Error propagation equations

Error propagation analysis is the estimation of error accumulation in a final result as a consequence of error in the individual components used to obtain the result. Given an equation explicitly expressing a result, the error propagation equation can be used to estimate the error in the result as a function of error in the other variables. 

Application of the Error Propagation Equation (Equation 11) gives ... 

This can be shown ly plication of the error propagation equation (Equation 11). If the error in the peak molecular weight M is represented by the error variance, s and we let... 

In thip appendix, a summary of the error propagation equations and objective functions used for standard characterization techniques are presented. These equations are Important for the evaluation of the errors associated with static measurements on the whole polymers and for the subsequent statistical comparison with the SEC estimates (see references 26 and 2J for a more detailed discussion of the equations). Among the models most widely used to correlate measured variables and polymer properties is the truncated power series model... 

Appendix 8 Compounds Recommended for the Preparation of Standard Solutions of Some Common Elements A-27 Appendix 9 Derivation of Error Propagation Equations A-29... 

If we assume that the residuals in Equation 2.35 (e,) are normally distributed, their covariance matrix ( ,) can be related to the covariance matrix of the measured variables (COV(sy.,)= LyJ through the error propagation law. Hence, if for example we consider the case of independent measurements with a constant variance, i.e. 

The error in equation 54-11 then propagated through to the rest of the equations for the Lorentzian distribution. The correct formulas are as follows ... 

Where no KIE is present, the measurement of the 5 value is a result of two measurements, each with their own associated precision. During correction for the added derivative carbon, where cr is the standard deviation associated with a given 5 determination, the errors propagate according to Equation (14.4). 

In derivatisation reactions with a KIE correction factors are first calculated this calculation introduces another step where errors propagate. The propagation of errors under these circumstances is calculated using Equation (14.5), where subscript s stands for the standard used in correction factor determination and sd stands for the derivatised standard. The magnitude of the errors associated with the correction factors themselves can be calculated using Equation (14.4), along with the precisions for each determination (Docherty et al. 2001). 

Equations (8.11) and (8.12) are approximate expressions for propagating the estimate and the error covariance, and in the literature they are referred to as the extended Kalman filter (EKF) propagation equations (Jaswinski, 1970). Other methods for dealing with the same problem are discussed in Gelb (1974) and Anderson and Moore (1979). 

There is no restriction in the derivation of this relationship that would prevent its extension to cases where. YelR", EA-e9Tx", Y and Ae9 m, and Ae9lm n with m n. Then, Eye91mx" and equation (4.2.31) is still valid. Error propagation is achieved by replacing the population parameters by the value estimated by sampling, e.g., x for the sample mean... 

In order to compute error propagation, we must evaluate the partial derivatives with respect to both the dependent and independent variables. This will be more clearly seen by differentiating equation (4.3.24)... 

The sensitivity of diffusion-model output to variations in input has been assessed by workers at Systems Applications, Inc., and at the California Department of Transportation. In each case, reports are in preparation and are therefore not yet available. It is important to distinguish between sensitivity and model performance. True physical or chemical sensitivity that is reflected by the simulation-model equations is a valid reflection of reality. But spurious error propagation through improper numerical integration techniques may be r arded as an artificial sensitivity. Such a distinction must be drawn carefully, lest great sensitivity come to be considered synonymous with unacceptable performance. 

For variables vAiose errors are not Independent, a general form of Equation 11 Incorporates covarleince as well as variance. Also, another form of the equation has been derived for non-random error (20). As will be seen below, certain ccxnputatlons, well known In SBC, cure now being found to be Intrinsically Imprecise because of error propagation. 

There are two main sources of error propagation in static measurements, errors due to successive dilutions and errors due to initial instrument offset. Other errors which are also applicable to SEC analysis are discussed in (J ). These errors can be propagated using the criteria presented here. If w is the intial mass of polymer and Vj is the amount of solvent added to obtain the desired concentration Ci, the dilution process can be represented by the following set of equations ... 

Applying the rules of error propagation and considering that x, is a function of o,-, cf x) can be approximated by the following equations ... 

Lorber, A. (1986), Error propagation and figures of merit for quantification by solving matrix equations, Anal. Chem., 58,1167. 

Equation (4.20) was proposed by Hoskuldsson [65] many years ago and has been adopted by the American Society for Testing and Materials (ASTM) [59]. It generalises the univariate expression to the multivariate context and concisely describes the error propagated from three uncertainty sources to the standard error of the predicted concentration calibration concentration errors, errors in calibration instrumental signals and errors in test sample signals. Equations (4.19) and (4.20) assume that calibrations standards are representative of the test or future samples. However, if the test or future (real) sample presents uncalibrated components or spectral artefacts, the residuals will be abnormally large. In this case, the sample should be classified as an outlier and the analyte concentration cannot be predicted by the current model. This constitutes the basis of the excellent outlier detection capabilities of first-order multivariate methodologies. 

The basic differences between spherical and cylindrical symmetry are in the propagation equations for the water and expln products, the equations of state and the shock front conditions remaining unchanged. Thus, even for acoustic waves, pressure for cylindrical waves varies as r-1/2 F(t—r/c0) where F is an undetermined function, as compared with r 1 F(t—r/c0), valid at any distance for acoustic spherical waves. The development of a finite amplitude theory will not therefore be as simply related to the actual state of affairs, and errors incurred in approximations used will be larger than for spherical waves... 

Figure 4-13 implements least-squares analysis, including propagation of error with Equation 4-27. Enter values of and y in columns B and C. Then select cells B 10 02. Enter the formula = LINEST(C4 C7,B4 B7,TRUE,TRUE) and press CONTROL+SHIFT+ENTER on a PC or COMMANDS + RETURN on a Mac. L1NEST returns m, b, sm, sb, R2, and sY in cells B10 C12. Write labels in cells A10 A12 and D10 D12 so you know what the numbers in cells B10 C12 mean. 

Errors in [HO] due to [T] measurement can be estimated by error propagation in equation 25, which yields... 

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