Parameter Estimation - Model Discrimination

Two examples are provided here to illustrate nonlinear parameter estimation, model discrimination, and analysis of variance. 

This paper describes the procedure and criteria used to evaluate commercially available software packages for kinetic modeling, and their capabilities for parameter estimation, model discrimination and design of experiments. Also the ease of use and other user-friendliness aspects receive attention. 

Parameter estimation is a procedure for taking the unit measurements and reducing them to a set of parameters for a physical (or, in some cases, relational) mathematical model of the unit. Statistical interpretation tempered with engineering judgment is required to arrive at realistic parameter estimates. Parameter estimation can be an integral part of fault detection and model discrimination. 

Because of these limitations, different models may appear to describe the unit operation equally well. Analysts must discriminate among various models with the associated parameter estimates that best meet the end-use criteria for the model development. There are three principal criteria forjudging the suitability of one model over another. In addition, there are ancillary criteria like computing time and ease of use that may also contribute to the decision but are not of general concern. 

A general method has been developed for the estimation of model parameters from experimental observations when the model relating the parameters and input variables to the output responses is a Monte Carlo simulation. The method provides point estimates as well as joint probability regions of the parameters. In comparison to methods based on analytical models, this approach can prove to be more flexible and gives the investigator a more quantitative insight into the effects of parameter values on the model. The parameter estimation technique has been applied to three examples in polymer science, all of which concern sequence distributions in polymer chains. The first is the estimation of binary reactivity ratios for the terminal or Mayo-Lewis copolymerization model from both composition and sequence distribution data. Next a procedure for discriminating between the penultimate and the terminal copolymerization models on the basis of sequence distribution data is described. Finally, the estimation of a parameter required to model the epimerization of isotactic polystyrene is discussed. 

Estimation of parameters. Model parameters in the selected model are then estimated. If available, some model parameters (e.g. thermodynamic properties, heat- and mass-transfer coefficient, etc.) are taken from literature. This is usually not possible for kinetic parameters. These should be estimated based on data obtained from laboratory expieriments, if possible carried out isothermal ly and not falsified by heat- and mass-transport phenomena. The methods for parameter estimation, also the kinetic parameters in complex organic systems, and for discrimination between models are discussed in more detail in Section 5.4.4. More information on parameter estimation the reader will find in review papers by Kittrell (1970), or Froment and Hosten (1981) or in the book by Froment and Bischoff (1990). 

The sequential experimental design can be made either for precise parameter estimation or for model discrimination purposes. 

If instead of precise parameter estimation, we are designing experiments for model discrimination, the best grid point of the operability region is chosen by maximizing the overall divergence, defined for dynamic systems as... 

Determination of Optimal Inputs for Precise Parameter Estimation and Model Discrimination... 

Froment, G.F., "Model Discrimination and Parameter Estimation in Heterogeneous Catalysis", AIChE. 1, 21 1041 (1975). 

With this book the reader can expect to learn how to formulate and solve parameter estimation problems, compute the statistical properties of the parameters, perform model adequacy tests, and design experiments for parameter estimation or model discrimination. 

Procedures on how to make inferences on the parameters and the response variables are introduced in Chapter 11. The design of experiments has a direct impact on the quality of the estimated parameters and is presented in Chapter 12. The emphasis is on sequential experimental design for parameter estimation and for model discrimination. Recursive least squares estimation, used for on-line data analysis, is briefly covered in Chapter 13. 

By slowly increasing the complexity of the models in this fashion, it was hoped that a model could be obtained that was just sufficiently complex to allow an adequate fit of the data. This conscious attempt to select a model that satisfies the criteria of adequate data representation and of minimum number of parameters has been called the principle of parsimonious parameterization. It can be seen from the table that the residual mean squares progressively decrease until entry 4. Then, in spite of the increased model complexity and increased number of parameters, a better fit of the data is not obtained. If the reaction order for the naphthalene decomposition is estimated, as in entry 5, the estimate is not incompatible with the unity order of entry 4. If an additional step is added as in entry 6, no improvement of fit is obtained. Furthermore, the estimated parameter for that step is negative and poorly defined. Entry 7 shows yet another model that is compatible with the data. If further discrimination between these two remaining rival models is desired, additional experiments must be conducted, for example, by using the model discrimination designs discussed later. The critical experiments necessary for this discrimination are by no means obvious (see Section VII). 

However, this particular experimental design only covered values of x3 up to 1.68 consequently, the saddle point is only predicted by the model and not exhibited by the data. This is the reason the lack-of-fit tests of Section IV indicated neither model 3 nor model 4 of Table XVI could be rejected as inadequately representing the data. As is apparent, additional data must be taken in the vicinity of the stationary point to confirm this predicted nature of the surface and hence to allow rejection of certain models. This region of experimentation (or beyond) is also required by the parameter estimation and model discrimination designs of Section VII. 

Fig. 28. General methods of selecting model discrimination or parameter estimation designs.

Once model discrimination has been accomplished (that is, from a large group of rival models one best model has been selected as being adequate), further experimentation can be conducted to improve parameter estimates. For such parameter-estimation experiments, one design criterion suggested is to choose experiments providing the most desirable posterior distribution of the parameters. Under certain assumptions (B15), the procedure reduces to the maximization of the determinant... 

The use of computers has made it possible to characterise models with large numbers of individual steps. Andersson and Lamb [25] used an analogue computer to estimate parameters in a model with 15 reactions which described naphthalene production by hydrodealkylation. Also, they were able to predict temperature distributions and effluent concentrations for a commercial reactor. Kurtz [26] took 200 simultaneous reactions into account in an experimental study of the gas-phase chlorination of methyl chloride. Model discrimination and parameter estimation for catalytic processes are discussed in a comprehensive review by Froment [27]. 

Most standard chemical engineering tests on kinetics [see those of Car-berry (50), Smith (57), Froment and Bischoff (19), and Hill (52)], omitting such considerations, proceed directly to comprehensive treatment of the subject of parameter estimation in heterogeneous catalysis in terms of rate equations based on LHHW models for simple overall reactions, as discussed earlier. The data used consist of overall reaction velocities obtained under varying conditions of temperature, pressure, and concentrations of reacting species. There seems to be no presentation of a systematic method for initial consideration of the possible mechanisms to be modeled. Details of the methodology for discrimination and parameter estimation among models chosen have been discussed by Bart (55) from a mathematical standpoint. 

The text reviews the methodology of kinetic analysis for simple as well as complex reactions. Attention is focused on the differential and integral methods of kinetic modelling. The statistical testing of the model and the parameter estimates required by the stochastic character of experimental data is described in detail and illustrated by several practical examples. Sequential experimental design procedures for discrimination between rival models and for obtaining parameter estimates with the greatest attainable precision are developed and applied to real cases. 

Note, however, that, in the case of fundamental models, there is not always a need to discriminate among rival models since, often, only a single model has been built up. Furthermore, the best criterion of the quality of a model is the consistency of fundamental parameter estimates with other values obtained by means of several methods under a large range of experimental conditions. Let us not be misled about the principle enemy the systematic errors both in experiments and in reaction and reactor models. 

For better model discrimination and/or parameter estimation, sequential methods for computer designed plans of experiments have been proposed [52], They take advantage of the information obtained from the previous experiments and plan the new experiments in the region of independent variables where the maximum difference of the dependent variable can be expected. 

The process of research in chemical systems is one of developing and testing different models for process behavior. Whether empirical or mechanistic models are involved, the discipline of statistics provides data-based tools for discrimination between competing possible models, parameter estimation, and model verification for use in this enterprise. In the case where empirical models are used, techniques associated with linear regression (linear least squares) are used, whereas in mechanistic modeling contexts nonlinear regression (nonlinear least squares) techniques most often are needed. In either case, the statistical tools are applied most fruitfully in iterative strategies. 

In this chapter the aspects of model selection/discrimination and parameter estimation and the experimental acquisition of kinetic data are not dealt with, since they fall fell outside its scope. Moreover, in interpreting the observed temperature dependency of the rate coefficients in this chapter it was assumed that we are dealing with intrinsic kinetic data. As will be shown in a Chapter 7, other, parasitic, phenomena of mass and heat transfer may interfere, disguising the intrinsic kinetics. Criteria will be presented there, however, to avoid this experimental problem. 

After the parameter estimation, the discarding of the mixing rules begins using the F-Test, that points out with 99% confidence whether or not the tested model is equivalent to the experimental variance. If there is an equivalence the discrimination goes to the next step on the other hand the model is discarded and the discrimination procedure continues for the remaining models. If no models is discarded three consecutive times the confidence is decreased down to 95% until one or more models are eliminated. 

The parameter estimation and the kinetic discrimination studies obtained from the numerical simulations are tedious, complex, and time-consuming. Complex softwares are required, and lengthy strategies are used to fit to the data. Moreover. the number of parameters describing the model complicates the interpretation of the experimental results. Simplifying approaches are recommended to make easier the analysis of the chromatographic profiles [371. 

Bayesian procedures are important not only for estimating parameters and states, but also for decision making in various fields. Chapters 6 and 7 include applications to model discrimination and design of experiments further applications appear in Appendix C. The theorem also gives useful guidance in economic planning and in games of chance (Meeden, 1981). 

The sequential procedures discussed above are appropriate when the investigator wants either to estimate parameters or to discriminate among rival models. However, often an investigator wants to do both. A design procedure for this purpose should emphasize discrimination until one model is gaining favor and should emphasize parameter estimation after that point. The criterion should guard against premature choice of a model, with consequent waste of effort on parameter estimation for an inferior model. 

Hill, W. J., W. G. Hunter, and D. W. Wichern. A joint design criterion for the dual problem of model discrimination and parameter estimation. Technometrics,... 

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