Nonlinear least squares regression

Now if the function is linear in the parameters, the derivative dyidaj does not contain the parameters, and the resulting set of equations can be solved for the parameters. If, however, the function is nonlinear in the parameters, the derivative contains the parameters, and the equations cannot in general be solved for the parameters. This is the basic problem in nonlinear least-squares regression. 

The method of least squares provides the most powerful and useful procedure for fitting data. Among other applications in kinetics, least squares is used to calculate rate constants from concentration-time data and to calculate other rate constants from the set of -concentration values, such as those depicted in Fig. 2-8. If the function is linear in the parameters, the application is called linear least-squares regression. The more general but more complicated method is nonlinear least-squares regression. These are examples of linear and nonlinear equations ... 

Toward these ends, the kinetics of a wider set of reaction schemes is presented in the text, to make the solutions available for convenient reference. The steady-state approach is covered more extensively, and the mathematics of other approximations ( improved steady-state and prior-equilibrium) is given and compared. Coverage of data analysis and curve fitting has been greatly expanded, with an emphasis on nonlinear least-squares regression. 

Studies interested in the determination of macro pharmacokinetic parameters, such as total body clearance or the apparent volume of distribution, can be readily calculated from polyexponential equations such as Eq. (9) without assignment of a specific model structure. Parameters (i.e., Ah Xt) associated with such an equation are initially estimated by the method of residuals followed by nonlinear least squares regression analyses [30],... 

The nonlinear least squares regression program PEST [79] was used to fit the proposed correlation relating the time invariant Sherwood number to overall Peclet numbers for circular pools given by Eq. (91) to the seven experimentally determined Sh values presented in Fig. 12b, in order to estimate the empirical coefficients fi, y2> and y3. The experimental, overall Sherwood number correlation applicable to circular TCE pool dissolution in water saturated, homogeneous porous media can be expressed by the following relationship ... 

Complex nonlinear least-squares regression (CNLS)... 

FIGURE 19.4 Relationsliip behA een plasma concentrations of tocainide and suppression of ventricular premature beats (VPBs) for four representative patients. The relationship betwreen VPB frequency and tocainide concentrations shown by the solid curves was obtained from a nonlinear least-squares regression analysis of the data using Equation 19.10. The estimate of n for each patient can be compared with the shape of the tocainide concentration-antiarrhythmic response curve. (Reproduced with permission from Meffin PJ, Winkle RA, Blaschke TF, Fitzgerald J, Harrison DC. Clin Pliarmacol Ther 1977 22 42-57.)... 

Among the nonlinear methods, there are, besides nonlinear least squares regression, i.e. polynomial regression, the nonlinear PLS method. Alternating Conditional Expectations ACE), SMART, and MARS. Moreover, some Artificial Neural Networks techniques have also to be considered among nonlinear regression methods, such as the back-propagation method. 

Computer programs that aid in the individualization of therapy are available for many different drugs. The most sophisticated programs use nonlinear regression to fit Cl and Vn to actual serum concentrations obtained in a patient. After drug doses and serum concentrations are entered into the computer, nonlinear least-squares regression programs adjust Cl and Vn until the sum of the squared error between actual (Cact) and computer-estimated concentrations (Cest) is at a minimum [E(Cest - Cact) ]. Once estimates of Cl and Vd are available, doses are calculated easily. 

VanderNoot69has studied poorly separated faradaic and diffusional processes. He has found that a complex, nonlinear, least-squares regression is capable of extracting kinetic information from impedance measurements when the ratio of the charge-transfer process time constant tf=... 

---