Nonlinear curve

FIGURE 11.12 Overextrapolation of data, (a) Nonlinear curve fitting techniques estimate an ordinate maximal asymptote that is nearly 100% beyond the last available data point, (b) The curve fitting procedure estimates a maximal asymptote much closer to the highest available data point. A useful rule is to reject fits that cause an estimated maximal asymptote that is >25% the value of the highest available data point. 

There are statistical methods to determine the verisimilitude of experimental data to models. One major procedure to do this is nonlinear curve fitting to dose-response curves predicted by receptor models. 

The total binding and nsb are plotted as a function of added radiolabel (as shown in Figure 12.1a), and fit simultaneously with nonlinear curve fitting... 

Data points are subjected to nonlinear curve fitting. For these data, Equation 12.5 is used to fit the curve with basal = 0. The fitting parameters for histamine and E-2-P are given in Table 12.4b. The curves are shown in Figure 12.5a. 

It is possible, in some situations, that two different phenomena which proceed at different rates with different temperature coefficients or activation energies will affect the physical properties. In such complex cases, it is not expect to obtain a linear relation between the logarithm of life and reciprocal absolute temperature. If one obtains a nonlinear curve, however, it may he possible to identify the reaction causing the nonlinearity and correct for it. When one can make such a correction, one obtains a linear relationship. 

Figure 9.14 shows a typical approach force curve along with schematic drawings of the relative positions of the SPM tip and the sample surface, as related to the force curve. At the start of the experiment, i.e., position A on the right-hand side of the figure, the tip is above the surface of the sample. As it approaches the surface the Z value decreases until at position B the tip contacts the surface. With further downward movement of the piezo the cantilever starts to be deflected by the force imposed on it by the surface. If the surface is much stiffer than the cantilever, we get a straight line with a slope of — 1, i.e., for every 1 nm of Z travel we get 1 nm of deflection (Une BC in Figure 9.14). If the surface has stiffness similar to that of the cantilever, the tip wUl penetrate the surface and we get a nonlinear curve with a decreased slope (line BD in Figure 9.14). The horizontal distance between the curve BD and the line BC is equal to the penetration at any given cantilever deflection or force. The piezo continues downward until a preset cantilever deflection is reached, the so-called trigger. The piezo is then retracted a predetermined distance, beyond the point at which the tip separates from the sample.

Since many equations and analysis procedures have been described in the literature, we present here just a few of the most commonly used equations. The solutions to these equations are obtained using a nonlinear curve fitting routine found in many commercially available statistical programs. 

A nonlinear fit weights the initial data points more heavily and gives a better description of the decline in oxamyl residues during the critical period when the residues are a concern in the evaluation of worker safety. The nonlinear curve fitting approach has been accepted by regulatory agencies for the determination of pesticide half-life determinations in soil when the decline data do not fit a linear first-order curve. 

The IC50 can thus be accurately determined by fitting the concentration-response data to Equation (5.1) through nonlinear curve-fitting methods. Some investigators prefer to plot data in terms of % inhibition rather than fractional activity. Using the mass-balance relationships discussed above, we can easily recast Equation (5.1) as follows ... 

The values of a and P in Eq. (125) are obtained by nonlinear curve-fitting techniques from which the uptake Pe and Ke values as a function of BSA concen-... 

Statistics available in the system include a large set of commonly used analysis techniques, as well as advanced nonlinear curve fitting techniques. Statistical results can be displayed numerically or graphically. 

A nonlinear curve fitting procedure of the experimental (Eq. 4.28) to the theoretical (Eq. 4.27) 2D autocovariance function can serve to perform some fundamental characterization of the 2D separation. The total volume (Vy) and the peak height dispersion (/a() can be readily measured in the chromatogram, thus the number of components (m) and the peak widths (a, and ay) can be estimated (Marchetti et al., 2004). 

Abstract. The subject of this research are the regularities of the particles motion in the electric and thermoelectric fields with distributed potential and the influence of temperature field to the particle motion trajectories in aggregate electric and thermal fields. The analytical solution of the problem of particle motion in thermoelectric field with distributed potential is produced. Common regularities of particle motion and trajectory changes in such fields are derived. It is shown that nonlinear curves give a nonconsiderable part of the trajectory within the macrostructures and so the trajectory shape doesn t considerably influence the electron flow transformation process. Conversely, the trajectory shape does influence the aforesaid processes in micro- and nanostructures defining the specific ways of transformation. 

By applying nonlinear curve fitting, we obtained the fuel requirements for the two generators explicitly in terms of MW produced. For generator 1 we have the fuel requirements for fuel oil in tons per hour (jtn)... 

Generally in toxicology, however, if we plot the log of a response (such as body weight) versus a linear scale of our dose or stimulus, we get one of four types of nonlinear curves. These are (Snedecor and Cochran, 1980)... 

Concret does not have well defined elastic and plastic regions due to its brittle nature. A maximum compressive stress value is reached at relatively low strains and is maintained for small deformations until crushing occurs. The stress-strain relationship for concrete is a nonlinear curve. Thus, the elastic modulus varies continuously with strain. The secant modulus at service load is normally used to define a single value for the modulus of elasticity. This procedure is given in most concrete texts. Masonry lias a stress-strain diagram similar to concrete but is typically of lower compressive strength and modulus of elasticity. 

Fig. 4 Interaction of HP-/3-CD with three /3-blockers (A) nonlinear curve fitting, (B) double reciprocal fit, (C) x-reciprocal fit, (D) y-reciprocal fit. Experimental conditions same as in Fig. 3.

Average values of complex mobilities obtained using nonlinear curve fitting assuming a 1 1 interaction were used to calculate the stoichiometric coefficients and stability constants through another nonlinear curve fitting [Eq. (3)]. The /x/K7) values were applied because they show a good approach... 

The increment in mechanical properties (tensile strength, 300% modulus, and Young s modulus) as a function of SAF is plotted in Fig. 39. In general, the higher level of SAF, which in turn indicates better exfoliation, results in high level of property enhancement. However, the level of increment with the increase in SAF is different in all three cases and follows a typical exponential growth pattern. The apparent nonlinear curve fitting of the experimental values presented in Fig. 39 is a measure of the dependence of mechanical properties on the proposed SAF function. 

There are many commercial programs available for fitting data to theoretical curves. They are extremely powerful but very dangerous. Used properly, they fit data directly to nonlinear curves without the need for transformation of data to linear equations that can distort the statistics, and they allow data to be fitted to complex equations that could not be solved by hand. But a computer just gives the best fit to the particular equation, y = f(x), that you choose, and your data might not follow that equation. 

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