Uncertainty propagation

Pauwels (1999) argues that the certified values of CRMs should be presented in the form of an expanded combined uncertainty according to the ISO Guide on the expression of uncertainty in measurement, so that coverage factor should always be clearly mentioned in order to allow an easy recalculation of the combined standard uncertainty. This is needed for uncertainty propagation when the CRM is used for calibration and the ISO Guide should be revised accordingly. The use of the expanded uncertainty has been pohcy in certification by NIST since 1993 (Taylor and Kuyatt 1994). 

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. 

Person S, Long TF. 1995. Conservative uncertainty propagation in environmental risk assessments. In Hughes JS, Biddinger GR, Mones E, editors. Environmental toxicology and risk assessment, 3rd vol, ASTM STP 1218. Philadelphia American Society for Testing and Materials, p 97-110. 

Proponents of the SI for chemistry must consider that proportionality is deeply embedded in chemical thinking.3 Many of the potentially most reliable analytical techniques - for instance isotope-dilution mass spectrometry - yield ratios in the first place. In complex series of ratio measurements the uncertainty propagation is more straightforward than when sums and differences from standards - such as for mass determinations - are involved. Consistent with the use of SI, the value of a ratio is called a measurement when numerator and denominator are multiplied by a unit and the related uncertainties have been evaluated. 

Olivieri, A.C., A simple approach to uncertainty propagation in preprocessed multivariate calibration, J. Chemom., 16, 207-217, 2002. 

In scientific calculations it is also useful to be able to estimate the precision of a procedure that involves several measurements by combining the precisions of the individual steps. That is, we want to answer the following question How do the uncertainties propagate when we combine the results of several different types of measurements There are many ways to deal with the propagation of uncertainty. We will discuss one simple method below. 

Cabaniss S. E. (1999) Uncertainty propagation in geochemical calculations non-linearity in solubility equilibria. Appl. Geochim. 14, 255-262. 

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,... 

Total uncertainty (received from uncertainty propagation) 1.8% 1.4%... 

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. 

S. S. IsukapalU, A. Roy, and P. G. Georgopoulos, Stochastic response surface methods (SRSMs) for uncertainty propagation application to environmental and biological systems. Risk Anal 18 351-363 (1998). 

A very important observation at this point is that an IDMS assay is in principle a physical measurement since it is a measurement of ratio of isotopes and not of a ratio of elements (as in classical analytical chemistry). Indeed two numbers of atoms are compared in a ratio determination and these atoms belong to the same element. Hence ail the chemical interferences, normal in a chemical assay, do not affect the result anymore. Combined with the fact that the requirement of being quantitative - essential and difficult in classical chemistry assay - must not be fulfilled (after spiking), this means that IDMS ranks higher in the hierarchy of methods than normal elemental assay methods since it is far less subject to potential chemical error sources. In other words its inherent potential for good precision and accuracy (i.e. small overall uncertainty) and - at least as important -the transparency of the uncertainty propagation in (Eqs. 4 and 5) give it the character of what some have called a "reference method" or even a definitive method". 

Stolarski, R. S., and Douglass, A. R. (1986) Sensitivity of an atmospheric photochemistry model to chlorine perturbations including consideration of uncertainty propagation, J. Geophys. Res., 91, 7853-7864. 

All the uncertainties u (x ) and covariances u (x , Xj) of the input estimates combine to produce the total uncertainty of the output estimate, y. The mathematical operation of combining the standard uncertainties and covariances of the input estimates x to obtain the standard uncertainty of the output estimate y is called propagation of uncertainty. The standard uncertainty of y obtained by uncertainty propagation is called the combined standard uncertainty of y and is denoted by Uc(y). The following general equation, which the GUM calls the law of... 

The uncertainty of the input quantities in this model is discussed below in Section 10.3.8. Some quantities, such as the decay factor D and the yield Y, are not directly observed but are calculated from other observed quantities. Their uncertainties are calculated by uncertainty propagation before being used in the equation above. 

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. 

The next example combines the concepts of uncertainty propagation as well as confidence intervals. The problem is related to measuring the viscosity of a transparent fluid in a falling ball viscometer, in which the time it takes a ball to cross two lines separated by a distance is measured. The ball falls through a tube filled with the fluid and is assumed to reach a steady velocity by the time it reaches the first Une. The velocity is determined by the geometry and density of the ball as well as the density and, most importantly, the viscosity of the liquid ... 

Two different methodologies can be used to estimate the combined standard uncertainty Ux, Uy, Uz one of them applies the law of uncertainty propagation taken in by the GUM, and the other calculates the standard measurement uncertainties by means of the Monte Carlo method through the simulation of random variables [15]. In this work we use the first one. 

Applying the law of uncertainty propagation the combined standard uncertainty for the 3D position of a measured point can be estimated as ttp = Ju. + + ui, where, taken into account the Eqns. (1)... 

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

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