Nonlinear Regression of Experimental Data

In the present example, it is required to derive empirical equations to predict the experimental shear rate, viscosity, and temperature data. The proposed equation is as follows  

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Solver Found a solution. All constraints and optimality conditions are satisFied. 

Enter the following equations, variables, initial parameters, and constraints into the appropriate windows. 

Ti iQCi riOTiifiti ic LirdL F Degrees f Radians O Grads  

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Obtaining Eft), t, and of from experimental tracer data involves determining areas under curves defined continuously or by discrete data. The most sophisticated approach involves die use of E-Z Solve or equivalent software to estimate parameters by nonlinear regression. In this case, standard techniques are required to transform experimental concentration versus time data into Eft) or F(t) data the subsequent parameter estimation is based on nonlinear regression of these data using known expressions for Eft) and F t) (developed in Section 19.4). In the least sophisticated approach, discrete data, generated directly from experiment or obtained from a continuous response curve, are... 

In 1991 Moffat, Jensen and Carr employed RRKM theory in the form of a nonlinear regression analysis of experimental data to estimate the high-pressure Arrhenius parameters for elimination of H2 from SM4 as logA = 15.79 0.5 s 1, E = 59.99 2.0 kcalmor1 and A//j(SiH2) = 65.5 1.0 kcalmorl62. 

A standard regression analysis can be performed on each set of data by applying a commercial regression software [95,96], The calculation can be also carried out by nonlinearly regressing each set of experimental data using directly the Langmuir equation for liquid-phase adsorption. 

Application of a least-squares method to the linearized plots (e.g., Scatchard and Hames) is not reasonable for analysis of drug-protein binding or other similar cases (e.g., adsorption) to obtain the parameters because the experimental errors are not parallel to the y-axis. In other words, because the original data have been transformed into the linear form, the experimental errors appear on both axes (i.e., independent and dependent variables). The errors are parallel to the y-axis at low levels of saturation and to the x-axis at high levels of saturation. The use of a double reciprocal plot to determine the binding parameters is recommended because the experimental errors are parallel to the y-axis. The best approach to this type of experimental data is to carry out nonlinear regression analysis on the original equation and untransformed data. 

POLYMATH can also be easily applied for regression of rate data (linear, polynomial, nonlinear regression). All one has to do is to type the experimental values in the computer, specify the model, enter the initial guesses of the parameters, and then push the computer button, and the best estimates of the parameter values along with 95 % confidence limits appear. The application of POLYMATH is described in some textbooks with many examples taken from practice [31 -33]. 

One motivation of performing simulations is the interpretation of experimental data, e.g. voltammograms or chronoamperometric data, by estimation of physical parameters. This can be achieved by fitting simulated curves to experimental ones in a nonlinear regression analysis process. The user provides a model to the simulation program and an objective function is then minimized by systematic variation of the model parameter. The best fit is achieved when a global minimum of the object... 

When experimental data is to be fit with a mathematical model, it is necessary to allow for the facd that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a hn-ear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just hnear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. The following description of maximum likehhood apphes to both linear and nonlinear least squares (Ref. 231). If each measurement point Uj has a measurement error Ayi that is independently random and distributed with a normal distribution about the true model y x) with standard deviation <7, then the probability of a data set is... 

Grunwald has shown applications of Eqs. (5-78) and (5-79) as tests of the theory and as mechanistic criteria. One way to do this, for a reaction series, is to estimate AG° and AG from thermodynamic data and from reasonable approximations and then to fit experimental rate data (AG values) to Eq. (5-78) by nonlinear regression. This yields estimates of AGq and AG (which are constants within the reaction series), and these are then used in Eq. (5-79) to obtain the transition state coordinates. 

When estimates of k°, k, k", Ky, and K2 have been obtained, a calculated pH-rate curve is developed with Eq. (6-80). If the experimental points follow closely the calculated curve, it may be concluded that the data are consistent with the assumed rate equation. The constants may be considered adjustable parameters that are modified to achieve the best possible fit, and one approach is to use these initial parameter estimates in an iterative nonlinear regression program. The dissociation constants K and K2 derived from kinetic data should be in reasonable agreement with the dissociation constants obtained (under the same experimental conditions) by other means. 

Finally, we should refer to situations where both independent and response variables are subject to experimental error regardless of the structure of the model. In this case, the experimental data are described by the set (yf,x,), i=l,2,...N as opposed to (y,Xj), i=l,2,...,N. The deterministic part of the model is the same as before however, we now have to consider besides Equation 2.3, the error in Xj, i.e., x, = Xj + ex1. These situations in nonlinear regression can be handled very efficiently using an implicit formulation of the problem as shown later in Section 2.2.2... 

This observation is expected from theory, as the observed thickness distributions are exactly the functions by which one-dimensional short-range order is theoretically described in early literature models (Zernike and Prins [116] J. J. Hermans [128]). From the transformed experimental data we can determine, whether the principal thickness distributions are symmetrical or asymmetrical, whether they should be modeled by Gaussians, gamma distributions, truncated exponentials, or other analytical functions. Finally only a model that describes the arrangement of domains is missing - i.e., how the higher thickness distributions are computed from two principal thickness distributions (cf. Sect. 8.7). Experimental data are fitted by means of such models. Unsuitable models are sorted out by insufficient quality of the fit. Fit quality is assessed by means of the tools of nonlinear regression (Chap. 11). 

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