Multiple Nonlinear Regression

In the previous four sections, the sum of squared residuals that was minimized was that given by Eq. (7.160). This was the sum of squared residuals determined from fitting one equation to measurements of one variable. However, most mathematical models may involve simultaneous equations in multiple dependent variables. For such a case, when more than one equation is fitted to multiresponse data, where there are v dependent variables in the model, the weighted sum of squared residuals is given by 

To minimize I by the Gauss-Newton method, we first linearize the models using Eq. (7.166) and combine with Eq. (7.176) to obtain 

Taking the partial derivative of O with respect to AZ , setting it equal to zero, and solving for Ab we obtain 

178) gives the correction of the parameter vector when fitting multiple dependent variables simultaneously. Eq. (7.178) becomes identical to Eq. (7.168) when v = I, that is, when only one dependent variable is fitted. When using the Marquardt method, the correction of the parameter vector is calculated from 

The denominator of Eq. (7.180) accounts for the possibility that each curve may have a different number of experimental points n, and weighs that accordingly. If the assumption that of is constant within one curve does not hold, then Eq. (7.180) can be extended so that weighting factor can be calculated at each point with the appropriate value of of. 

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This process is partially overlapped with the next process, the j3 relaxation. To analyze the loss permittivity in the subglass zone in a more detailed way, the fitting of the loss factor permittivity by means of usual equations is a good way to get confidence about this process [69], Following procedures described above Fig. 2.42 represent the lost factor data and deconvolution in two Fuoss Kirwood [69] as function of temperature at 10.3 Hz for P4THPMA. In Fig. 2.43 show the y and relaxations that result from the application of the multiple nonlinear regression analysis to the loss factor against temperature. The sum of the two calculated relaxations is very close to that in the experimental curve. 

After the responses were obtained, they were subjected to multiple nonlinear regression and optimization using the software Design-Expert (V6.0 Stat-Ease, Minneapolis, MN). Second-order polynomialmodels were applied to correlate these variables. Only the estimates of coefficients with significant levels higher than 90% (i.e., p < 0.10) were included in the final model. An F-test was used to evaluate the significance of the models. 

As an example, the data from Ketchum (37) for the rate of phosphate absorption as a function of both phosphate and nitrate concentration can be satisfactorily fit with a product of two Michaelis-Menton expressions. The resulting fit, obtained by a multiple nonlinear regression analysis, is shown in Figure 5. The Michaelis constants that result are 28.4 p.g NOa-N/liter and 30.3 pg P04-P/liter, with a saturated absorption rate of 15.1 X 10 8 pg P04-P/cell-hr. This approximation to the growth rate behavior as a function of more than one nutrient must be regarded as only a first approximation, however, since the complex interaction reported between the nutrients is neglected. 

FIGURE 16.5 Celecoxib inhibition of CYP2C19-catalyzed (5)-mephenytoin 4 -hydroxylation in human liver microsomes Y ax = 0.40 0.02 nmol/min/mg, = 13.8 1.9 xM, Ki = 3.2 0.4 xM, RSS = 0.002, and = 0.981, respectively. All kinetic values were determined by multiple nonlinear regression (3-dimension) using the velocity equation of competitive inhibition in Table 16.3. 

Method of Solution The Marquardt method using the Gauss-Newton technique, described in Sec. 7.4.4, and the concept of multiple nonlinear regression, covered in Sec. 7.4.5, have been combined together to solve this example. Numerical differentiation by forward finite differences is used to evaluate the Jacobian matrix defined by Eq. (7.164). 

Least squares multiple nonlinear regression using the Marquardt and Gauss-Newton methods. The program can fit simultaneous ordinary differential equations and/or algebraic equations to multiresponse data. 

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... 

Nonlinearity is a subject the specifics of which are not prolifically or extensively discussed as a specific topic in the multivariate calibration literature, to say the least. Textbooks routinely cover the issues of multiple linear regression and nonlinearity, but do not cover the issue with full-spectrum methods such as PCR and PLS. Some discussion does exist relative to multiple linear regression, for example in Chemometrics A Textbook by D.L. Massart et al. [6], see Section 2.1, Linear Regression (pp. 167-175) and Section 2.2, Non-linear Regression, (pp. 175-181). The authors state,... 

POLYMATH. AIChE Cache Corp, P O Box 7939, Austin TX 78713-7939. Polynomial and cubic spline curvefitting, multiple linear regression, simultaneous ODEs, simultaneous linear and nonlinear algebraic equations, matrix manipulations, integration and differentiation of tabular data by way of curve fit of the data. 

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 ... 

Fig. 1.1. Examples for standard curves resulting from multiple determinations of different amounts of BSA. Line with circies protocol according to Lowry et al. Soiid iine. nonlinear regression dotted iine linear regressions wavelength 720 nm. Line with squares BCA protein determination. Soiid iine nonlinear regression dotted iine linear regression wavelength 562 nm). Findings means an example for graphical evaluation...

We cannot simply take logs of both sides of the equation as the disturbance is additive rather than multiplicative. So, we must heat the model as a nonlinear regression. The linearized equation is... 

A particnlarly easy type of least-sqnares analysis called multiple linear regression is possible for fitting data with a low-order polynomial, and this technique can be used for many of the experiments in this book. The nse of spreadsheet programs, as discussed in Chapter III, is strongly recommended in snch cases. In the case of more complicated nonlinear fitting procednres, other techniqnes are described in Chapter XXL... 

Models can be generated using stepwise addition multiple linear regression as the descriptor selection criterion. Leaps-and-bounds regression [10] and simulated annealing (ANNUN) can be used to find a subset of descriptors that yield a statistically sound model. The best descriptor subset found with multiple linear regression can also be used to build a computational neural network model. The root mean square (rms) errors and the predictive power of the neural network model are usually improved due to the higher number of adjustable parameters and nonlinear behavior of the computational neural network model. 

These later two models of bioavailability as a continuous variable are linear since they used stepwise multiple linear regression (M LR) as the modeling tool. An obvious alternative, which may offer improved performance, is a nonlinear technique and such a model using an artificial neural network (ANN) was reported by Turner and colleagues [30], This study employed 167 compounds characterized by several descriptor types, ID, 2D, and 3D, and resulted in a 10-term model. Although the predictive performance was judged adequate, it was felt that the model was better able to differentiate qualitatively between poorly and highly bioavailable compounds. 

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