Model discriminative

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. 

Model discrimination is a procedure for developing a suitable description of the unit performance. The techniques are drawn from the mathematics hterature where the goodness-of-fit of various proposed models are compared. Unfortunately, the various proposed models will usually describe a unit s performance equally well. Model discrimination is better accomplished when raw or adjusted measurements from many, unique operating conditions provide the foundation for the comparisons. 

While many data are suggestive of chain length dependence, the data are not usually suitable for or have not been tested with respect to model discrimination. Values of ,u have been determined for a variety of small monomeric radicals to be ca I09 M s 1.4 Taking kt0 as Jk,lj and a as 1.0 in the geometric expression yields values of ,iJ as shown in Figure 5.4a.49 Use of the Smoluchowski mean or the harmonic mean approximation prediets a shallower dependence of k 1 on the chain length (Figure 5.4b). All expressions yield the same dependence for j=i. 

It has been argued that for a majority of copolymerizations, composition data can be adequately predicted by the terminal model copolymer composition equation (eqs. 5-9). However, in that composition data are not particularly good for model discrimination, any conclusion regarding the widespread applicability of the implicit penultimate model on this basis is premature. 

It is also possible to process copolymer composition data to obtain reactivity ratios for higher order models (e.g. penultimate model or complex participation, etc.). However, composition data have low power in model discrimination (Sections 7.3.1.2 and 7.3.1.3). There has been much published on the subject of the design of experiments for reactivity ratio determination and model discrimination.49 "8 136 137 Attention must be paid to the information that is required the optimal design for obtaining terminal model reactivity ratios may not be ideal for model discrimination.49... 

NMR spectroscopy has made possible the characterization of copolymers in terms of their monomer sequence distribution. The area has been reviewed by Randall,100 Bovey,139 Tonelli,101 Hatada140 and others. Information on monomer sequence distribution is substantially more powerful than simple composition data with respect to model discrimination,25,49 Although many authors have used the distribution of triad fractions to confirm the adequacy or otherwise of various models, only a few25 58,141 have used dyad or triad fractions to calculate reactivity ratios directly. 

The important aspect of this problem is that while the penultimate model involves a four dimensional parameter space, the model discrimination problem can be reduced to a two dimensional space by dealing with functions of the original parameters. This approach requires that probabilities for array locations in the four dimensional r, r/, rj, rj ) space be mapped to array locations in the ((ri-r/), (rj-rj )) space. 

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

The positive values obtained in practically all cases indicate that all these models may be plausible representations of the data and indeed, the correlation coefBcients, R, are greater than 0.9. Thus, statistical compliance is not a sufficient basis for model discrimination. Specifically, the thermodynamic consistency of the estimates, as proposed by Boudart et al. [3], is appropriate further scrutinizing criterion during kinetic modelling and has been gainfully employed in other reactions [4-6]. 

Kinetic Model Discrimination. To discriminate between the kinetic models, semibatch reactors were set up for the measurement of reaction rates. The semi-batch terminology is used because hydrogen is fed to a batch reactor to maintain a constant hydrogen pressme. This kind of semi-batch reactor can be treated as a bateh reactor with a constant hydrogen pressme. The governing equations for a bateh reactor, using the product formation rate for three possible scenarios, were derived, as described in reference (12) with the following results ... 

The results of the kinetic study and model discrimination show that insertion of SM is rate-controlling. Two reasons may explain why this step is ratecontrolling. First, the protection group in om SM is very bulky, making the reaction slow, which is consistent with literature data (8) showing the size effect on reactivity. Second, a free aniline group in the SM could bond with Rh and reduce the catalyst reactivity. 

Some data fitting results are displayed in Figures 12.1 and 12.3. The general conclusion is that both models describe the behaviours of the main components, lactose and lactitol very well, both for sponge nickel and ruthenium catalysts. In this respect, no real model discrimination is possible. Both models also describe equally well the behaviour of lactobionic acid (D), including its concentration maximum when the reversible step is included (ks) (Figure 12.3). 

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

Selection of Optimal Sampling Interval and Initial State for Model Discrimination... 

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... 

The use of time stages of varying lengths in iterative dynamic programming (Luus, 2000) may indeed provide a computationally acceptable solution. Actually, such an approach may prove to be feasible particularly for model discrimination purposes. In model discrimination we seek the optimal inputs, u(t), that will maximize the overall divergence among r rival models given by Equation 12.23. 

As a third example let us consider the growth kinetics in a chemostat used by Kalogerakis (1984) to evaluate sequential design procedures for model discrimination in dynamic systems. We consider the following four kinetic models for biomass growth and substrate utilization in the continuous baker s yeast fermentation. 

Buzzi-Ferraris, G., P. Forzatti, G. Emig and H. Hofmann, "Sequential Experimental Design for Model Discrimination in the Case of Multiple Responses", Chem. Eng. Sci., 39(1), 81-85 (1984). 

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. 

Frank IE (1989) Classification models discriminant analysis, SIMCA, CART. Chemom Intell Lab Syst 5 247... 

To examine the fitness of the kinetic model CAER used for study of the water effect, we conducted model discrimination using the kinetic data of 15% Co/Si02 by comparing values of the standard function of mean absolute relative residual (MARR), which is simply defined as... 

Based on the experimental data and some speculations on detailed elementary steps taking place over the catalyst, one can propose the dynamic model. The model discriminates between adsorption of carbon monoxide on catalyst inert sites as well as on oxidized and reduced catalyst active sites. Apart from that, the diffusion of the subsurface species in the catalyst and the reoxidation of reduced catalyst sites by subsurface lattice oxygen species is considered in the model. The model allows us to calculate activation energies of all elementary steps considered, as well as the bulk... 

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