Parameters, of a model

One other remark of Vineyard s in 1972, made with evident feeling, is worth repeating here Worthwhile computer experiments require time and care. The easy understandability of the results tends to conceal the painstaking hours that went into conceiving and formulating the problem, selecting the parameters of a model. 

To ht the parameters of a model, there must be at least as many data as there are parameters. There should be many more data. The case where the number of data equals the number of points can lead to exact but spurious fits. Even a perfect model cannot be expected to fit all the data because of experimental error. The residual sum-of-squares is the value of after the model... 

Salazar-Sotelo D., A. Boireaut and H. Renon, "Computer Calculations of the Optimal Parameters of a Model for the Simultaneous Representation of Experimental, Binary and Ternary Data", Fluid Phase Equilibria, 27, 383-403 (1986). 

After each peak has been described by the parameters of a model function, the convolution in Eq. (8.13) can be carried out analytically. As a result, equations are obtained that describe the effects of crystal size, lattice distortion, and instrumental broadening38 on the breadth of the observed peak. Impossible is in this case the separation of different kinds of lattice distortions. 

Useful fine ratios to pin down the basic parameters of a model for the optical lines are (Dopita 1977) ... 

Instead of converting the step or pulse responses of a system into frequency response curves, it is fairly easy to use classical least-squares methods to solve for the best values of parameters of a model that fit the time-domain data. 

Polarizability Effects. The next model demonstrates that an additivity scheme can be combined with other forms of mathematical relations to extract the fundamental parameters of a model from primary information. And furthermore, it shows than an additivity scheme useful for the estimation of a global molecular proparty can be modified to obtain a local, site specific property. 

After fitting the parameters of a model to a set of measurement data, criteria for the good-ness-of-fit are required. There will always be some differences between the measured data and the values calculated from the model. These differences may be due to the following... 

First-order error analysis is a method for propagating uncertainty in the random parameters of a model into the model predictions using a fixed-form equation. This method is not a simulation like Monte Carlo but uses statistical theory to develop an equation that can easily be solved on a calculator. The method works well for linear models, but the accuracy of the method decreases as the model becomes more nonlinear. As a general rule, linear models that can be written down on a piece of paper work well with Ist-order error analysis. Complicated models that consist of a large number of pieced equations (like large exposure models) cannot be evaluated using Ist-order analysis. To use the technique, each partial differential equation of each random parameter with respect to the model must be solvable. 

With due deference to the myriad mathematics dissertations and journal articles on the subject of optimization, f will briefly mention some of the general approaches to finding an optimum and then describe the recommended methods of experimental design in some detail. There are two broad classes that define the options systems that are sufficiently described by a priori mathematical equations, called models, and systems that are not explicitly described, called model free. Once the parameters of a model are known, it is often quite trivial, via the miracles of differentiation, to find the maximum (maxima). 

For molecular transport the parameters of a model are to be determined by ab initio methods or considered as semi-empirical. This is a compromise, which allows us to consider complex molecules with a relatively simple model. 

A further reduction of experimental effort may be achieved by the selection of special designs developed, for example, by HARTLEY [1959], BOX and BEHNKEN [1960], WESTLAKE [1965], and others. In these designs the ratio of experiments to the number of coefficients necessary is reduced almost to unity. (This situation is somewhat different from regression analysis or random selection of experiments where, in principle, k experiments or measurements are sufficient to estimate k parameters of a model. In experimental design the optimized number of experiments is derived from statistical consideration to encompass as much variation of the factors as possible.)... 

Two modeling methods are current. The first is determining the parameters of a model of the form ... 

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. 

SA determines the change in a response R as a result of a perturbation in one of the parameters P of the model. Parameters of a model can be any conceivable ones. For example, in a MD simulation, parameters could be all factors appearing in the intermolecular potential. Since the magnitude of various parameters can be very different, it is common to compute a normalized sensitivity coefficient (NSC) defined as... 

Druaux, C., Lubbers, C., Lubbers, S., Charpentier, C., VoUley, A. (1995). Effects of physicochemical parameters of a model wine on the binding of a model wine on the binding of gamma-decalactone on bovine serum-albumin. Food Chem., 53, 203-207. 

The methods for identifying the parameters of a model can be classified in terms of the complexity of the mathematical model and constraints accepted for its parameters. 

Table 3.16 Classification of the methods for identifying the parameters of a model.

Identification of the Parameters of a Model by the Steepest Slope Method... 

Such maps are primarily used to refine a trial structure, to find a part of the structure that may not yet have been identified or located, to identify errors in a postulated structure, or to refine the positional and displacement parameters of a model structure. A difference map is very useful for analyses of the crystal structures of small molecules. It is also very useful in studies of the structures of crystalline macromolecules, since it can be used to find the location of substrate or inhibitor molecules that have been soaked into a crystal once the macromolecular structure is known. A formula like that in Equation 9.1.5 is then used. When a structure determination is complete, it is usual to compute a difference electron-density map to check that the map is flat, and approximately zero at all points. 

As briefly mentioned in the previous chapter, the determination of a crystal structure may be considered complete only when multiple pattern variables and crystallographic parameters of a model have been fully refined against the observed powder diffraction data. Obviously, the refined model should remain reasonable from both physical and chemical standpoints. The refinement technique, most commonly employed today, is based on the idea suggested in the middle 1960 s by Rietveld. The essence of Rietveld s approach is that experimental powder diffraction data are utilized without extraction of the individual integrated intensities or the individual structure factors, and all structural and instrumental parameters are refined by fitting a calculated profile to the observed data. 

Although it requires equipment for numerical calculations, a simulation and fitting method is of particular interest for more complex mechanisms. It can be employed to analyze thermal or photochemical reisomerizations and reactions involving several photoisomers as well as photodegradation processes. This approach is also referred to as inverse treatment as it proceeds back from the experimental data to the parameters of a model. A block diagram ofthe method of simulation and fitting is presented in Figure 3. 

Regression analysis A statistical technique for determining the parameters of a model see also least-squares method. 

Reference [766] reports selected parameters of a model of surface charging of MX-80 bentonite. 

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