Total error propagation

In a forward-marching method such as Euler s method, we are more interested in the total error propagation over multiple usage of the algorithm than in the local one-step error. If we let Cj be the error between the approximate solution, rcj, and exact solution, x ti) to the differential equation, then... 

It is not always possible to tell strictly the difference between random and systematic deviations, especially as the latter are defined by random errors. The total deviation of an analytical measurement, frequently called the total analytical error , is, according to the law of error propagation, composed of deviations resulting from the measurement as well as from other steps of the analytical process (see Chap. 2). These uncertainties include both random and systematic deviations, as a rule. 

Propagation of error Mathematical technique for computing the total error of a model prediction by calculating the error for each term in the model, and propagating the errors through the model into the total error of prediction. 

According to the law of error propagation (see Eq. 4-21), the total variance of the analytical result can be expressed as ... 

Figure 11 shows the contour plots of the 2D 51V EXSY spectrum of a two-site system (40 mM total vanadate at pH 10.9) in which V exchanges with V2. Quantification of the EXSY spectrum and calculation of the error propagated to the rate constants from the integration precision gives a 25% error on the rate constant. The results (both the rate constants and the errors) correspond nicely to the results obtained from a ID magnetization transfer experiment on the same sample (27). The EXSY spectrum of a sample containing 12.5 mM total vanadate at 1.0... 

The experimental data were interpreted by using mononuclear hydrolysis constants only. The total Zr concentration could not be used as a variable to distinguish between mononuclear or polynuclear hydrolysis products because the concentration was fixed by the solubility of Zr(OH)4(s). The stability constants listed in Table A-5 were obtained. These constants were converted, with error propagation, to hydrolysis constants, logio Pq x, as given in Table A-6. 

Subsequently the total fluctuation is calculated by error propagation of the noise and the blank ... 

The analysis error includes the error by the variance of the sample from the average and the measuring error. Based on the law of error propagation, the total error in the value of results is... 

This back-propagation of the error is repeated layer by layer, until each unit in the network has received an error signal that describes its relative contribution to the total error. 

The ionosphere has an important effect on a position error. Propagation time differs from a geometric distance between the satelhte and the receiver. This physical phenomena is caused by the wave refraction and the Faraday rotation. These phenomena depend on a number of electrically charged particles in a unit volume of the ionosphere (TEC, Total Electron Content [elections/m ]). Several models are used for this error suppression (Klobuchar for example). These errors could be mitigated with the knowledge of a difference among measurements at two or more frequencies with a help of Equations (2) (3) (Bradford, 2009). 

We will discuss errors that affect the irradiance later these include the geometry and reflections in our test setup. The formulas for propagation of component errors are developed in Chapter 11. Equations 2.19 and 2.20 are worst case predictions the RSS of the individual terms is a more reliable statistical measure of the total error. 

In this example, an initial steady-state solution with a = 0 is propagated downstream. At the fourth axial position, the concentration in one cell is increased to 16. This can represent round-off error, a numerical blunder, or the injection of a tracer. Whatever the cause, the magnitude of the upset decreases at downstream points and gradually spreads out due to diffusion in the y-direction. The total quantity of injected material (16 in this case) remains constant. This is how a real system is expected to behave. The solution technique conserves mass and is stable. 

The scaled total electron density pobs = p obs/k, where p obs is the density on the experimental scale. Using the expression for propagation of errors given in Eq. 

Insisting that the propagation of error formula be used wherever appropriate in laboratory reports, we find it easier to introduce the total differential once the topic comes up during the thermodynamics portion of the course. The similarity between the propagation of error formula and the total differential provides a more intuitive model for our students. Because our order of topics delays thermodynamics to later in the semester, we have time to emphasize the more concrete example used to determine the uncertainty in a measurement. 

Propagation of errors using isotope dilution ICP-MS has been considered to determine how to optimize the measurements [201]. Comparison of analysis results from external calibration versus isotope dilution can be used to assess the quality of external calibration results and the effectiveness of internal standards with external calibration [202,203]. Because isotope ratio precision depends on the total ion count rate, the use of high-efficiency sample introduction to generate larger signals can improve isotope ratio precision and, therefore, analysis precision [204]. 

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