Nonlinear regression analysis

If an analytical solution is available, the method of nonlinear regression analysis can be applied this approach is described in Chapter 2 and is not treated further here. The remainder of the present section deals with the analysis of kinetic schemes for which explicit solutions are either unavailable or unhelpful. First, the technique of numerical integration is introduced. 

Section 5.1 shows how nonlinear regression analysis is used to model the temperature dependence of reaction rate constants. The functional form of the reaction rate was assumed e.g., St = kab for an irreversible, second-order reaction. The rate constant k was measured at several temperatures and was fit to an Arrhenius form, k = ko exp —Tact/T). This section expands the use of nonlinear regression to fit the compositional and temperature dependence of reaction rates. The general reaction is... 

The partial differential equations describing the catalyst particle are discretized with central finite difference formulae with respect to the spatial coordinate [50]. Typically, around 10-20 discretization points are enough for the particle. The ordinary differential equations (ODEs) created are solved with respect to time together with the ODEs of the bulk phase. Since the system is stiff, the computer code of Hindmarsh [51] is used as the ODE solver. In general, the simulations progressed without numerical problems. The final values of the rate constants, along with their temperature dependencies, can be obtained with nonlinear regression analysis. The differential equations were solved in situ with the backward... 

The final values of the rate constants along with their temperature dependencies were obtained with nonlinear regression analysis, which was applied to the differential equations. The model fits the experimental results well, having an explanation factor of 98%. Examples of the model fit are provided by Figures 8.3 and 8.4. An analogous treatment can be applied to other hemicelluloses. 

Bates DM, Watts DG. Nonlinear regression analysis and applications, New York, Wiley, 1988. 

Bates, D.M. and Watts. D.G., 1991, Nonlinear Regression Analysis and its Applications , J.Wiley, New York. 

The plot of pH against titrant volume added is called a potentiometric titration curve. The latter curve is usually transformed into a Bjerrum plot [8, 24, 27], for better visual indication of overlapping pKiS or for pffjS below 3 or above 10. The actual values of pKa are determined by weighted nonlinear regression analysis [25-27]. 

The unknown model parameters will be obtained by minimizing a suitable objective function. The objective function is a measure of the discrepancy or the departure of the data from the model i.e., the lack of fit (Bard, 1974 Seinfeld and Lapidus, 1974). Thus, our problem can also be viewed as an optimization problem and one can in principle employ a variety of solution methods available for such problems (Edgar and Himmelblau, 1988 Gill et al. 1981 Reklaitis, 1983 Scales, 1985). Finally it should be noted that engineers use the term parameter estimation whereas statisticians use such terms as nonlinear or linear regression analysis to describe the subject presented in this book. 

Sachs, W.H., "Implicit Multifunctional Nonlinear Regression Analysis". Technometrics. 18, 161-173(1976). 

Schaper, K.-J., Simultaneous determination of electronic and lipophilic properties [pKa, P(ion), P(neutral)] for acids and bases by nonlinear regression analysis of pH-dependent partittion measurements, J. Chem. Res. (S) 357 (1979). 

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. 

In a well-behaved calibration model, residuals will have a Normal (i.e., Gaussian) distribution. In fact, as we have previously discussed, least-squares regression analysis is also a Maximum Likelihood method, but only when the errors are Normally distributed. If the data does not follow the straight line model, then there will be an excessive number of residuals with too-large values, and the residuals will then not follow the Normal distribution. It follows, then, that a test for Normality of residuals will also detect nonlinearity. 

A major limitation of the linearized forms of the Michaelis-Menten equation is that none provides accurate estimates of both Km and Vmax. Furthermore, it is impossible to obtain meaningful error estimates for the parameters, since linear regression is not strictly appropriate. With the advent of more sophisticated computer tools, there is an increasing trend toward using the integrated rate equation and nonlinear regression analysis to estimate Km and While this type of analysis is more complex than the linear approaches, it has several benefits. First, accurate nonbiased estimates of Km and Vmax can be obtained. Second, nonlinear regression may allow the errors (or confidence intervals) of the parameter estimates to be determined. 

Bates, D. M. and D. G. Watts. Nonlinear Regression Analysis and Its Applications. Wiley, New York (1988). 

To form the process model, regression analysis was carried out. The alkylate yield x4 was a function of the olefin feed xx and the external isobutane-to-olefin ratio jc8. The relationship determined by nonlinear regression holding the reactor temperatures between 80-90°F and the reactor acid strength by weight percent at 85-93 was... 

The separation of synthetic red pigments has been optimized for HPTLC separation. The structures of the pigments are listed in Table 3.1. Separations were carried out on silica HPTLC plates in presaturated chambers. Three initial mobile-phase systems were applied for the optimization A = n-butanol-formic acid (100+1) B = ethyl acetate C = THF-water (9+1). The optimal ratios of mobile phases were 5.0 A, 5.0 B and 9.0 for the prisma model and 5.0 A, 7.2 B and 10.3 C for the simplex model. The parameters of equations describing the linear and nonlinear dependence of the retention on the composition of the mobile phase are compiled in Table 3.2. It was concluded from the results that both the prisma model and the simplex method are suitable for the optimization of the separation of these red pigments. Multivariate regression analysis indicated that the components of the mobile phase interact with each other [79],... 

To verify such a steric effect a quantitative structure-property relationship study (QSPR) on a series of distinct solute-selector pairs, namely various DNB-amino acid/quinine carbamate CSPpairs with different carbamate residues (Rso) and distinct amino acid residues (Rsa), has been set up [59], To provide a quantitative measure of the effect of the steric bulkiness on the separation factors within this solute-selector series, a-values were correlated by multiple linear and nonlinear regression analysis with the Taft s steric parameter Es that represents a quantitative estimation of the steric bulkiness of a substituent (Note s,sa indicates the independent variable describing the bulkiness of the amino acid residue and i s.so that of the carbamate residue). For example, the steric bulkiness increases in the order methyl < ethyl < n-propyl < n-butyl < i-propyl < cyclohexyl < -butyl < iec.-butyl < t-butyl < 1-adamantyl < phenyl < trityl and simultaneously, the s drops from -1.24 to -6.03. In other words, the smaller the Es, the more bulky is the substituent. The obtained QSPR equation reads as follows ... 

Therefore, plotting the ALIS MS response from a titration series versus the total ligand concentration yields a saturation binding curve that can be fit to this equation by nonlinear regression analysis to yield the of the ligand of interest. 

Duplicate injections shown. (A) Fitting the data by nonlinear regression analysis yields a of 5.6 1.0 pM. (B) Data from A, plotted as a sigmoidal curve to better show the fit at low titrant concentrations. (C) Residuals plotted as absolute and (D) as percent of signal. 

Fig. 6.13 Measurement of FAC data for a range of pyrimethamine concentrations applied to sol-gel-entrapped dehydrofolate reductase (A) overlay of breakthrough curves and (B) nonlinear regression analysis of the fit to the measured breakthrough volumes from A [28]. Adapted with permission from the American Chemical Society.

Martin H) has written a perceptive analysis of the possible ways in which an ionized species may behave in various models and contribute to or be responsible for a given activity. QSAR studies that have dealt with ion-pair partitioning include a study of fibrinolytics ( ) and the effect of benzoic acids on the K ion flux in mollusk neurons ( ). Schaper (10) recently reanalyzed a large number of absorption studies to include terms for the absorption of ionized species. Because specific values were not available for log Pj, he let the relation between log Pi and log P be a parameter in a nonlinear regression analysis. In most cases he used the approximation that the difference between the two values is a constant in a given series. This same assumption was made in the earlier studies (, ) Our work suggests that the pKa of an acid can influence this differential (see below). The influence of structure on the log P of protonated bases or quaternary ammonium compounds is much more complex (11,12) and points out the desirability of being able to easily measure these values. 

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