Selection of the most plausible model

General considerations Several methods are available for model selection. We shall only be concerned with statistical methods, which in their simplest form are based on selection of the most plausible model from among a nnmber of candidates. It must be emphasized that these methods do not nniqnely select a candidate as the only acceptable model, bnt converge to one that is statistically the most acceptable. 

Initial rapid Increase followed by a slow increase 

The basic steps in model selection Having narrowed down the field by the initial rate method, the following procedure can then be used to select the most probable model from the surviving list of contenders. 

Write down all possible mechanisms and the corresponding rate equations. 

Linearize the equations as illustrated below for a typical case. 

---

Outline a methodology for the selection of the most plausible model for the solid catalyzed reactor design. 

The plausibility P(Cj V, U) can be used not only for selection of the most plausible class of models, but also for response prediction based on all the model classes. Let Q denote a quantity to be predicted, e.g., first story drift. Then, the PDF of Q given the data V can be calculated from the law of total probability as follows ... 

The idea of finding the best model was extended by Jones et al. (1994). A dynamic programming algorithm was used to select the most plausible model, and the same authors also presented an ambitious method to predict the three-dimensional structure of the a-helical membrane proteins (Taylor etal., 1994). Finally, HMMs were used to model the overall structure of the membrane topology by two groups of researchers (Sonn-hammer et al., 1998 Tusnady and Simon, 1998). 

Note that the first four model classes possess similar plausibility, implying that the Bayesian model selection method does not have a strong preference on the most plausible model class. This is in contrast to the previous case in the Tangshan region, in which the plausibility of the optimal model class is over 0.7. With the data of Xinjiang, a multi-model predictive formula can be used as follows ... 

Bayesian model class selection is utilized for selecting the most plausible model class from a set of Nc dynamic model class candidates C, C2, , Cvc by considering their plausibility P Cj D) conditional rai the available set of dynamic measurement D (Beck and Yuen 2004 Yuen 2010b) ... 

A perfect selective proton transfer mechanism is easily conceivable and its role appears particularly suggestive in relation to the paradigmatic frame of reference of the chemiosmotic theories, 13, However, we may fairly ask the question whether experimental proofs and theoretical argumentation are nowadays sufficiently conclusive in order to retain the predominant proton pump as the most plausible model for steady state electrogenic transfer in these cells. 

Influenced by the mind of forward modeling problems, it is easily directed to adopt complicated model classes so as to capture various complex physical mechanisms. However, the more complicated the model class is utilized, the more uncertain parameters are normally induced unless extra mathematical constraints are imposed. In the former case, the model output may not necessarily be accurate even if the model well characterizes the physical system since the combination of the many small errors from each uncertain parameter can induce a large output error. In the latter case, it is possible that the extra constraints induce substantial errors. Therefore, it is important to use a proper model class for system identification purpose. In this chapter, the Bayesian model class selection approach is introduced and applied to select the most plausible/suitable class of mathematical models representing a static or dynamical (structural, mechanical, atmospheric,...) system (from some specified model classes) by using its response measurements. This approach has been shown to be promising in several research areas, such as artificial neural networks [164,297], structural dynamics and model updating [23], damage detection [150] and fracture mechanics [151], etc. 

Let T> denote the input-output or output-only data from a physical system or phenomenon. The goal is to use T> to select the most plausible/suitable class of models representing the system out of Nc given classes of models Ci, C2,..., Cjvc- Since probability may be interpreted as a measure of plausibility based on specified information [63], the probability of a class of models conditional on the set of dynamic data T> is required. This can be obtained by using the Bayes theorem as follows ... 

To summarize, in the Bayesian approach to model selection, the model classes are ranked according to p V Cj)P(Cj U) for 7 = 1,2,..., Nc, where the most plausible class of models representing the system is the one which gives the largest value of this quantity. The evidence p V Cj) can be calculated for each class of models using Equation (6.11) where the likelihood p V 9, Cj) is evaluated using the methods presented in Chapters 2-5. The prior distribution P Cj U) over all the model classes Cj, j = 1,2,..., Nc, can be used for other concerns, such as computational demand. However, it is out of the scope of this book and uniform prior plausibilities are chosen, leaving the Ockham factor alone to penalize the model classes. 

To summarize, the enhancement in regioselectivity, due to the small amount of modifier added, was interrelated with the enantioselectivity (cjf) and can be most plausibly explained by similar interactions on the catalyst surface which are responsible for the enantiodifferentiation. The substrate-modifier interaction on the catalyst surface coupled with the coverage dependent adsorption modes of the modifier and reactant explain the enhancement of rs as well as es and their dependence on modifier concentration. In the light of presented data it is evident that one needs to incorporate the coverage dependent adsorption modes in the kinetic model for a correct description of the rs and es, otherwise the maximum in selec-tivities and selectivity dependence on modifier concentration caiuiot be described. 

---