On the Classification of Models

When the grouping criterion is given by the mathematical complexity of the process model (models), we can distinguish  

Chemical Engineering. Tanase G. Dobre and Jose G. Sanchez Marcano Copyright 2007 WILEY-VCH Verlag GmbH Co. KGaA, Weinheim ISBN 978-3-527 30607-7 

For the mathematical models based on transport phenomena as well as for the stochastic mathematical models, we can introduce new grouping criteria. When the basic process variables (species conversion, species concentration, temperature, pressure and some non-process parameters) modify their values, with the time and spatial position inside their evolution space, the models that describe the process are recognized as models with distributed parameters. From a mathematical viewpoint, these models are represented by an assembly of relations which contain partial differential equations The models, in which the basic process variables evolve either with time or in one particular spatial direction, are called models with concentrated parameters. 

When one or more input process variable and some process and non-process parameters are characterized by means of a random distribution (frequently normal distributions), the class of non-deterministic models or of models with random parameters is introduced. Many models with distributed parameters present the state of models with random parameters at the same time. 

The models associated to a process with no randomly distributed input variables or parameters are called rigid models. If we consider only the mean values of the parameters and variables of one model with randomly distributed parameters or input variables, then we transform a non-deterministic model into a rigid model. 

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We now consider a type of analysis in which the data (which may consist of solvent properties or of solvent effects on rates, equilibria, and spectra) again are expressed as a linear combination of products as in Eq. (8-81), but now the statistical treatment yields estimates of both a, and jc,. This method is called principal component analysis or factor analysis. A key difference between multiple linear regression analysis and principal component analysis (in the chemical setting) is that regression analysis adopts chemical models a priori, whereas in factor analysis the chemical significance of the factors emerges (if desired) as a result of the analysis. We will not explore the statistical procedure, but will cite some results. We have already encountered examples in Section 8.2 on the classification of solvents and in the present section in the form of the Swain et al. treatment leading to Eq. (8-74). 

The values can also be used to validate rank by examining the impact of different ranks on the classification of the validation samples. When the rank is too large, ovcrfitting occurs and the validation samples will be incorrectly excluded from the cia.s.s for which the SIM<1A model is consmicted. 

Two types of regulatory approvals exist for medical devices in the United States, 510(k) notification and premarket approval (PMA). The types of tests required for approval depend on the classification of the medical device. 510(k) notification involves marketing a device that is substantially equivalent to a device on the market prior to 1976. All devices introduced after 1976 that are not substantially equivalent to devices on the market before 1976 are automatically classified as Class 3 devices and require PMA (16). For a device fo be considered subsfantially equivalenf to a device on the market before 1976, it must have the same intended use, no new technological characteristics, and have the same performance as one or more devices on the market prior to 1976. In addition, all medical devices must be sterilized either by end-sterilization or by some other acceptable means that can be validated, which means that any test done in cell culture or in an animal model must be conducted on a device that has been validated to be sterile. Sterility validation is conducted on all medical devices as described in the literature (17). 

Givehchi, A. and Schneider, G. (2004) Impact of descriptor vector scaling on the classification of drugs and nondrugs with artificial neural networks./. Mol. Model., 10, 204—211. 

FIGURE 8 Plot of K20 versus Si02 for various models of bulk continental crustal composition, superimposed on the classification of calc-alkaline volcanic rocks. The model favored here is shown in the heavy black circle. A variety of other compositions have been proposed on the basis of plate tectonic models (e.g., andesite, ocean island basalt open crosses) and various seismic and geological models (open squares). These various models predict a wide range of heat-producing element abundances (K, Th, U) and thus can be tested from heat-flow data. 

Using the information just given, the overall kinetics of the PTC cycle in a two-phase system can be determined. Considering the complexity of the systems, several approaches to LLPTC modeling have been taken (Evans and Palmer, 1981 Lele et al., 1983 Chen et al., 1991 Wu, 1993, 1996 Bhattacharya, 1996). All these are based essentially on the classification of fluid-fluid reactions into four regimes, as described in Chapters 14 and 15. 

The relationship between metal ionic characteristics and the maximum biosorption capacity was estabUshed using QSAR models based on the classification of metal ions (soft, hard, and borderhne ions). Ten kinds of metal were selected and the waste biomass of Saccharomyces cerevisiae obtained from a local brewery was used as biosorbent. Eighteen parameters of physiochemical characteristics of metal ions were selected and correlated with Ths suggestion was made that classification of metal ions could improve the QSAR models and different characteristics were significant in correlating with ax, such as polarizing power Z /r or the first hydrolysis constant logRo or ionization potential IP. 

From among the many reaction classification schemes, only a few are mentioned here. The first model concentrates initially on the atoms of the reaction center and the next approach looks first at the bonds involved in the reaction center. These are followed by systems that have actually been implemented, and whose performance is demonstrated. 

The possibilities for the application for neural networks in chemistry arc huge [10. They can be used for various tasks for the classification of structures or reactions, for establishing spcctra-strncturc correlations, for modeling and predicting biological activities, or to map the electrostatic potential on molecular surfaces. 

Modelling of steady-state free surface flow corresponds to the solution of a boundary value problem while moving boundary tracking is, in general, viewed as an initial value problem. Therefore, classification of existing methods on the basis of their suitability for boundary value or initial value problems has also been advocated. 

In the second section a classification of the different kinds of polymorphism in polymers is made on the basis of idealized structural models and upon consideration of limiting models of the order-disorder phenomena which may occur at the molecular level. The determination of structural models and degree of order can be made appropriately through diffraction experiments. Polymorphism in polymers is, here, discussed only with reference to cases and models, for which long-range positional order is preserved at least in one dimension. 

A useful tool in the interpretation of SIMCA is the so-called Coomans plot [32]. It is applied to the discrimination of two classes (Fig. 33.18). The distance from the model for class 1 is plotted against that from model 2. On both axes, one indicates the critical distances. In this way, one defines four zones class 1, class 2, overlap of class 1 and 2 and neither class 1 nor class 2. By plotting objects in this plot, their classification is immediately clear. It is also easy to visualize how certain a classification is. In Fig. 33.18, object a is very clearly within class 1, object b is on the border of that class but is not close to class 2 and object c clearly belongs to neither class. 

When the MLF is used for classification its non-linear properties are also important. In Fig. 44.12c the contour map of the output of a neural network with two hidden units is shown. It shows clearly that non-linear boundaries are obtained. Totally different boundaries are obtained by varying the weights, as shown in Fig. 44.12d. For modelling as well as for classification tasks, the appropriate number of transfer functions (i.e. the number of hidden units) thus depends essentially on the complexity of the relationship to be modelled and must be determined empirically for each problem. Other functions, such as the tangens hyperbolicus function (Fig. 44.13a) are also sometimes used. In Ref. [19] the authors came to the conclusion that in most cases a sigmoidal function describes non-linearities sufficiently well. Only in the presence of periodicities in the data... 

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