Proportional error region

Figure 14-35 An example of weighted Deming regression analysis for the comparison of drug assays. A, The solid line is the estimated weighted Deming regression line, the dashed curves indicate the 95% confidence region, and the dotted line is the line of identity. B, A plot of residuals standardized to unit standard deviation.The homogeneous scatter supports the assumed proportional error model and the assumption of linearity.

Compression may be achieved if some regions of the time-frequency space in which the data are decomposed do not contain much information. The square of each wavelet coefficient is proportional to the least-squares error of approximation incurred by neglecting that coefficient in the reconstruction ... 

As illustrated in Fig. 5, large errors on logP are produced when approximated by Eq. (14). Indeed, in the pH region where the ionized species dominate, the proportion of neutral molecules in the aqueous phase becomes rapidly negligible. The ions are then liable to partition into the organic phase provided they form an ion pair with an electrolyte present in the aqueous phase or a species of equal charge crosses the interface in the opposite direction in order to maintain electroneutrality. 

The differential equation is evaluated at certain collocation points. The collocation points are the roots to an orthogonal polynomial, as first used by Lanczos [Lanczos, C.,/. Math. Phys. 17 123-199 (1938) and Lanczos, C., Applied Analysis, Prentice-Hall (1956)]. A major improvement was proposed by Villadsen and Stewart [Villadsen, J. V., and W. E. Stewart, Chem. Eng. Sci. 22 1483-1501 (1967)], who proposed that the entire solution process be done in terms of the solution at the collocation points rather than the coefficients in the expansion. This method is especially useful for reaction-diffusion problems that frequently arise when modeling chemical reactors. It is highly efficient when the solution is smooth, but the finite difference method is preferred when the solution changes steeply in some region of space. The error decreases very rapidly as N is increased since it is proportional to [1/(1 - N)]N 1. See Finlayson (2003) and Villadsen, J. V., and M. Michelsen, Solution of Differential Equation Models by Polynomial Approximations, Prentice-Hall (1978). 

A finite element method based on these functions would have an error proportional to Ax2. The finite element representations for the first derivative and second derivative are the same as in the finite difference method, but this is not true for other functions or derivatives. With quadratic finite elements, take the region from x,.i and x,tl as one element. Then the interpolation would be... 

Optimal sampling. As was pointed out earlier, the error in the derivative of A is proportional to ak j Jnk(N). The optimal sampling is therefore obtained when ak/ Jnk(N) is constant as a function of k. In regions where ak is large, additional sample points should be added to compensate. This is often a small effect but in some special cases is worth considering. In order to obtain the optimal sampling, the potential energy should be corrected as... 

An estimate of the probable errors in the correction factors and cutoff values follows. From Equation 3 one sees that the fractional errors in both are of the same order of magnitude as the fractional error in the velocity, Av/v, averaged over the region of motion. There are three main contributions to this error. One comes from the approximation to the Davies equations (8 and 10). The average fractional error is of the order of Av/v —5%, the minus sign occurring since Equations 9 and 10 underestimate the true values of Re and v. The other error contributions come from the approximations for air density and viscosity. One sees from Equations 7-9 that the first-order term in v is independent of p and has a 1/rj dependence. The second-order term is directly proportional to P. Since this term contributes a maximum of 30% to the velocity and the maximum error in p is 8%, this contribution to Av/v should be... 

For many films exposed in the region of density <2.5, the optical density of a diffraction spot is often proportional to exposure and inversely proportional to the log Q of transmission coefficient. Thus, accuracy of reflection measurements tended to be reasonably uniform over the full range of density. The technique proved usable in estimating diffraction data on patterns on low background and reflections of moderate intensity, generally of photographic density < 1.0 and has been found to yield an error of 15-20% (14). 

If the analysis of a dynamic NMR spectrum is carried out by an iterative least-squares fitting method, the results are accompanied by estimates of the errors. These are proportional to the square root of the sum of the squares of the deviations of the theoretical spectrum from the experimental one, as well as to the sensitivity of the sum to changes in the value of the parameter considered within the region where the sum attains a minimum. These estimates constitute a measure of the effects, on the resulting parameter values, of random errors. They do not include any effects due to systematic errors such as those involved in the assumed values of certain parameters. Moreover, because of the nonlinearity of the least-squares fitting procedure employed, estimates of the errors have only an approximate statistical significance (Section IV.B.2 and reference 67). 

It is obvious, therefore, that ignoring Dl would result with over-estimating C-derived estimate of terrestrial carbon sink (or under-estimate a source). The error can be significant, in the order of 0.6PgCyr globally, and proportionally larger in regions such as the tropics (Ciais et al., 1999). 

This eirgument implies that as the gap thickness h decreases with fixed error in the gap, the boundary to region III should shift upward in proportion to h. This seems to agree with the experiments. If this argument is correct, then from the concentration-dependence of this boundary one can infer the concentration-dependence of 5 it must then scale as 5 a 0. ... 

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