Nonlinear curve modeling

For other nonlinear curve modelling techniques see D. Satyajit, Risk Management and Financial Derivatives, (New York McGraw-Hill, Inc., 1998). 

Also, the excellent properties of the robust procedures are demonstrated at constmcting the nonlinear regression models for the two-atomic system potential energy curves. 

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 complete solution with k3 6 0 is described by De Bortoli et al. (1996). The properties of different models were compared by Colombo and Bortoli (1992). However, the calculation of emission rates using physically based models requires nonlinear curve fitting and a sufficient number of data points. Large errors in parameter estimates can result from rough chamber data and/or wrong models (Salthammer, 1996). 

Traditional approaches to experimental data processing are largely based on linearization and/or graphical methods. However, this can lead to problems where the model describing the data is inherently nonlinear or where the linearization process introduces data distortion. In this case, nonlinear curve-fitting techniques for experimental data should be applied. 

Model-free adaptive (MFA) control does not require process models. It is most widely used on nonlinear applications because they are difficult to control, as there could be many variations in the nonlinear behavior of the process. Therefore, it is difficult to develop a single controller to deal with the various nonlinear processes. Traditionally, a nonlinear process has to be linearized first before an automatic controller can be effectively applied. This is typically achieved by adding a reverse nonlinear function to compensate for the nonlinear behavior so that the overall process input-output relationship becomes somewhat linear. It is usually a tedious job to match the nonlinear curve, and process uncertainties can easily ruin the effort. 

FIGURE 12.5 Experimental and simulated (solid and dashed curves) BTCs of S04 effluent concentrations from the Bs horizon (column Bs-I, input S04 (C0) of 0.005 M). Simulations are for a range of n values where a nonlinear-equilibrium model was assumed. 

There is considerable controversy over the shape of the dose-response curve at the chronic low dose levels important for environmental contamination. Proposed models include linear models, nonlinear (quadratic) models, and threshold models. Because risks at low dose must be extrapolated from available data at high doses, the shape of the dose-response curve has important implications for the environmental regulations used to protect the general public. Detailed description of dosimetry models can be found in Cember (1996), BEIR IV (1988), and Harley (2001). 

There are instances where data are compared to models that predict linear relationships between ordinates and abscissae. Before the widespread availability of computer programs allowing nonlinear fitting techniques, linearizing data was a common practice because it yielded simple algebraic functions and calculations. However, as noted in discussions of Scatchard analysis (Chapter 4) and double reciprocal analysis (Chapter 5), such procedures produce compression of data points, abnormal emphasis on certain data points, and other unwanted aberrations of data. For these reasons, nonlinear curve fitting is... 

Fig. 10-10. Observed P effluent concentrations from the Al horizon of a sandy soil with predicted curves determined using a one-site, nonlinear, nonequilibrium model with and without a sink term for irreversible sorption unil immobilization [from Mansell et al. (1977a), with permission].

The most frequently used nonlinear calibration curve models [18] are the four- and five-parameter logistic models (4PL and 5PL). For example, the four-parameter logistic model is expressed mathematically as follows ... 

The underlying processes behind these dynamics are not well understood. One descriptive model that has been proposed for bark beetles and other eruptive herbivores is known as dual equilibria theory (Fig. 4.5). According to this view, population growth rates follow the standard discretized nonlinear curve... 

Therefore, although the formalism of the model may be similar to the examples Melchers gives, there may exist other mechanisms and assumptions to explain the nonlinear curve. 

Figure 7,2 Extraction of rate constants by fitting the data to the three-component model. Two representative peptides (HC V254-267 and HC E233-M244) of native (control) and reduced gC2 are taken as examples. Deuterium levels were corrected for back-exchange. Nonlinear curve fitting were performed on KaleidaCraph (Synergy Software). Optimized parameters are shown In the Inserted tables...

The nonlinear curve-fitting process described previously (Section 7.3.2) needs the user to select an appropriate model, which is often arbitrary. The semilogarithm method also needs the user to divide the points into arbitrary linear sections. As a result, the obtained rate constants by different methods often do not match one another. A more objective method for finding the distribution of rate constants is needed. 

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