Curve-fitting model functions

FIGURE 11.14 Data set consisting of a control dose-response curve and curves obtained in the presence of three concentrations of antagonist. Panel a curves fit to individual logistic functions (Equation 11.29) each to its own maximum, K value, and slope. Panel b curves fit to the average maximum of the individual curves (common maximum) and average slope of the curves (common n) with only K fit individually. The F value for the comparison of the two models is 2.4, df = 12,18. This value is not significant at the 95% level. Therefore, there is no statistical support for the hypothesis that the more complex model of individual maxima and slopes is required to fit the data. In this case, a set of curves with common maximum and slope can be used to fit these data. 

Alivisatos and coworkers reported on the realization of an electrode structure scaled down to the level of a single Au nanocluster [24]. They combined optical lithography and angle evaporation techniques (see previous discussion of SET-device fabrication) to define a narrow gap of a few nanometers between two Au leads on a Si substrate. The Au leads were functionalized with hexane-1,6-dithiol, which binds linearly to the Au surface. 5.8 nm Au nanoclusters were immobilized from solution between the leads via the free dithiol end, which faces the solution. Slight current steps in the I U) characteristic at 77K were reflected by the resulting device (see Figure 8). By curve fitting to classical Coulomb blockade models, the resistances are 32 MQ and 2 G 2, respectively, and the junction... 

A majority of the models incorporate what are essentially curve fitting parameters or functions. Some (C11, K12) are more pertinent to the pressed, briquetted, or tableted beds of particles rather than to granulated ensembles of particles, even though the distinction between the two kinds of pellets is necessarily somewhat arbitrary. 

Figure 9. Data reduction and data analysis in EXAFS spectroscopy. (A) EXAFS spectrum x(k) versus k after background removal. (B) The solid curve is the weighted EXAFS spectrum k3x(k) versus k (after multiplying (k) by k3). The dashed curve represents an attempt to fit the data with a two-distance model by the curve-fitting (CF) technique. (C) Fourier transformation (FT) of the weighted EXAFS spectrum in momentum (k) space into the radial distribution function p3(r ) versus r in distance space. The dashed curve is the window function used to filter the major peak in Fourier filtering (FF). (D) Fourier-filtered EXAFS spectrum k3x (k) versus k (solid curve) of the major peak in (C) after back-transforming into k space. The dashed curve attempts to fit the filtered data with a single-distance model. (From Ref. 25, with permission.)...

Such a function exhibits peaks (Fig. 9C) that correspond to interatomic distances but are shifted to smaller values (recall the distance correction mentioned above). This finding was a major breakthrough in the analysis of EXAFS data since it allowed ready visualization. However, because of the shift to shorter distances and the effects of truncation, such an approach is generally not employed for accurate distance determination. This approach, however, allows for the use of Fourier filtering techniques which make possible the isolation of individual coordination shells (the dashed line in Fig. 9C represents a Fourier filtering window that isolates the first coordination shell). After Fourier filtering, the data is back-transformed to k space (Fig. 9D), where it is fitted for amplitude and phase. The basic principle behind the curve-fitting analysis is to employ a parameterized function that will model the... 

Another method of predicting human pharmacokinetics is physiologically based pharmacokinetics (PB-PK). The normal pharmacokinetic approach is to try to fit the plasma concentration-time curve to a mathematical function with one, two or three compartments, which are really mathematical constructs necessary for curve fitting, and do not necessarily have any physiological correlates. In PB-PK, the model consists of a series of compartments that are taken to actually represent different tissues [75-77] (Fig. 6.3). In order to build the model it is necessary to know the size and perfusion rate of each tissue, the partition coefficient of the compound between each tissue and blood, and the rate of clearance of the compound in each tissue. Although different sources of errors in the models have been... 

The data collected are subjected to Fourier transformation yielding a peak at the frequency of each sine wave component in the EXAFS. The sine wave frequencies are proportional to the absorber-scatterer (a-s) distance /7IS. Each peak in the display represents a particular shell of atoms. To answer the question of how many of what kind of atom, one must do curve fitting. This requires a reliance on chemical intuition, experience, and adherence to reasonable chemical bond distances expected for the molecule under study. In practice, two methods are used to determine what the back-scattered EXAFS data for a given system should look like. The first, an empirical method, compares the unknown system to known models the second, a theoretical method, calculates the expected behavior of the a-s pair. The empirical method depends on having information on a suitable model, whereas the theoretical method is dependent on having good wave function descriptions of both absorber and scatterer. 

In the introduction we mentioned that it is sometimes necessary to develop a model for the objective function using cost data. Curve fitting of the costs of fabrication of heat exchangers can be used to predict the cost of a new exchanger of the same class with different design variables. Let the cost be expressed as a linear equation... 

Alternative methods and algorithms may be used, such as the model-independent approach to compare similarity limits derived from multi-variate statistical differences (MSD) combined with a 90% confidence interval approach for test and reference batches (21). Model-dependent approaches such as the Weibull function use the comparison of parameters obtained after curve fitting of dissolution profiles. See Chapters 8 and 9 for further discussion of these methods. 

In the text which follows we shall examine in numerical detail the decision levels and detection limits for the Fenval-erate calibration data set ( set-B ) provided by D. Kurtz (17). In order to calculate said detection limits it was necessary to assign and fit models both to the variance as a function of concentration and the response (i.e., calibration curve) as a function of concentration. No simple model (2, 3 parameter) was found that was consistent with the empirical calibration curve and the replication error, so several alternative simple functions were used to illustrate the approach for calibration curve detection limits. A more appropriate treatment would require a new design including real blanks and Fenvalerate standards spanning the region from zero to a few times the detection limit. Detailed calculations are given in the Appendix and summarized in Table V. 

In Figures 8 and 9 are shown the data for the dependence of the characteristic film buildup time t on Apg and U. In accord with the model, t is found to be independent of U, with only a very weak dependence on Apg indicated. This latter result could in part be a function of experimental inaccuracy. The data reduction for t introduces no assumptions beyond that needed to draw the exponential flux decline curves such as those shown in Figures 2 and 3. However, an error analysis shows that the maximum errors relative to the exponential curve fits occur at the earlier times of the experiment. This is seen in the typical error curve plotted in Figure 10. The error analysis indicates that during the early fouling stage the relatively crude experimental procedure used is not sufficiently accurate or possibly that the assumed flux decline behavior is not exponential at the early times. In any case, it follows that the accuracy of the determination of 6f is greater than that for t. 

Figure 13 Charge exchange cross sections due to and (upper left panel), He (lower left panel), He (lower right panel), and He° (upper right panel) impact on water vapor. The curves are fitted to experimental data [199,211-215] by various model functions and extrapolated where data are lacking.

A convenient way to illustrate the behavior of the model for the example of ZnS deposition is to plot the measured deposition rate, r(d, ZnS), as a function of the incident-flux rate of one element when the incident-flux rate of the second element is fixed. An example is shown in Figure 13 for the deposition of ZnS as a function of the incident-flux rate of sulfur at a substrate temperature of 200 °C. Experimental data points and curves representing the best-fit model predictions are shown for each of four zinc incident-flux rates. A nonlinear least-square procedure was used to obtain the following values for the model parameters that best fit equation 40 to the experimental data 8(Zn) = 0.6-0.7, 8(S) = 0.5-0.7, and K(ZnS) > 1015 cm2-s/ZnS. 

All experiments in this study were carried out under conditions where C02 and styrene are miscible. The solubilities of C02 and ethylbenzene (a model for styrene) in HDPE were determined at 80 °C and 243 bar. The HDPE samples were immersed in either pure C02 or a 36 wt % ethylbenzene/C02 solution within pressure vessels under these conditions for various times. Figure 10.1 shows results of a typical desorption experiment to determine the mass uptake of ethylbenzene for a given soak time the equilibrium mass uptake was found to be 4% and this was reached after approximately 5 h. Figure 10.2 illustrates the mass uptakes as a function of soak time the diffusivity of ethylbenzene in C02-swollen HDPE under these conditions was calculated by curve fitting to be 9.23 x 10 7 cm2/s. Attempts to determine the equilibrium mass uptake of neat ethylbenzene in HDPE at 80 °C failed because ethylbenzene dissolves polyethylene under these conditions. 

---