Discrimination of models

The discrimination among rival models has to take into account the fact that, in general, when the number of parameters of a model increases, the quality of fit, evaluated by the sum S(a) of squared deviations, increases, but that, at the same time, the size of confidence regions for parameters also increases. Thus, there is, in most cases, a compromise between the wish to lower both the residuals and the confidence intervals for parameters. The simplest way to achieve the discrimination of models consists of comparing their respective experimental error variances. Other methods and examples have been given in refs. 25, 32 and 195—207. 

Finally, we should remark that sequential methods for the optimal discrimination of models and the optimal estimation of parameters are discussed in refs. 25 and 32, for example. 

The cracking experiments have been modeled according to the five lump model with different deactivation orders. The parameters of the model have been calculated using a Levenberg-Marquard minimization routine written in Fortran. Discrimination of models was based on ... 

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). 

Discrimination between exposed and unexposed areas in this process requires the selection of thia zolidine compounds that do not readily undergo alkaline hydrolysis in the absence of silver ions. In a study of model compounds, the rates of hydrolysis of model /V-methyl thia zolidine and A/-octadecyl thiazolidine compounds were compared (47). An alkaline hydrolysis half-life of 33 min was reported for the /V-methyl compound, a half-life of 5525 min (3.8 days) was reported for the corresponding V/-octadecyl compound. Other factors affecting the kinetics include the particular silver ligand chosen and its concentration (48). Polaroid Spectra film introduced silver-assisted thiazolidine cleavage to produce the yellow dye image (49), a system subsequentiy used in 600 Plus and Polacolor Pro 100 films. 

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. 

The results of the Debye theory reproduced in the lowest order of perturbation theory are universal. Only higher order corrections are peculiar to the specific models of molecular motion. We have shown in conclusion how to discriminate the models by comparing deviations from Debye theory with available experimental data. 

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... 

A second kind of model is briefly treated, based on isotopic mass balance arguments, and it is shown that large isotopic discrimination during methano-genesis in ruminants may account for data trends when comparing herbivores and carnivores. A third class of model is sketched at the level of biochemical flows, where some fundamental points are made concerning points where the isotopic composition of metabolites may be altered. The relevance of this to nitrogen isotopic enrichment is considered. 

L CHECK, VAUDITY OF MODEL BV CONOUCTiNO EKRERIVENTS UNDER DISCRIMINATING CONDITIONS ... 

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. 

O Neil, GA Wisnudel, MB Torkelson, JM, Gel Effect in Free Radical Polymerization Model Discrimination of Its Cause, AIChE Journal 44, 1226, 1998. 

The way in which the family of models is selected depends on the main purpose of the exercise. If the purpose is just to provide a reasonable description of the data in some appropriate way without any attempt at understanding the underlying phenomenon, that is, the data-generating mechanism, then the family of models is selected based on its adequacy to represent the data structure. The net result in this case is only a descriptive model of the phenomenon. These models are very useful for discriminating between alternative hypotheses but are totally useless for capturing the fundamental characteristics of a mechanism. On the contrary, if the purpose of the mode-... 

The combined use of a continuous flow system and a spectrophotometer for sample screening to discriminate between synthetic and natural colorants is also available. With a very simple flow system on a column packed with natural materials, one can discriminate natural and synthetic colorants. The natural (not retained) ones can be determined in the first step and the synthetic (retained) ones in the second step after their elution. For yellow, red, green, blue, and brown, natural or synthetic colorants were chosen as models. The specific maximum wavelength for each color (400,530, and 610 mn, respectively) was selected by a diode array system. A complete discrimination of natural and synthetic colorants was obtained for concentrations of natural colorants (in the absence of synthetic ones) up to 2000 (yellow), 2000 (red), and 10,000 (brown) times that of the detection limits (DLs) of synthetic additives. This method was applied to screen fruit drinks and candies. ... 

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). 

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. 

In SIMCA, we can determine the modelling power of the variables, i.e. we measure the importance of the variables in modelling the elass. Moreover, it is possible to determine the discriminating power, i.e. which variables are important to discriminate two classes. The variables with both low discriminating and modelling power are deleted. This is more a variable elimination procedure than a selection procedure we do not try to select the minimum number of features that will lead to the best classification (or prediction rate), but rather eliminate those that carry no information at all. 

Taylor PDF, De Bievre P, Walder AJ, Entwistle A (1995) Vahdation of the analytical linearity and mass discrimination correction model exhibited by a Multiple Collector Inductively Coupled Plasma Mass Spectrometer by means of a set of synthetic uranium isotope mixtures. J Anal At Spectrom 10 395-398... 

More than ten years ago, when distributing the popular set of models published in D Antona Mazzitelli (1994) the authors added a readme file asking observers to use the tracks as tests , more than as a benchmark for observations, in the hope that observations could throw some light on our models, which, especially in regard to convection, were -and are- parametric. After that, many other sets of tracks have been computed. In this paper we try to summarize which are the key ingredients of the published models, and whether we can discriminate between numerical uncertainties and physical uncertainties. 

V. Chan, Y. Chong, L. Cheung, J. Vielmetter, and D.H. Farkas, Bioelectronic detection of point mutations using discrimination of the H63D polymorphism of the Hfe gene as a model. Mol. Diagn. 5, 321-328 (2000). 

The investigation of dynamics by NMR methods usually requires a model with an underlying analytical theory to simulate experimental data. The rationalization of the model and the discrimination of alternatives demand a critical check of the case under investigation. 

In the field of DNA sequencing instruments, Perkin-Elmer [26] switched from the PMT-based model 373 sequencer to the CCD-based model 377, which allowed the simultaneous discrimination of four or more colors in emission, in a single lane of an electrophoretic gel. The CCD made possible the use of a specific fluorescent label for each type of base and throughput was improved more than four times [27], Excitation uses an argon laser. 

According to X-ray and neutron diffraction structures [3, 4] the binding of CO to the heme leads to a bent FeCO unit. The Fe-C-O angle is, however, found to be linear in synthetic models of the protein (hiomimetic molecules). Because of this, it was originally thought that the FeCO distortion was responsible for the well known discrimination of the protein against CO - the affinity ratio C0/02 is lower in the protein than in biomimetic systems [1]. In... 

The utility of the overall dependence of the reaction rate upon temperature appears to be slight, perhaps in some cases allowing a discrimination between adsorption and surface reaction classes of models. The study of the residuals of various estimated parameters as a function of temperature can clearly indicate model inadequacies (Section IV) and, in some cases, can lead to model modifications correcting these model inadequacies (Section V). 

Recently certain diagnostic parameters have been exploited to allow a discrimination among several rival models. These diagnostic parameters can be grouped into two broad classes—those that are inherently present in the model, and those that are introduced solely for the purpose of model discrimination. 

If this linear analysis is to be used, the experimental conversion-spacetime data should first be taken at several pressure levels. Using the C2 analysis alone, then, the plots of Ct versus total pressure should be made for a preliminary indication of model adequacy. If several models are found to provide near-linear Cx plots, the complete linear analysis using the C2 plots should assist in the discrimination among the remaining rival models. If a model is adequate, both the Cl and C2 points should be correctable by a straight line with a common intercept, as demanded by Eqs. (85) and (86). If only one model is found to be adequate following the initial Cl analysis, the complete Ct and C2 analysis should still be carried out on this model to verify its ability to fit the high conversion data. 

It is partly the fault of statistics that experimenters have misconstrued the value of the number and precision of data points relative to the value of the location of the points. The importance of the location of the data in the model specification stage can be seen from Fig. 1, which represents literature data (M3) on sulfur dioxide oxidation. The dashed and solid lines represent the predicted rates of two rival models, and the points are the results of two series of experimental runs. It can be seen that neither a greater number of experimental points nor data of greater precision will be of major assistance in discriminating between the two rival models, if data are restricted to the total pressure range from 2 to 10 atm. These data simply do not place the models in jeopardy, as would data below 2 atm and greater than 10 atm total pressure. This is presumably the problem in the water-gas shift reaction, which is classical in terms of the number of models proposed, each of which adequately represent given sets of data. 

Figure 30 portrays the grid of values of the independent variables over which values of D were calculated to choose experimental points after the initial nine. The additional five points chosen are also shown in Fig. 30. Note that points at high hydrogen and low propylene partial pressures are required. Figure 31 shows the posterior probabilities associated with each model. The acceptability of model 2 declines rapidly as data are taken according to the model-discrimination design. If, in addition, model 2 cannot pass standard lack-of-fit tests, residual plots, and other tests of model adequacy, then it should be rejected. Similarly, model 1 should be shown to remain adequate after these tests. Many more data points than these 14 have shown less conclusive results, when this procedure is not used for this experimental system.

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