Uncertainties error propagation

In fact, the error term in eqn (2.1), a,-, produces uncertainty in the estimators bg and bj. In order for these estimates to be useful, it is necessary to know how important their uncertainties are. As in any uncertainty (error propagation) calculation, the variance of the data has to be propagated to the estimates. Unfortunately, the LS method does not provide unbiased estimators for the variance of the y-values (cr unless there is no lack of fit between the data and the line. ... 

Reinhart and Rippin (1986) proposed two methods for design under uncertainty (1) introduction of a penalty function for the probability of exceeding the available production time, whereby the probability can be generated by standard error propagation techniques for technical or commercial uncertainties, and (2) the Here and Now method. 

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. 

Traditionally, analytical chemists and physicists have treated uncertainties of measurements in slightly different ways. Whereas chemists have oriented towards classical error theory and used their statistics (Kaiser [ 1936] Kaiser and Specker [1956]), physicists commonly use empirical uncertainties (from knowledge and experience) which are consequently added according to the law of error propagation. Both ways are combined in the modern uncertainty concept. Uncertainty of measurement is defined as Parameter, associated with the result of a measurement that characterizes the dispersion of the values that could reasonably be attributed to the measurand (ISO 3534-1 [1993] EURACHEM [1995]). 

A realistic uncertainty interval has to be estimated, namely by considering the statistical deviations as well as the non-statistical uncertainties appearing in all steps of the analytical process. All the significant deviations have to be summarized by means of the law of error propagation see Sect. 4.2. 

Added effects of the y-threshold, if there are uncertainties in B and A, are to produce (generally small) uncertainties in a and xq. The latter is given by error propagation ... 

Each stated uncertainty in this and other tables represents one estimated standard error, propagated to parameters from uncertainties of measurements of wave numbers the uncertainties of the latter measurements were provided by authors of papers [91,93] reporting those data, and the weight of each datum in the non-linear regression was taken as the reciprocal square of those uncertainties. As the reduced standard deviation of the fit was 0.92, so less than unity, the authors... 

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... 

Since the uncertainties in and > 238 independent and random, then the uncertainty in the age of the zircon can be calculated using error propagation for independent random errors as... 

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 following is a conversion table for absorbance and transmittance, assuming no reflection. Included for each pair is the percent error propagated into a measured concentration (using the Beer-Lambert law), assuming an uncertainty in transmittance of+0.005.1 The value of transmittance that will give the lowest percent error in concentration is 3.368. Where possible, analyses should be designed for the low uncertainty area. 

All of the considerations discussed lead naturally to the question of what price the analyst pays for this less-than-ideal spike/sample ratio. In most cases, error in the measurement of Rm makes the largest contribution to analytical uncertainty the isotopic compositions of sample and spike are usually well known in comparison to Rm. The matter of error propagation in isotope dilution analyses has been extensively treated by Adriaens et al., [13], and Patterson et al. used Monte Carlo simulation to study the problem [14]. Using propagation of error laws, Heumann derived the following relationship with which to calculate tfopt, the optimum spike-to-sample ratio (neglecting cost and availability) [8] ... 

A parameter such as a rate constant is usually obtained as a consequence of various arithmetic manipulations, and in order to estimate the uncertainty (error) in the parameter we must know how this error is related to the uncertainties in the quantities that contribute to the parameter. For example, Eq. (2-33) for a pseudo-first-order reaction defines k, which can be determined by a semilogarithmic plot according to Eq. (2-6). By a method to be described later in this section the uncertainty in obs (expressed as its variance o ) can be estimated. However, is related to the desired parameter k by k - kc, and presumably some uncertainty is associated with Cb. Thus, we need to know how the errors in k and cb are propagated into the rate constant k. 

Figure 12 shows the left-hand side of Equation (25) evaluated for all 481 points available to Wood and Blundy, plotted against temperature. As can be seen, the simple activity-composition relations appear to remove most melt-compositional dependence of partition coefficient. Using normal error propagation we found that uncertainties in the strain-radius correction contribute 2,762 J to the standard error of the fit and that the errors introduced by the thermodynamic approximations are 2,219 J, equivalent to about a 20% error in D. This seems rather minor when compared to the order of magnitude, or more, uncertainties in some partition coefficients (Table 1). 

In practice, measured isotope ratios are reported as mean and standard error. For analyses with stable ion-beam intensities, the within-run statistical errors are typically similar to those predicted by counting statistics. Uncertainties in sample weight, spike weight, spike concentration, chemical blanks, and filament blanks are typically small compared to the analytical uncertainty, but are also included in the error propagation through Eq. (1). 

The calculation of the uncertainties of individual strain components follows the same procedures, except that the details of the error propagation through the equations defining the strain components may differ and, if Equation (5) is used to fit the data, Fr is replaced in Equation (8) with the cube of the lattice parameter. A worked example is provided in the Appendix. Consideration of the form of Equations (7), (8), and (9) suggests that in general the absolute uncertainties in the calculated strains will be smallest at the pressures closest to the phase transition because both the strains and Vv,v will be smaller than at points further away (Fig. 3b). 

Calculated values are given in Table A4. The uncertainty estimates are obtained by standard error propagation through these equations of the uncertainties in the monoclinic unit-cell parameters and the uncertainties in the extrapolated values of the trigonal unitcell parameters (Table A3). Thus, for example. 

Uncertainty can be assessed in various ways, and often a combination of procedures is necessary. In principle, uncertainty can be judged directly from measurement comparisons or indirectly from an analysis of individual error sources according to the law of error propagation ( error budget )- Measurement comparison may consist of a method comparison study with a reference method based on patient samples according to the principles outlined previously or by measurement of certified matrix reference materials (CRMs). 

It is important to note that the error propagation in taking a number to a power is different from the error propagation in multiplication. For example, consider the uncertainty in the square of 4.0 ( 0.2). Here, the relative error in the result (16.0) is given by Equation 6-13 ... 

D is the Debye-Huckel term in molal units and / , the ionic strength converted to molal units by using the conversion factors listed in [76BAE/MES] (p.439). The following list gives the details of this calculation. The resulting uncertainties in log p are obtained based on the rules of error propagation as described in Section C.6.2. 

Estimate some coefficient of uncertainty, such as the standard deviation, for each liqjortant model coefficient and then use analytical procedures for "error propagation" to propagate this uncertainty through the analysis. 

In the discussion of Schwendeman s paper laurie re-emphasized the point that the Costain uncertainties should always be added to the Kraitchman coordinate in propagating the uncertainty to an error estimate for a derived parameter, schwen-deman agreed that the Costain uncertainty refers to the absolute value of the Kraitchman coordinate, which is known to err by being too small, nelson observed that as a result these uncertainties should be classified as systematic errors. schwendeman agreed and pointed out that in his own method of analysis the Costain uncertainties were propagated in such a manner as to produce asymmetric uncertainties that, in turn, reflect systematic errors. 

Table II.1. Molecular quadrupole moments calculated with the molecular Zeeman parameters and Eq. (II. 1) H. The experimental uncertainties follow from standard error propagation and do not reflect systematic errors introduced, for instance, through the neglect of vibrations. Also listed for comparison are values calculated from atom dipoles and values calculated from INDO-wavefunc-tions 10). Only the quadrupole moments of the most abundant isotopic species are listed in each case. The values are given in units of 10 28 esu cm and are referred to the principal axis system of the moment of inertia tensor. The structure references are given in Refs. 9) and i )...

First, the error limits given in Table II. 1 follow from the experimental uncertainties by standard error propagation and do not account for possible deficiencies of the rigid rotor model (see below). 

---