Propagation of variance

This argument obviously can be generalized to any number of variables. Equation (2-65) describes the propagation of mean square error, or the propagation of variances and covariances. 

The calibration hierarchy is the sequence of calibrations from a reference to the final measuring system, where the outcome of each calibration depends on the outcome of the previous calibration [13]. This hierarchy requires that for measurements incorporating more than one input quantity in the measurement model (e.g., pH, 7), each input quantity must itself be metrologically traceable. In addition, each measurement and derived quantity is listed with an evaluated uncertainty that captures the uncertainties of the measurements and of the calibration hierarchy. Also, because the propagation of variances is additive, measurement uncertainty increases throughout the calibration hierarchy fiomthe RM (which is ideally a certified reference material aka CRM) to the sample. A statement describing the uncertainty is essential, as a measured quantity value unaccompanied by a measurement uncertainty is not only useless, but it is potentially dangerous because the measured value may be misinterpreted or misused. 

To assess the significance of the features of the charge density, the propagation of observational errors must be examined. Given the standard deviations in the observations, and assuming a diagonal variance-covariance matrix of the observations, we may write, for the covariance between the densities at points A and B,... 

The variance for the weight fraction w(i) can be obtained from equation 1H and the variance for M(i) can be calculated directly from equation 18. Note that the propagation of error analysis can be readily extended to other averages and and it can also be used to account for the errors associated with the calibration of columns and detectors. 

The description of an object in the sense of environmental investigation may be the determination of the gross composition of an environmental compartment, for example the mean state of a polluted area or particular location. If this is the purpose, the number of individual samples required and the required mass or size of these increments have to be determined. The relationship between the variance of sampling and that of analysis must be known and both have to be optimized. The origin of the variance of the samples can be investigated by the study of variance contribution of the different steps of the analytical process by means of the law of error propagation (Eq. 4-21) according to Section 4.3.4. 

For many mathematical operations, including addition, subtraction, multiplication, division, logarithms, exponentials and power relations, there are exact analytical expressions for explicitly propagating input variance and covariance to model predictions of output variance (Bevington, 1969). In analytical variance propagation methods, the mean, variance and covariance matrix of the input distributions are used to determine the mean and variance of the outcome. The following is an example of the exact analytical variance propagation approach. If w is the product of x times y times z, then the equation for the mean or expected value of w, E(w), is ... 

As mentioned above, the expanded covariance matrix rni of the results contains a zero-filled row and column when a Cartesian coordinate has been kept fixed. A judiciously chosen variance G2 11) can be entered on the respective diagonal of rm, prior to the transformation (Eq. 61) if the fixed coordinate is afflicted with an (estimated) error and the propagation of this error to any of the derived internal coordinates is to be studied. For this reason it may even be practical to carry along in the expanded vector of variables jj an atom that has never been substituted (and will hence drop out of the fit by the application of E), but whose position can be estimated and is required for the calculation of certain bond lengths and angles involving atoms that were substituted. 

A statistical analysis based on General Linear Model (GLM) was developed to analyse the influence of loading and off-axis angle on damage of composite laminates. The void content in the composite specimens acted as stress raiser resulting in cracks initiation, propagation and failure as tensile loads progressively applied. Analysis of Variance (ANOVA) was performed for void contents to check the statistical differences caused by the experimental errors. 

In order to carry out the weighting of y values, we need crj, the variance of y. Applying the propagation of errors treatment (Eq. 2-66) to the function y = In we have... 

The estimation of the error of a computed result R from the errors of the component terms or factors A, B, and C depends on whether the errors are determinate or random. The propagation of errors in computations is summarized in Table 26-2. The absolute determinate error e or the variance V = s for a random error is transmitted in addition or subtraction. (Note that the variance is additive for both a sum and a difference.) On the other hand, the relative determinate error ejx or square of the relative standard deviation (sJxY is additive in multiplication. The general case R = f A,. ) is valid only if A, B,C,... are independently variable it is... 

Analytic expressions are available for assessing the propagation of errors through linear systems. Such approaches can be used as well when the variances are sufficiently small that the system can be linearized about its expectation value. Numerical techniques are generally needed to assess the propagation of errors through nonlinear systems. 

To consider the effect of signal averaging on the noise level we must refer to the propagation of errors. The variance associated with the sum of independent errors is equal to the sum of their variances, i.e. 

In estimating the overall uncertainty, it may be necessary to take each source of variance and treat it separately to obtain the contribution from the source. Each of these is referred to as an uncertainty component. Expressing this as a standard deviation, an uncertainty component is known as a standard uncertainty. For a measured result y, the total uncertainty, called combined standard uncertainty and denoted as uc(y), is an estimated standard deviation equal to the positive square root of the total variance obtained by combining all the uncertainty components, using a propagation law of components. 

In the next two sections we encountered the problem of propagation of experimental imprecision through a calculation. When the calculation involves only one parameter, taking its first derivative will provide the relation between the imprecision in the derived function and that in the measured parameter. In general, when the final result depends on more than one independent experimental parameter, use of partial derivatives is required, and the variance in the result is the sum of the variances of the individual parameters, each multiplied by the square of the corresponding partial derivative. In practice, the spreadsheet lets us find the required answers in a numerical way that does not require calculus, as illustrated in the exercises. While we still need to understand the principle of partial differentiation, i.e., whatit does, at least in this case we need not know how to do it, because the spreadsheet (and, specifically, the macro PROPAGATION, see section 10.3) can simulate it numerically. 

Often, the quantity of interest in an experiment is not measured directly, but is computed via a mathematical equation or model. Eor example, the overall heat-transfer coefficient (say, C) of a specific heat exchanger might be determined indirectly by measuring the inlet and outlet temperatures. Repeated experiments provide an estimate of the variance of U, but this variance does not account for possible experimental errors (e.g., the thermocouple errors in the temperature measurements). The error in each measurement accumulates in the overall error of a calculated quantity in a manner known as propagation of error. Measurement error can arise from random variability or instrument sensitivity. There is, however, a mathematical approach to deal with these errors. 

The most prominent examples of oscillatory low latitude wind systems are the quasi-biennial oscillation (QBO) in the equatorial stratosphere and the semi-annual oscillation (SAO) at the equatorial stratopause (Figure 3.35). The quasi-biennial oscillation (QBO) (Veryard and Ebdon, 1961 Reed et ah, 1961) is characterized by the downward propagation of westerly and easterly jets from the upper stratosphere to the lower stratosphere near 70 hPa, almost without loss in amplitude, at a rate of typically 1 km per month (Figure 3.36). The period of the QBO varies between 22 and 34 months (28 months on average), and the amplitude of the westerly and easterly jets is typically 15 m/s and -30 m/s, respectively. The QBO is the main cause of interannual variance of the zonal wind in the equatorial stratosphere. Detailed reviews are given by Dunkerton and Delisi (1985) and Baldwin et al. (2001). 

Standard errors and confidence intervals for functions of model parameters can be found using expectation theory, in the case of a linear function, or using the delta method (which is also sometimes called propagation of errors), in the case of a nonlinear function (Rice, 1988). Begin by assuming that 0 is the estimator for 0 and X is the variance-covariance matrix for 0. For a linear combination of observed model parameters... 

The variances or the uncertainties in each reading are additive. See propagation of error. Chapter 3. 

A term that is sometimes useful in statistics is the variance. This is the square of the standard deviation, We shall use this in determining the propagation of error and in the F test below (Section 3.13). 

Applying the principles of propagation of error (absolute variances in numerator additive, relative variances in the division step additive), we calculate that x = 0.2 5 O.OI4 ppm. 

The variance of Oj and 02 is obtained by using the principle of propagation of error presented in Chap. 2. 

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