Model discrimination validation

The practical value of the quantitative theory of radical copolymerization depends to a great extent on the adequacy of the applied kinetic model to the real systems. Hence, in Sect. 6 we shall discuss the issues of model discrimination and also the problems of reliability and validity of the calculations of the model parameters with an account of the potentialities of the modern experimental techniques. 

It is important to recognize that results of this study are based on models which involve several assumptions. The validity of some of these assumptions may be questionable under certain circumstances. However, gas-liquid reaction systems involve complex interacting events, which are difficult to describe precisely. There is also a paucity of experimental data on the behavior of gas-liquid reactors. In view of all this, we believe that studies of this type can be valuable qualitatively for the purpose of model discrimination. 

Heterogeneously catalyzed reactions are usually studied under steady-state conditions. There are some disadvantages to this method. Kinetic equations found in steady-state experiments may be inappropriate for a quantitative description of the dynamic reactor behavior with a characteristic time of the order of or lower than the chemical response time (l/kA for a first-order reaction). For rapid transient processes the relationship between the concentrations in the fluid and solid phases is different from those in the steady-state, due to the finite rate of the adsorption-desorption processes. A second disadvantage is that these experiments do not provide information on adsorption-desorption processes and on the formation of intermediates on the surface, which is needed for the validation of kinetic models. For complex reaction systems, where a large number of rival reaction models and potential model candidates exist, this give rise to difficulties in model discrimination. 

Internal vapor and/or liquid composition data rarely are available, but such data are the best possible for model discrimination and validation. It is often relatively easy to match even a simple model only to product compositions. In the absence of composition profiles, the internal temperature profile can often be as useful provided that it is known to which phase a measured temperature pertains. The table below compares the few available measured tray temperatures with those computed during the simulation. The agreement is quite good. 

Other model selection and/or discrimination tools include the posterior predictive check (PPG) and cross-validation (27,32,33). The PPG is useful for examination of the ability of the model to predict accurately certain features of the observed data (e.g., maximum concentration). Although PPG is not strictly a model discrimination technique, as it does not compare the predictive performance between models but rather evaluates the predictive performance of a single model, it does have useful characteristics that are discussed in more detail in Section 5.3.3. Gross-validation is considered accurate but is computer intensive and generally considered not to be suitable for small data sets (32). 

RTD experiments showed that the fixed-bed almost behaves like a plug-flow reactor and the infrared cell like a continuous stirred tank reactor. This fixed-bed is described by the tanks-in-series model, using 9 tanks for the catalyst compartment. The two kinetic models (Equations 1-6) are able to describe the stop-effect experiments at 180 and 200°C, and the considerations made in this work are valid for both temperatures. However, for the sake of clarity, only model discrimination at 180°C will be presented here. In the experimental conditions used here, both models can be simplified the first adsorption step is considered as irreversible, and instantaneous equilibrium is assumed for the second one. With these hypothesis the total number of kinetic parameters is reduced from five (ki, Li, k2, k.2 and ks) to three (ki, K2 and ks), and the models can be expressed as follows ... 

As can be seen from the above, the shape of the resolved rotational structure is well described when the parameters of the fitting law were chosen from the best fit to experiment. The values of estimated from the rotational width of the collapsed Q-branch qZE. Therefore the models giving the same high-density limits. One may hope to discriminate between them only in the intermediate range of densities where the spectrum is unresolved but has not yet collapsed. The spectral shape in this range may be calculated only numerically from Eq. (4.86) with impact operator Tj, linear in n. Of course, it implies that binary theory is still valid and that vibrational dephasing is not yet... 

E. Marengo and R. Todeschini, Linear discriminant hierarchical clustering a modeling and cross-validable divisive clustering method. Chemom. Intell. Lab. Syst., 19 (1993) 43-51. 

The VolSurf method was used to produce molecular descriptors, and PLS discriminant analysis (DA) was applied. The statistical model showed two significant latent variables after cross-validation. The 2D PLS score model offers a discrimination between the permeable and less permeable compounds. When the spectrum color is active (Fig. 17.2), red points refer to high permeability, whereas blue points indicate low permeability. There is a region in the central part of the plot with both red and blue compounds. In this region, and in between the two continuous lines, the permeability prediction is less reliable. The permeability model... 

Cabrera et al. [50] modeled a set of 163 drugs using TOPS-MODE descriptors with a linear discriminant model to predict p-glycoprotein efflux. Model accuracy was 81% for the training set and 77.5% for a validation set of 40 molecules. A "combinatorial QSAR" approach was used by de Lima et al. [51] to test multiple model types (kNN, decision tree, binary QSAR, SVM) with multiple descriptor sets from various software packages (MolconnZ, Atom Pair, VoSurf, MOE) for the prediction of p-glycoprotein substrates for a dataset of 192 molecules. Best overall performance on a test set of 51 molecules was achieved with an SVM and AP or VolSurf descriptors (81% accuracy each). 

Many more such curves were measured in subsequent years, some of which were reported by Abragam (17). When Abragam s work was published it was already quite clear that the dispersion curves could become a valid tool for the study of molecular dynamics, thus laying down the foundation for variable field NMR relaxometry. In principle, the dispersion curves are potentially powerful tools to discriminate between various molecular dynamics models. 

The use of cross-validation should be widely recommended for crystallographic model refinement. Its simplicity to understand for the nonexpert and its power in discriminating models that are consistent with the experimental data make it indispensable. However, in modern crystallography model refinement is often falsely seen as the... 

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