Data fitting procedures

Generally, inaccuracies can also be expected at low photon counts (N < 100). Besides comparatively large statistical fluctuations, also a bias in lifetime is introduced by the data fitting procedure [37],... 

Equations of an Arrhenius type are commonly used for the temperature-dependent rate constants ki = kifiexp(—E i/RT). The kinetics of all participating reactions are still under investigation and are not unambiguously determined [6-8], The published data depend on the specific experimental conditions and the resulting kinetic parameters vary considerably with the assumed kinetic model and the applied data-fitting procedure. Fradet and Marechal [9] pointed out that some data in the literature are erroneous due to the incorrect evaluation of experiments with changing volume. 

The solid line in Fig. 2 shows the permeability-pressure curve of C02 in polycarbonate calculated by fitting the simplified permeability expression, eq. (14), to experimental data of Wonders and Paul (15). The data fitting procedure, described elsewhere (1 1), gives Do = 1.09 x 10 8 cm2/sec and p = 0.065 cm3(polymer)/cm3(STP). Fig. 2 shows a good agreement between the experimental data and eq. (14) over the entire pressure range. 

FIGURE 7.9 Reaction A —+B—+C. Results of the data-fitting procedure visualized in terms of Beer-Lambert s law in its matrix notation C x A = Y. 

To assess the effect of the assumption, we have repeated the entire data-fitting procedure for two other values of E, 13.5 and 18 K cal/mole. For the lower value, the procedure does not work. The model in effect says that if the activation energy were that low, the observed hot spots could not have occurred. For E - 18, however, a criterion equivalent to Equation (15) can be derived in the same way as before. We give the result, without details. ... 

Many of the data-fitting procedures demonstrated so far are based on least-squares analysis. In the next chapter we will encounter Fourier transformation, which can often provide an alternative approach to solving such problems. 

The data fitting procedure can be carried out depending on the requirements of different corrosion process models, including the simplest case of two activation controlled reactions (anodic and cathodic), as well as more complex systems with the participation of a greater number of reactions proceeding with mixed activation-diffusion control. 

The specific expressions for Es m), AE t),g(m), and a T,e) were obtained from data fitting procedure and are listed in that work (Harper et al. 1997). 

Mathematical analysis of the T-chrom relationship. The curve e f(T) is sigmoidal with an inflexion corresponding to the point of maximum temperature sensitivity of the equilibrium. However, these parameters cannot be determined reliably from this curve. Rather than use a classical data fitting procedure which can theoretically determine all 4 parameters, it is preferable to separate the spectral parameters from the thermodynamic parameters and then carry out data fitting on a restricted... 

The data analysis approach of Hirsh et al. is similar to the one of S. S. Eaton, G. R. Eaton and co-workers in their studies of Fe(iii)-induced RE on iron-containing proteins.As mentioned, our data analysis procedure, described above, and the approach of Eatons and Hirsh are both based on essentially the same underlying theory and approximations. Still, in our opinion, the new procedure offers some important advantages. The most significant modification in the new procedure is the division of the relaxation trace for Dy(iii)-nitroxide pairs by the reference trace measured on isolated nitroxide radicals in the same environment. First of all, this makes it possible to remove all relaxation pathways except of the RE pathway from the analysis. Of course, this relies on the assumption of non-correlated relaxation rate distributions for different relaxation pathways, but essentially the same assumption is done indirectly in the other approach as well. Second, the RE time trace obtained after the division allows one to visually inspect and estimate the depth of the intramolecular RE effect, the steepness and curvature of the decay due to the intermolecular RE and the overall quality and sufficient length of the measured data. As the data fitting procedure is an ill-posed problem, these additional controls allow for better confidence and more precise error estimates. Finally, it is more informative to analyse the pure Ak T) and not the complete l/Tis T) dependence. Such a plot can be built within each of the two approaches. On the AA (7 diagram it is easiest to determine the maximum RE value, which is, as discussed, least sensitive to the... 

---