Relative error structure

Figure 1.18. Mean residual pattern characteristic of a heterogeneous, or relative, error structure in the experimental data.

Compared with the experimental values for which was noted a high level of measurement error, a level of agreement was found that is not worse than the disparities found for a lot of compounds, which were the subject of independent measurement. Note in particular the good estimates obtained with two compounds that have relatively complex structures, such as styrene oxide and glycidyl acrylate. Nevertheless, there are two estimates that seem sufficiently different from the experimental values to require explanation. 

When an online interpolator is used to estimate the uncertain term, the interpolation error g can be kept bounded, provided that a suitable interpolator structure is chosen [26, 28], Among universal approximators, Radial Basis Function Interpolators (RBFIs) provide good performance in the face of a relatively simple structure. Hence, Gaussian RBFs have been adopted, i.e.,... 

Of importance for any experimental technique which is to be used to fit some complex reaction model is the way in which experimental errors influence the result [93]. The error structure for the EHD method utilising the RDE has been analysed in detail by Orazem et al. [94]. These authors showed that information could reliably and accurately be extracted even at high modulation frequencies (up to 20 Hz). In principle the determination of Sc should only require data at low modulation frequency. They demonstrated that extraction of accurate values for Sc required data which had been recorded over a relatively wide frequency range and had been weighted according to a reliable model for the errors. They also showed that the EHD (7 - Cl) response could be fitted empirically to the form ... 

In the case of benzenoid hydrocarbons, Eq. (35) reproduces E with an average relative error of only 0.4%. This, in turn, means that more than 99,5% of E is determined by the simple structural parameters n and m and that variations in E-values of isomers are fairly (but not negligibly) small. 

Phase-sensitive detection is accurate and can be relatively inexpensive. Modem instrumentation uses more than one reference signal and can mitigate the bias in the error structure. 

Errors in variables methods are particularly suited for parameter estimation of copolymerization models not only because they provide a better estimation in general but also, because it is relatively easy to incorporate error structures due to the different techniques used in measuring copolymer properties (i.e. spectroscopy, chromatography, calorimetry etc.). The error structure for a variety of characterization techniques has already been identified and used in conjunction with EVM for the estimation of the reactivity ratios for styrene acrylonitrile copolymers (12). 

Most methods to determine molecular structures try to compensate for or correct the effect of the vibration-rotation terms. As a result, there are now multitudes of methods available with as many special notations as methods. Their isotope dependence varies in practice from zero to full (for vq structures). Their error limits are sometimes difficult to determine. The goals of some of the newer methods are internal consistency and maximum relative errors (difference to Vg structure) of 0.1% or less. 

Measurement error causes double-trouble attenuation of the slope and increased error about the regression line. However, when more complex error structures are assumed, such as when X is not an unbiased estimate of x or the variance of 8 depends on x, then it is possible for the opposite effect to occur, e.g., i is inflated (Car-roll, Ruppert, and Stefanski, 1995). Rarely are these alternative measurement error models examined, however. The bottom line is that measurement error in the predictors leads to biased estimates of the regression parameters, an effect that is dependent on the degree of measurement error relative to the distribution of the predictors. 

After single dose administration, unreliable estimates (> 15% relative error from true value and >35% standard deviation) of the structural model parameters and most variance components were observed when the degree of random error was more than 30%, whereas biased residual variance was observed when the random error was more than 20% under single dose administra-... 

For 0.42 fm < a < 0.62 fm the relative error in the hyperfine structure obtained using this approximation is at most 10. Alternatively, the calculated hyperfine splittings can be written in terms of 6 r ) and Sa ... 

According to Armstrong and Fildes (1995), the objective of a forecast accuracy measure is to provide an informative and clear understanding of the error distribution. Theoretically, when the forecast errors are randomly structured, the form of the forecasts is independent of the selected accuracy measure. Otherwise, it is generally accepted that there is no single best accuracy measure, and deciding on the assessment method is essentially subjective. In this study, a simple form of relative error (E) is selected as the forecast accuracy measure, since it offers a number of desirable properties ... 

Since the factor qt + qj will dominate qy we see that the relative error when falsely assuming independence is much larger for parallel systems than for series systems. When considering more complex structures, these facts indicate that there is reason to prioritise dependencies between components in parallel over components in series. In the following, we test this idea on a somewhat more complex system. 

Fig. 4 Errors in AE values relative to structure IV, in kcal/ mol, compared to CCSD(T)/ CBS//B2PLYP-D/aug-cc-pVTZ for structures I-III of FP...

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