivity analysis, assures the modeller that an underestimated uncertainty factor will not bias the analysis. The results of thousands of runs are used differently to identify important reactions and to estimate model error bars. 

While few researches on the use of BNs in system rehability and safety analysis take epistemic uncertainty or uncertainty propagation into account, one exception is an interesting work by Simon et al. (Simon et al. 2008) which concerns how a BN can be modified to a so called evidential network to handle epistemic uncertainties. Evidential network is a combination of evidence theory and BN. In this section the basic concepts of D-S evidence theory and evidential network are introduced. 

The Best Estimate Plus Uncertainties (BEPU) analysis is recommended by IAEA in addition or alternatively to the deterministic approach in the safety analysis of nuclear components and systems (IAEA 2009). The aim of the BEPU analysis is to determine a quantile of an output measure of interest (noted R in the following) with a certain level of confidence (usually the 95% quantile obtained with a 95% confidence, denoted Rg gf and to verify that this quantile is below an acceptable limit (acceptance criteria). In order to obtain this quantile, the uncertainty space of input parameters is sampled at random according to their combined probability distribution and a code calculation is performed for each sampled set of parameters. The number of code calculations is determined by the requirement to estimate a tolerance and confidence interval for the quantity of interest. Wilks formula (Wilks 1941) (or Wald formula (Wald 1943) when several criteria must be respected simultaneously) is used to determine the number of calculations to obtain the uncertainty bands and the associated quantile with a given confidence level. In classical BEPU analysis, there is no separation between the aleatory variables and the epistemic variables the epistemic variables, which are often model uncertainties, are generally modeled by uniform probability distributions within intervals provided by expert opinion and propagated in the same way that the aleatory variables by Monte Carlo simulation. 

In the safety analysis of complex systems, the treatment of uncertainty must distinguish the aleatory uncertainties, which represent the intrinsic randomness of the phenomena, from the epistemic uncertainties resulting from a lack of knowledge. This separation has to be done through a two-level uncertainty propagation the internal level concerning the aleatory variables and the external level the epistemic variables. For the propagation of epistemic uncertainties, it appears more appropriate to use extra-probabilistic approaches such as interval analysis or Dempter-Shafer Theory of Evidence (DSTE). 

This type of analysis constitutes a form of level 1 purely epistemic uncertainty propagation approach. Hence we do not consider a level 2 aleatory/epistemic uncertainty propagation approach (e.g. the probability of frequency approach) see Aven et al. (2014, p. 59-78) for a detailed description of the meaning of a level 1 or level 2 uncertainty propagation approach. Furthermore, we leave out of the consideration the goodness of the risk index R(X), i.e. to what extent the index appropriately characterises risk. Of course, having a well-founded uncertainty characterisation of Xmay be less important if R( represents a poor characterisation of risk. 

The result of this uncertainty analysis - or uncertainty propagation - is then a distribution function of the core damage frequency, as shown in the lower right subfigure. The uncertainty in the core damage frequency is here emphasized by a blue, double-sided horizontal arrow (Fig. 10). 

An estimation of the experimental uncertainty was determined for both g and X. Since both of these parameters were related to measured quantities having their own level of uncertainty, an error propagation analysis [22] was performed to estimate the uncertainty range (at the 95% confidence level) of the computed values for g and X. 

A propagation of uncertainty also helps in deciding how to improve the uncertainty in an analysis. In Example 4.7, for instance, we calculated the concentration of an analyte, obtaining a value of 126 ppm with an absolute uncertainty of 2 ppm and a relative uncertainty of 1.6%. How might we improve the analysis so that the absolute uncertainty is only 1 ppm (a relative uncertainty of 0.8%) Looking back on the calculation, we find that the relative uncertainty is determined by the relative uncertainty in the measured signal (corrected for the reagent blank)... 

The effect of an uncertainty in potential on the accuracy of a potentiometric method of analysis is evaluated using a propagation of uncertainty. For a membrane ion-selective electrode the general expression for potential is given as... 

Combine the above with the internal events analysis to obtain plant risk and propagate the uncertainties... 

The uncertainty in the measurement of elution time / or elution volume of an unretained tracer is another potential source of error in the evaluation of thermodynamic quantities for the chromatographic process. It can be shown that a small relative error in the determination of r , will give rise to a commensurate relative error in both the retention factor and the related Gibbs free energy. Thus, a 5% error in leads to errors of nearly 5% in both k and AG. An analysis of error propagation showed that if the... 

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. 

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