Nonlinear versus Linear Models

Nonlinear versus Linear Models If V, F, and k are constant, then Eq. (8-1) is an example of a linear differential equation model. In a linear equation, the output and input variables and their derivatives appear to only the first power. If the rate of reaction were second-order, then the resulting dynamic mass balance would be 

Since cA appears in this equation to the second power, the equation is nonlinear. 

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The nonlinear character of log has not often been discussed previously. Nevertheless, Jorgensen and Duffy [26] argued the need for a nonlinear contribution to their log S regression, which is a product of H-bond donor capacity and the square root of H-bond acceptor capacity divided by the surface area. Indeed, for the example above their QikProp method partially reflects for this nonlinearity by predichng a much smaller solubility increase for the indole to benzimidazole mutation (0.45 versus 1.82 [39, 40]). Abraham and Le [41] introduced a similar nonlinearity in the form of a product of H -bond donor and H -bond acceptor capacity while all logarithmic partition coefficients are linear regressions with respect to their solvation parameters. Nevertheless, Abraham s model fails to reflect the test case described above. It yields changes of 1.8(1.5) and 1.7(1.7) [42] for the mutations described above. 

Current methods for supervised pattern recognition are numerous. Typical linear methods are linear discriminant analysis (LDA) based on distance calculation, soft independent modeling of class analogy (SIMCA), which emphasizes similarities within a class, and PLS discriminant analysis (PLS-DA), which performs regression between spectra and class memberships. More advanced methods are based on nonlinear techniques, such as neural networks. Parametric versus nonparametric computations is a further distinction. In parametric techniques such as LDA, statistical parameters of normal sample distribution are used in the decision rules. Such restrictions do not influence nonparametric methods such as SIMCA, which perform more efficiently on NIR data collections. 

In Fig. 3.14a, the dimensionless limiting current 7j ne(t)/7j ne(tp) (where lp is the total duration of the potential step) at a planar electrode is plotted versus 1 / ft under the Butler-Volmer (solid line) and Marcus-Hush (dashed lines) treatments for a fully irreversible process with k° = 10 4 cm s 1, where the differences between both models are more apparent according to the above discussion. Regarding the BV model, a unique curve is predicted independently of the electrode kinetics with a slope unity and a null intercept. With respect to the MH model, for typical values of the reorganization energy (X = 0.5 — 1 eV, A 20 — 40 [4]), the variation of the limiting current with time compares well with that predicted by Butler-Volmer kinetics. On the other hand, for small X values (A < 20) and short times, differences between the BV and MH results are observed such that the current expected with the MH model is smaller. In addition, a nonlinear dependence of 7 1 e(fp) with 1 / /l i s predicted, and any attempt at linearization would result in poor correlation coefficient and a slope smaller than unity and non-null intercept. 

Step 3. Assess functional form for the covariates to be entered into the model (e.g., linear versus nonlinear). 

A markedly nonlinear trend in the exploratory plot of the empirical logit versus exposure or a covariate of interest would indicate that the assumptions of the linear logistic model are not met. In this case, a more complex nonlinear model can be evaluated using a shght modification to the models already presented. Let s suppose that the plot of the empirical logit versus AUC had a distinctly concave trend, curving upward similarly at each end of the observed AUC distribution. With appropriate precautions and care in interpretation, especially outside the range of the data, a quadratic model may be implemented ... 

Figure 1.12 Scatter plot of data used to illustrate bias versus variance tradeoff. The dashed-dotted line is the fit to a one-exponent model. The dashed line is the fit to a two-exponent model and the solid line is the line of fit to a three-exponent model. Data were fit using nonlinear linear regression with weights equal to inverse concentration.

In this case the expression W/Fm versus/(x) was linear in two groups containing the parameters, so that linear regression was possible when the sum of squares on WjF Q was minimized. When the objective function was based on the conversion itself, an implicit equation had to solved and the regression was nonlinear. Only approximate confidence intervals can then be calculated from a linearization of the model equation in the vicinity of the minimum of the objective function. 

Nowak, M. D. (1993), Linear versus nonlinear material modeling of the scapholunate ligament of the human wrist, in H. D. Held, C. A. Biebbia, R. D. Ciskowski, H. Power (eds). Computational Biomedicine, pp. 215-222, Computational Mechanics Publications, Boston. 

Furthermore, let gi () denote the piecewise linear approximation of the nonlinear flow versus effort relation of the t-tti diode according to Shockley s equation implemented by the switch model Sw Z),. 

LWR has been recommended to counteract the common inherent nonlinearity existing between laboratory values and the spectral absorbances. Essentially, segments of the NIR versus laboratory value regression computed are used to give a much better approximation to linearity than a regression line model for the whole dataset. 

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