Nonlinear modeling

Subroutine REGRES. REGRES is the main subroutine responsible for performing the regression. It solves for the parameters in nonlinear models where all the measured variables are subject to error and are related by one or two constraints. It uses subroutines FUNG, FUNDR, SUMSQ, and SYMINV. 

C. D. Lutes, Nonlinear Modeling and Initial Condition Estimation for Identifying the Aerothermodynamic Environment of the Space Shuttle Orbiter, Masters thesis. Air Eorce Institute of Technology, WPAPB, Ohio, Jan. 1984. 

A nonlinear model would involve one or more of the coefficients in nonlinear form such as the Antoine equation (eq. 8) or... 

Verneuil et al. (Verneuil, V.S., P. Yan, and F. Madron, Banish Bad Plant Data, Chemical Engineering Progress, October 1992, 45-51) emphasize the importance of proper model development. Systematic errors result not only from the measurements but also from the model used to analyze the measurements. Advanced methods of measurement processing will not substitute for accurate measurements. If highly nonlinear models (e.g., Cropley s kinetic model or typical distillation models) are used to analyze unit measurements and estimate parameters, the Hkelihood for arriving at erroneous models increases. Consequently, resultant models should be treated as approximations. 

Several methods are used to fit rate models, the two most common of which often give erroneous results. The first is the transformation of a proposed rate model to achieve a model form that is linear in the parameters. An example is the nonlinear model ... 

The procedure is to use Eqs. (2-102) and the nonlinear model function. Preliminary parameter estimates Go, bo, are needed. The resulting parameter values... 

Neural networks can also be classified by their neuron transfer function, which typically are either linear or nonlinear models. The earliest models used linear transfer functions wherein the output values were continuous. Linear functions are not very useful for many applications because most problems are too complex to be manipulated by simple multiplication. In a nonlinear model, the output of the neuron is a nonlinear function of the sum of the inputs. The output of a nonlinear neuron can have a very complicated relationship with the activation value. 

Lindsey JK. Nonlinear models in medical statistics, Oxford Statistical Science Series 24, 2001. 

Davidian M, Giltinan DM. Nonlinear models for repeated measurement data. Chapman Hall/CRC, 1995. 

Numerous QSAR tools have been developed [152, 154] and used in modeling physicochemical data. These vary from simple linear to more complex nonlinear models, as well as classification models. A popular approach more recently became the construction of consensus or ensemble models ( combinatorial QSAR ) combining the predictions of several individual approaches [155]. Or, alternatively, models can be built by rurming the same approach, such as a neural network of a decision tree, many times and combining the output into a single prediction. 

NN models for the three datasets contained the same number of descriptors as the MLR models, yet no more than two descriptors in each model were the same in both NN and MLR models. No descriptor was found in common with all models, although, each model contained a descriptor that relied on H-bonding in some manner. Nonlinear modeling from the NN approach gave better representation of the data than the linear models from MLR the value for the three datasets was 0.88, 0.98 and 0.90, respectively. 

J. Zhang, J.-H. Jiang, P. Liu, Y.-Z. Liang and R.-Q. Yu, Multivariate nonlinear modelling of fluorescence data by neural network with hidden node pruning algorithm. Anal. Chim. Acta, 344(1997) 29 0. 

Figure 12 Residuals plot for the nonlinear model 5.2.1 Hamaker equation...

Using the nonlinear model substituting 0.915 p.gcm (the DFR value from the model 1 day after application) into Equation (6) yields a dose of 1.05 mgkg day" and an MOL from Equation (5) of 47, below the required value of 100 for margin of exposure. Performing the same calculation on the day 2 data gives a dose of 0.344 mg kg day and an MOE of 145, which is above the level of 100 required to establish a safe re-entry level. Therefore, a 2-day period is adequate to ensure worker safety. The observed values of 0.936 and 0.234 qg cm for days 1 and 2 match closely the values predicted by the model. 

The scope of this book deals primarily with the parameter estimation problem. Our focus will be on the estimation of adjustable parameters in nonlinear models described by algebraic or ordinary differential equations. The models describe processes and thus explain the behavior of the observed data. It is assumed that the structure of the model is known. The best parameters are estimated in order to be used in the model for predictive purposes at other conditions where the model is called to describe process behavior. 

As seen in Chapter 2 a suitable measure of the discrepancy between a model and a set of data is the objective function, S(k), and hence, the parameter values are obtained by minimizing this function. Therefore, the estimation of the parameters can be viewed as an optimization problem whereby any of the available general purpose optimization methods can be utilized. In particular, it was found that the Gauss-Newton method is the most efficient method for estimating parameters in nonlinear models (Bard. 1970). As we strongly believe that this is indeed the best method to use for nonlinear regression problems, the Gauss-Newton method is presented in detail in this chapter. It is assumed that the parameters are free to take any values. 

Let us consider the following nonlinear model (Bard, 1970). Data for the model are given in Table 4.3. 

Data on the thermal isomerization of bicyclo [2,1,1] hexane were measured by Srinivasan and Levi (1963). The data are given in Table 4.4. The following nonlinear model was proposed to describe the fraction of original material remaining (y) as a function of time (x,) and temperature (x2). The model was reproduced from Draper and Smith (1998)... 

A suitable transformation of the model equations can simplify the structure of the model considerably and thus, initial guess generation becomes a trivial task. The most interesting case which is also often encountered in engineering applications, is that of transformably linear models. These are nonlinear models that reduce to simple linear models after a suitable transformation is performed. These models have been extensively used in engineering particularly before the wide availability of computers so that the parameters could easily be obtained with linear least squares estimation. Even today such models are also used to reveal characteristics of the behavior of the model in a graphical form. 

Approximate inference regions for nonlinear models are defined by analogy to the linear models. In particular, the (I-a)I00% joint confidence region for the parameter vector k is described by the ellipsoid,... 

The above experimental design constitute an excellent set of "preliminary experiments" for nonlinear models with several unknown parameters. Based on the analysis of these experiments we obtain estimates of the unknown parameters that we can use to design subsequent experiments in a rational manner taking advantage of all information gathered up to that point. 

Watts (1994) dealt with the issue of confidence interval estimation when estimating parameters in nonlinear models. He proceeded with the reformulation of Equation 16.19 because the pre-exponential parameter estimates "behaved highly nonlinearly." The rate constants were formulated as follows... 

Schwetlick, H. and V. Tiller, "Numerical Methods for Estimating Parameters in Nonlinear Models with Errors in the Variables", Technometrics, 27, 17-24 (1985). 

Methods based on linear projection transform input data by projection on a linear hyperplane. Even though the projection is linear, these methods may result in either a linear or a nonlinear model depending on the nature of the basis functions. With reference to Eq. (6), the input-output model for this class of methods is represented as... 

Linear PCR can be modified for nonlinear modeling by using nonlinear basis functions 0m that can be polynomials or the supersmoother (Frank, 1990). The projection directions for both linear and nonlinear PCR are identical, since the choice of basis functions does not affect the projection directions indicated by the bracketed term in Eq. (22). Consequently, the nonlinear PCR algorithm is identical to that for the linear PCR algorithm, except for an additional step used to compute the nonlinear basis functions. Using adaptive-shape basis functions provides the flexibility to find the smoothed function that best captures the structure of the unknown function being approximated. 

On some occasions, the body does not behave as a single homogeneous compartment, and multicompartment pharmacokinetics are required to describe the time course of drug concentrations. In other instances certain pharmacokinetic processes may not obey first-order kinetics and saturable or nonlinear models may be required. Additionally, advanced pharmacokinetic analyses require the use of various computer programs, such as those listed on the website http //www.boomer.org/pkin/soft.html. 

To solve this problem, we obviously have to linearize the equations. In vector form, the nonlinear model is... 

The nature of the optimization problem can mm out to be linear or nonlinear depending on the mass transfer model chosen14. If a model based on a fixed outlet concentration is chosen, the model turns out to be a linear model (assuming linear cost models are adopted). If the outlet concentration is allowed to vary, as in Figure 26.35a and Figure 26.35b, then the optimization turns out to be a nonlinear optimization with all the problems of local optima associated with such problems. The optimization is in fact not so difficult in practice as regards the nonlinearity, because it is possible to provide a good initialization to the nonlinear model. If the outlet concentrations from each operation are initially assumed to go to their maximum outlet concentrations, then this can then be solved by a linear optimization. This usually... 

Zhang, X., 1995. Algorithms for optimal scheduling using nonlinear models. PhD thesis, University of London, London. 

Constraints (4.1), (4.2), (4.3), (4.4), (4.5) and (4.6) constitute a nonconvex nonlinear model due to constraints (4.3) and (4.4), which involve bilinear terms. Nonconvexity, and not necessarily nonlinearity, is a disadvantageous feature in any model, since global optimality cannot be guaranteed. Therefore, if can be avoided, it should. This is achieved by either linearizing the model or using convexification techniques where applicable. In this instance, the first option was proven possible as shown below. 

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