Parametric modeling advantages

The models used can be either fixed or adaptive and parametric or non-parametric models. These methods have different performances depending on the kind of fault to be treated i.e., additive or multiplicative faults). Analytical model-based approaches require knowledge to be expressed in terms of input-output models or first principles quantitative models based on mass and energy balance equations. These methodologies give a consistent base to perform fault detection and isolation. The cost of these advantages relies on the modeling and computational efforts and on the restriction that one places on the class of acceptable models. 

The control points are defined by the basis set of points P. These control points define the parametric bicubic patches which form the surface model. Advantages of the parametric bicubic surface include continuity of position, slope, and curvature at the points where two patches meet. All the points on a bicubic surface are de by cubic equations of two parameters s and t, where s and t vary from 0 to 1. The equation for x s,t) is ... 

A few programs are now available that allow the efficient simultaneous data analysis from a population of subjects. This approach has the significant advantage that the number of data points per subject can be small. However, using data from many subjects, it is possible to complete the analyses and obtain both between- and within-subject variance information. These programs include NONMEM and WinNON-MIX for parametric (model dependent) analyses and NPEM when non-parametric (model independent) analyses are required. This approach nicely complements the Bayesian approach. Once the population values for the pharmacokinetic parameters are obtained, it is possible to use the Bayesian estimation approach to obtain estimates of the individual patient s pharmacokinetics and optimize their drug therapy. 

The advantage of parametric models is that estimates obtained from such models will tend to have lower variability than those obtained from semiparametric and nonparametric models. 

The advantage of semiparametric and nonparametric models is that they have more robustness than parametric models. If the true data model does not match the model being fit by the investigator, then estimates from the semiparametric and nonparametric models will have less associated bias than that from a parametric model. 

The structure of this chapter is as follows. In the next section a comparison of the FBR and PBMR will be presented using a simplified ID modeL Advantages and drawbacks of the PBMR will be illustrated. Subsequently, a more detailed 2D model will be developed to study important aspects of radial mass and heat transfer, as well as scale-up problems that might occur in a PBMR. In the last section a short outlook to more sophisticated 3D membrane reactor models is given. Such models are still not suitable for extensive parametric studies. However, they enable a deeper investigation of local velocity and concentration profiles that develop in such reactors. 

Editing (changing, correcting) in parametric modeling is one of the biggest advantages over conventional CAD. 

While accurate, direct calculation of f-state energy structure is not generally feasible for lanthanide and actinide ions, parametric models for f-state energy-level structure determination have been developed that take advantage of relationships established by relativistie Hartree-Fock calculations (Crosswhite and Crosswhite 1984). Works by Cowan (1981) and Szasz (1992) should be consulted for additional information on theoretical atomic spectroscopy and relativistic Hartree-Fock calculations. 

Based on both theoretical knowledge and experimental observations, parametric models of the source and noise can be chosen. The model parameters are then estimated from training data. This method has some important advantages ... 

Parametric models, and their respective signal-based (identification) methods, are known to be characterized by a number of important advantages, such as representation parsimony, improved accuracy and resolution. 

Without truncation, the FCI and full coupled-cluster functions contain the same number of parameters since there is then one connected cluster amplitude for each determinant. In this special case, the Cl and coupled-cluster models provide linear and nonlinear parametrizations of the same state and there is then no obvious advantage in employing the more complicated exponential parametrization. The advantages of the cluster parametrization become apparent only upon tmncation and are related to the fact that, even at the truncated level, the coupled-cluster state contains contributions from all determinants in the FCI wave function, with weights obtained from the different excitation processes leading to the determinants. 

The level of accuracy that can be achieved by these different methods may be viewed as somewhat remarkable, given the approximations that are involved. For relatively small organic molecules, for instance, the calculated AGsoivation is now usually within less than 1 kcal/mole of the experimental value, often considerably less. Appropriate parametrization is of key importance. Applications to biological systems pose greater problems, due to the size and complexity of the molecules,66 156 159 161 and require the use of semiempirical rather than ab initio quantum-mechanical methods. In terms of computational expense, continuum models have the advantage over discrete molecular ones, but the latter are better able to describe solvent structure and handle first-solvation-shell effects. 

The residuals discussed thus far have been associated with some dependent variable, such as the reaction rate r. It is particularly advantageous in pinpointing the type of defect present in an inadequate model to expand this definition to include parametric residuals. The parametric residual, then, is simply the difference between a value of a given parameter estimated from the data and that predicted from a model. For example, the dots in Fig. 17 represent the logarithm of the alcohol adsorption constants measured in alcohol dehydrogenation experiments from isothermal data at each of several temperature levels (FI). The solid line represents the expectation that these... 

Such applications of NN as a predictive method make the artificial neural networks another technique of data treatment, comparable to parametric empirical modeling by, for example, numerical regression methods [e.g., 10,11] briefly mentioned in section 16.1. The main advantage of NN is that the network needs not be programmed because it learns from sets of experimental data, which results in the possibility of representing even the most complex implicit functions, and also in better modeling without prescribing a functional form of the actual relationship. Another field of... 

Unfortunately, a detailed comparison of the continuum models is available only at the semiempirical level.54,55 Because the SMx models are specially parametrized to describe free energy of hydration, it is not surprising that they are the best for reproducing this value. A detailed discussion of the advantages and limitations of different types of solvation models with regard to the various types of approximations and different types of organic molecules can be found in references 53-55. 

While the advantages of parametric controllers are well established, a key challenge for their wider applicability is the ability to derive parametric controllers from arbitrary large scale and complex mathematical models. In this context. Model Order Reduction [5] can be a useful tool, since it could lead to an approximate model of reduced size, and complexity and of sufficient accuracy. 

Table 10.3 provides an example of the cyclic steady-state performance of the SERP concept using a 6 1 H20 + CH4 feed gas at a pressure of 11.4psig and a temperature of 490°C.S1 The process can directly produce an essentially COx-free H2 product (-94.4% H2 + 5.6% CH4), which is suitable for H2 fuel-cell use. The conversion of CH4 to H2 was -73.0%. The table also shows the equilibrium compositions of the H2 product from a conventional plug-flow reforming reactor operating under identical conditions. Both the H2 conversion and product purity were rather poor in the latter case, which demonstrates the advantage of the SERP concept. Theoretical models of the above-described SERP concept and its variations for H2 production by SMR have been developed, and theoretical parametric studies of the process have been conducted by various authors.62,63... 

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