Regression analysis nonlinear least squares method

Since the nonlinear least-squares method requires initial guesses to start the procedure, three different initial trials were performed (1) (0,0), (2) (1,1), and (3) the values obtained from the Lineweaver-Burk plot in Example 4.2.4. All three initial trials give the same result (and thus the same relative error). Note the large differences in the values obtained from the nonlinear analysis versus those from the linear regression. If the solutions are plotted along with the experimental data as shown below, it is clear that the Lineweaver-Burk analysis does not provide a good fit to the data. 

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. 

Non-linear methods are used not only for the optimization as illustrated above but also in regression analysis when fitting functions which are nonlinear with respect to their coefficients. For example, an application of the least squares method for the estimation of coefficients a and b of the function y = a(l-exp(-bx)) leads to NLP. In Section 1.4.3 NLP has been used for the estimation of unknown parameters of a fibre migration model. 

Changing the regression method by using either weighted, least-squares analysis, if the variance is not homoscedastic, or nonlinear, least-squares analysis to determine the parameter values. 

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. 

In this section, on the one hand, methods that are used to estimate intrinsically nonlinear parameters by means of nonlinear regression (NLR) analysis will be introduced. On the other hand, we will learn about methods that are based on nonpara-metric, nonlinear modeling. Among those are nonlinear partial least squares (NPLS), the method of alternating conditional expectations (ACE), and multivariate adaptive regression splines (MARS). 

The mathematical solution of the pharmacokinetic model depicted by Figure 5 Is described by Equation 5, where K12 and K23 are first order rate constants analogous to Ka and Ke, respectively. This solution was applied to the data and "best fit" parameters estimated by Iterative computational methods. The "fit" of the data to the kinetic model was analyzed by least squares nonlinear regression analysis ( ). 

To take account of interactions between individual components (association, nonlinearities), calibration using multivariate data analysis is often also carried out with mixtures rather than pure substances. Despite this fact, limitations to this method of assessment are encountered quickly. Therefore, the so-called inverse method using the g-matrix is employed, and either principal component regression (PCR) or the partial least squares (PLS) method is used [6, [114], [116]. In both methods, calibration is carried out not with pure substances, but with various mixtures, which must cover the expected concentration range of all components. Within limits, this can allow for non-linearities ... 

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