Modeling initial parameter estimates

A key factor in modeling is parameter estimation. One usually needs to fit the established model to experimental data in order to estimate the parameters of the model both for simulation and control. However, a task so common in a classical system is quite difficult in a chaotic one. The sensitivity of the system s behavior to the initial conditions and the control parameters makes it very hard to assess the parameters using tools such as least squares fitting. However, efforts have been made to deal with this problem [38]. For nonlinear data analysis, a combination of statistical and mathematical tests on the data to discern inner relationships among the data points (determinism vs. randomness), periodicity, quasiperiodicity, and chaos are used. These tests are in fact nonparametric indices. They do not reveal functional relationships, but rather directly calculate process features from time-series records. For example, the calculation of the dimensionality of a time series, which results from the phase space reconstruction procedure, as well as the Lyapunov exponent are such nonparametric indices. Some others are also commonly used ... 

Initial parameter estimates for the first-order rate constant were arbitrarily set equal to 2/h and inhalational bioavailability was set equal to 0.9. After fitting the model, the estimate of ka was 660 per hour. Hence, in essence, all the drug reached the systemic compartment almost immediately. The model was modified, removing the dosing compartment, and allowing the drug to be treated as an intravenous model with variable bioavailability (Model 3 in Fig. 5.5). Under this model the bioavailability of the inhaled dose was 62.6 4.3%. 

If poor initial parameter estimates are used, it becomes likely that the nonlinear regression will not be able to find an acceptable fit, or it may stride off on a mathematical tangent and arrive at a nonsensical fit to the data. It turns out that the values determined by traditional methods of residual analyses typically offer excellent starting estimates for the model parameters. Hence the traditional data analysis methods described in this chapter remain highly useful, even when nonlinear regression curve fitting is employed as the final step of parameter evaluation. 

One crucial problem in using the nonlinear least squares technique is that estimates of the totally unknown parameters should be supplied to begin the computation. Unfortunately, the ease of convergence, the converged parameter values, and the sum of squares of residuals at convergence depend heavily on these initial parameter estimates. Needless to say, difficulties in finding good initial estimates increase With the number of parameters involved and the complexity of rate models. 

As mentioned earlier non-linear regression is an iterative process and, provided the initial parameter estimates are not too poor and the model is not under-determined by the data, will converge to a unique minimum yielding the best fit parameters. With more complex models it is often necessary to fix certain parameters (either rate constants, equilibrium constants or complete spectra) particularly if they are known through independent investigations and most fitting applications will allow this type of constraint to be applied. 

For the initial, first-order Box-Jenkins model, the parameter estimates and their standard deviation are ... 

The proposed model function shown in Figure 9.44 is the model described in the referenced source for the data set. The function definitions and initial parameter estimations are shown in Listing 9.12 along with the evaluated model parameters with standard errors printed in the selected output listing. The model has four parameters, but the data range covers a sufficient range of the functional dependency to obtain good estimates of all the parameters. 

Thus, tlie focus of tliis subsection is on qualitative/semiquantitative approaches tliat can yield useful information to decision-makers for a limited resource investment. There are several categories of uncertainties associated with site risk assessments. One is tlie initial selection of substances used to characterize exposures and risk on tlie basis of the sampling data and available toxicity information. Oilier sources of uncertainty are inlierent in tlie toxicity values for each substance used to characterize risk. Additional micertainties are inlierent in tlie exposure assessment for individual substances and individual exposures. These uncertainties are usually driven by uncertainty in tlie chemical monitoring data and tlie models used to estimate exposure concentrations in tlie absence of monitoring data, but can also be driven by population intake parameters. As described earlier, additional micertainties are incorporated in tlie risk assessment when exposures to several substances across multiple patliways are suimned. 

Topaz was used to calculate the time response of the model to step changes in the heater output values. One of the advantages of mathematical simulation over experimentation is the ease of starting the experiment from an initial steady state. The parameter estimation routines to follow require a value for the initial state of the system, and it is often difficult to hold the extruder conditions constant long enough to approach steady state and be assured that the temperature gradients within the barrel are known. The values from the Topaz simulation, were used as data for fitting a reduced order model of the dynamic system. 

Obtaining Kinetic Samples for Reactive Extrusion. To develop and test kinetic models, homogeneous samples with a well defined temperature-time history are required. Temperature history does not necessarily need to be isothermal. In fact, well defined nonisothermal histories can provide very good test data for models. However, isothermal data is very desirable at the initial stages of model building to simplify both model selection and parameter estimation problems. 

The structure of such models can be exploited in reducing the dimensionality of the nonlinear parameter estimation problem since, the conditionally linear parameters, kl5 can be obtained by linear least squares in one step and without the need for initial estimates. Further details are provided in Chapter 8 where we exploit the structure of the model either to reduce the dimensionality of the nonlinear regression problem or to arrive at consistent initial guesses for any iterative parameter search algorithm. 

If we have very little information about the parameters, direct search methods, like the LJ optimization technique presented in Chapter 5, present an excellent way to generate very good initial estimates for the Gauss-Newton method. Actually, for algebraic equation models, direct search methods can be used to determine the optimum parameter estimates quite efficiently. However, if estimates of the uncertainty in the parameters are required, use of the Gauss-Newton method is strongly recommended, even if it is only for a couple of iterations. 

Using the initial rate data given above do the following (a) Determine the parameters, kR, kH and KA for model-A and model-B and their 95% confidence intervals and (b) Using the parameter estimates calculate the initial rate and compare it with the data. Shah (1965) reported the parameter estimates given in Table 16.14. 

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