Discrimination between Models

In many process-design calculations it is not necessary to fit the data to within the experimental uncertainty. Here, economics dictates that a minimum number of adjustable parameters be fitted to scarce data with the best accuracy possible. This compromise between "goodness of fit" and number of parameters requires some method of discriminating between models. One way is to compare the uncertainties in the calculated parameters. An alternative method consists of examination of the residuals for trends and excessive errors when plotted versus other system variables (Draper and Smith, 1966). A more useful quantity for comparison is obtained from the sum of the weighted squared residuals given by Equation (1). 

More complex situations may also be envisaged and it should always be borne in mind that the tit of experimental data to a simple model provides support for but does not prove that model. The power of the experiment to discriminate between models has to be considered. 

For homonuclear molecules s = / — j takes only even values whereas j is even for para modification and odd for ortho modification of the molecules. With a proper choice of fitting parameters any fitting law reproduces experimental line width rather well. Hence the good fit to their -dependence may not be considered as a criterion of quality of a fitting law. To discriminate between models it is necessary to gain agreement with experimental data on te or xE, which are much more... 

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 various approaches to discriminate between models and to test a given model s adequacies have been treated above, and focus is given here on the design criterion A simple approach can be followed to determine the maximum divergence between model predictions Based on statistical considerations for two models, a simple expression was derived to measure the divergence D(x ) at the experimental settings x of the experimental grid... 

Choose the best model. Again, the sum of squares for mode (d) is significantly higher than in model (b) (o-j = 0.042 versus o-p = 0.297). Hence, we choose model Cb) as our choice to fit the data. For cases when the sums of squares are relatively close togedier, we can use the F-test to discriminate between models to learn if one model is statistically better than another. ... 

Although these NMR data clearly support a dynamical model for disorder in P-cristobalite, they are not sensitive to whether the motions of adjacent oxygens are correlated (as required for a model of re-orienting twin domains), or, whether the motion is continuous or a hopping between discrete positions they indicate only that the path of each oxygen traces a pattern with 3-fold or higher symmetry over times of the order 4.7-10 s. Thus, these results cannot discriminate between models based on RUMs or dynamical twin domains, and place only a lower limit on the timescale of the motions. A tighter restriction... 

It is important to note that PPC does not provide a method for discriminating between models. PPC is included in this section because it does provide evidence for assessment of a given model and therefore has some useful model selection properties. It is possible therefore that a model could be rejected as a possible candidate for describing how the current data arose using a PPC format. 

The methods reviewed above address primarily hierarchical models but an issue often arises concerning competing nonhierarchical models. That is, which model is the preferred These models are most often not independent. However, a test statistic can be used to discriminate between models, which is the difference of the minimized objective functions (log-UkeUhood differences, LLDs) for the two nonhierarchical models (18). In the next section the approach for obtaining the test statistic for comparing the two nonhierarchical models (18) is described. 

We conclude that interpretation of even the most precise experiments on fluorescence decay in synthetic polymers will be difficult, given the complexities of such systems. However, with physically well-defined systems, adequate discrimination between models should be possible. In less well defined systems, such as co-polymers of vinyl aromatic monomers in fluid solution, discrimination between models may be more intractable, but some progress can be made if extensive, and complete, experiments are attempted. 

Distribution measurements such as those made in this investigation are certainly most important in interpreting nitration results. Clearly more solubility information for the aromatic hydrocarbon in the acid phase is needed for the purpose of discrimination between models. The distribution measurements... 

In the early days of fluidized bed reactor modelling, experimental testing of models consisted of measuring only overall conversions over severely limited ranges of such variables as particle size, temperature, bed depth, superficial gas velocity and catalyst activity. Since most models had at least one parameter which could be fitted, and since much of the work was for low conversions where predictions are insensitive to the model adopted because of control by kinetic rather than hydrodynamic factors, it was claimed that each model was successful. In the past decade, there has been considerable effort to discriminate between models based on more extensive measurements than conversion alone, such as concentration... 

The t test on parameters, described in Sec. 7.3.2, is useful in establishing whether a model contains an insignificant parameter. This information can be used to make small adjustments to models and thus discriminate between models that vary from each other by one or two parameters. This test, however, does not give a criterion for testing the adequacy of this model. The residual sum of squares, calculated by Eq. (7.160), contains two components. One is due to the scatter in the experimental data and the other is due to the lack of fit of the model. In order to test the adequacy of the fit of a model, the sum of squares must be partitioned into its components. This procedure is called analysis of variance, which is summarized in Table 7.2. To maintain generality, we examine a set of nonlinear data and assume the availability of multiple values of the dependent variable y j at each point of the independent variable jc, (see Fig. 7.12). 

Selecting the best distribution function is not a trivial task. A wide variety of statistical data can be used in this duty, including standard deviations, R, Akaike and Bayesian Information Criteria, and even CPU time, which are aU presented in Table 12.23. It is well accepted that correlation coefficients are not very useful in discriminating between models. In this study, the correlation coefficients were very close to unity (0.986-0.999) for all of the functions. To highlight this point, only the Alpha distribution function exhibited a value of R lower than 0.99. 

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