Parameters nonlinear regression

The formulation of the parameter estimation problem is equally important to the actual solution of the problem (i.e., the determination of the unknown parameters). In the formulation of the parameter estimation problem we must answer two questions (a) what type of mathematical model do we have and (b) what type of objective function should we minimize In this chapter we address both these questions. Although the primary focus of this book is the treatment of mathematical models that are nonlinear with respect to the parameters nonlinear regression) consideration to linear models linear regression) will also be given. 

The pathogens (103 to 104 spores mL 1) caused significant shoot growth inhibition within 25 to 30 h and seedling death within 40 to 50 h. Stem collapse time, as a function of various spore concentrations, was also a useful bioassay parameter. Nonlinear regression analysis86 was used to model stem collapse time as a function of spore concentration (Fig. 16.2, solid lines). The trend used for this model was an exponential decay model of the form ... 

For fitting the binary interaction parameters nonlinear regression methods are applied, which allow adjusting the parameters in such a way that a minimum deviation of an arbitrary chosen objective function F is obtained. For this job, for example, the Simplex-Nelder-Mead method (21j can be applied successfully. The Simplex-Nelder-Mead method in contrast to many other methods [22] is a simple search routine, which does not need the first and the second derivate of the objective function with respect to the different variables. This has the great advantage that computational problems, such as "underflow or overflow with the arbitrarily chosen initial parameters can be avoided. 

VLE data are correlated by any one of thirteen equations representing the excess Gibbs energy in the liquid phase. These equations contain from two to five adjustable binary parameters these are estimated by a nonlinear regression method based on the maximum-likelihood principle (Anderson et al., 1978). 

In Figure 2, a double-reciprocal plot is shown Figure 1 is a nonlinear plot of as a function of [S]. It can be seen how the least accurately measured data at low [S] make the deterrnination of the slope in the double-reciprocal plot difficult. The kinetic parameters obtained in this example by making linear regression on the double-reciprocal data ate =1.15 and = 0.25 (arbitrary units). The same kinetic parameters obtained by software using nonlinear regression are = 1.00 and = 0.20 (arbitrary units). 

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... 

Kinetic studies at several temperatures followed by application of the Arrhenius equation as described constitutes the usual procedure for the measurement of activation parameters, but other methods have been described. Bunce et al. eliminate the rate constant between the Arrhenius equation and the integrated rate equation, obtaining an equation relating concentration to time and temperature. This is analyzed by nonlinear regression to extract the activation energy. Another approach is to program temperature as a function of time and to analyze the concentration-time data for the activation energy. This nonisothermal method is attractive because it is efficient, but its use is not widespread. ... 

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. 

The sum of squares as defined by Equation 7.8 is the general form for the objective function in nonlinear regression. Measurements are made. Models are postulated. Optimization techniques are used to adjust the model parameters so that the sum-of-squares is minimized. There is no requirement that the model represent a simple reactor such as a CSTR or isothermal PER. If necessary, the model could represent a nonisothermal PFR with variable physical properties. It could be one of the distributed parameter models in Chapters 8 or 9. The model... 

Repeat Problem 7.1 using the entire set. First do a preliminary analysis using linear regression and then make a final determination of the model parameters using nonlinear regression. 

Referring to Example 14.9, Vermeulen and Fortuin estimated aU the parameters in their model from physical data. They then compared model predictions with experimental results and from this they made improved estimates using nonlinear regression. Their results... 

Bates DM, Hamilton DC, Watts DG. Calculation of intrinsic and parameter-effects curvatures for nonlinear regression models. Commun Stat Simul Comput 1983 12 469-77. 

Relaxation data may be analyzed by two general methods a two-parameter, linear regression and a three-parameter, nonlinear, fitting procedure. " " The first method requires an accurate experimental determination of Mo, which is both difficult and time-consuming. Furthermore, the... 

Nonlinear regression of the data provides the parameter estimates (shown in Table 2) associated with the models listed in Table 1. 

The structure of such models can be exploited in reducing the dimensionality of the nonlinear parameter estimation problem since, the conditionally linear parameters, kl5 can be obtained by linear least squares in one step and without the need for initial estimates. Further details are provided in Chapter 8 where we exploit the structure of the model either to reduce the dimensionality of the nonlinear regression problem or to arrive at consistent initial guesses for any iterative parameter search algorithm. 

As seen in Chapter 2 a suitable measure of the discrepancy between a model and a set of data is the objective function, S(k), and hence, the parameter values are obtained by minimizing this function. Therefore, the estimation of the parameters can be viewed as an optimization problem whereby any of the available general purpose optimization methods can be utilized. In particular, it was found that the Gauss-Newton method is the most efficient method for estimating parameters in nonlinear models (Bard. 1970). As we strongly believe that this is indeed the best method to use for nonlinear regression problems, the Gauss-Newton method is presented in detail in this chapter. It is assumed that the parameters are free to take any values. 

Having the smoothed values of the state variables at each sampling point and having estimated analytically the time derivatives, n we have transformed the problem to a usual nonlinear regression problem for algebraic models. The parameter vector is obtained by minimizing the following LS objective function... 

Development of a distributed parameter model will rely on data obtained in vivo. Time and spatial dependencies of drug concentration in a target organ are used as the basis to estimate parameters by nonlinear regression analysis. Distribu-... 

The MO concentrations versus time profiles were fitted to second order polynomial equations and the parameters estimated by nonlinear regression analysis. The initial rates of reactions were obtained by taking the derivative at t=0. The reaction is first order with respect to hydrogen pressure changing to zero order dependence above about 3.45 MPa hydrogen pressure. This was attributed to saturation of the catalyst sites. Experiments were conducted in which HPLC grade MIBK was added to the initial reactant mixture, there was no evidence of product inhibition. 

Nonlinear calibration is carried out by nonlinear regression where two types have to be distinguished (1) real (intrinsic) nonlinear regression and (2) quasilinear (intrinsic linear) regression. The latter is characterized by the fact that only the data but not the regression parameters are nonlinear. Typical examples are polynomials... 

In addition to the three methods described above, nonlinear regression methods or other transform approaches may be used to determine the dispersion parameter. For a more complete treatment of the use of transform methods, consult the articles by Hopkins et al. (15) and Ostergaard and Michelsen (14). 

If the structural entities are lamellae, Eq. (8.80) describes an ensemble of perfectly oriented but uncorrelated layers. Inversion of the Lorentz correction yields the scattering curve of the isotropic material I (5) = I (s) / (2ns2). On the other hand, a scattering pattern of highly oriented lamellae or cylinders is readily converted into the ID scattering intensity /, (53) by ID projection onto the fiber direction (p. 136, Eq. (8.56)). The model for the ID intensity, Eq. (8.80), has three parameters Ap, dc, and <7C. For the nonlinear regression it is important to transform to a parameter set with little parameter-parameter correlation Ap, dc, and oc/dc. When applied to raw scattering data, additionally the deviation of the real from the ideal two-phase system must be considered in an extended model function (cf. p. 124). 

Models are fitted to scattering data by means of nonlinear regression [270] and related computer programs [154,271], The quality of the parameterization (by structural parameters) and of the fit are estimated. The best fitting model is accepted. The found values for the structural parameters are plotted vs. environmental parameters of the experiment and discussed. Environmental parameters that come into question are, for example, the materials composition, its temperature, elongation, or the elapsed time. 

Linear and nonlinear regressions of data for estimation of rate parameters ... 

Statistical methods can be applied to obtain values of parameters in both linear and nonlinear forms (i.e., by linear and nonlinear regression, respectively). Linearity with respect to the parameters should be distinguished from, and need not necessarily be associated with, linearity with respect to the variables ... 

In equation 3.4-18, the right side is linear with respect to both the parameters and the variables, j/the variables are interpreted as 1/T, In cA, In cB,.. . . However, the transformation of the function from a nonlinear to a linear form may result in a poorer fit. For example, in the Arrhenius equation, it is usually better to estimate A and EA by nonlinear regression applied to k = A exp( —EJRT), equation 3.1-8, than by linear regression applied to Ini = In A — EJRT, equation 3.1-7. This is because the linearization is statistically valid only if the experimental data are subject to constant relative errors (i.e., measurements are subject to fixed percentage errors) if, as is more often the case, constant absolute errors are observed, linearization misrepresents the error distribution, and leads to incorrect parameter estimates. 

As introduced in sections 3.1.3 and 4.2.3, the Arrhenius equation is the normal means of representing the effect of T on rate of reaction, through the dependence of the rate constant k on T. This equation contains two parameters, A and EA, which are usually stipulated to be independent of T. Values of A and EA can be established from a minimum of two measurements of A at two temperatures. However, more than two results are required to establish the validity of the equation, and the values of A and EA are then obtained by parameter estimation from several results. The linear form of equation 3.1-7 may be used for this purpose, either graphically or (better) by linear regression. Alternatively, the exponential form of equation 3.1-8 may be used in conjunction with nonlinear regression (Section 3.5). Seme values are given in Table 4.2. 

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