Complex systems data fitting problems

Under elevated pressures, the rocksalt stmcture transforms into the CsCl structure. Changes in lattice constants and measurements of other physical properties have provided much quantitative information for empirical fits to equations of state. Modem theoretical tools are used for obtaining a deeper understanding of charge being transferred back from the anion towards the cation in the aUcah hahdes. However, as can be seen from calculations of their cohesive properties, even nowadays there are problems to be solved for such simple stmctures. The data for the halide ions, in particular, are quite useful and may be transferred to other halide systems, and give good predictive values for more complex systems. 

From a postulated reaction mechanism (the model), a rate equation can be derived and used to analyse the experimental data. If the obtained fit is not statistically significant, the scheme is rejected. In complex systems, several schemes can produce compatible rate expressions and the problem of model discrimination is of primary importance. 

Weaknesses These programs only work for dimerization and 1 1 complexation NMR data. In addition, as they are written around the Solver routine in Excel , it would probably be difficult to modify them for more complex systems. The program does not give any estimation of the uncertainties of the fitting process. Finally, it shonld be noted that it was written for Excel 2004 on Mac OS X—Windows users may experience some problems in running it (this author s attempts to run it on Excel 2010 on a Win 7 computer were not successful). 

A more quantitative prediction of activity coefficients can be done for the simplest cases [18]. However, for most electrolytes, beyond salt concentrations of 0.1 M, predictions are a tedious task and often still impossible, although numerous attempts have been made over the past decades [19-21]. This is true all the more when more than one salt is involved, as it is usually the case for practical applications. Ternary salt systems or even multicomponent systems with several salts, other solutes, and solvents are still out of the scope of present theory, at least, when true predictions without adjusted parameters are required. Only data fittings are possible with plausible models and with a certain number of adjustable parameters that do not always have a real physical sense [1, 5, 22-27]. It is also very difficult to calculate the activity coefficients of an electrolyte in the presence of other electrolytes and solutes. Even the definition is difficult, because electrolyte usually dissociate, so that extrathermodynamical ion activity coefficients must be defined. The problem is even more complex when salts are only partially dissociated or when complex equilibriums of gases, solutes, and salts are involved, for example, in the case of CO2 with acids and bases [28, 29]. 

Empirical Models. The use of empirical models is widespread in both academia and indnstry. There are for example a large number of statistical potential fnnctions that have been developed to address problems such as protein folding or protein-ligand docking. The appeal of snch models is that they can be designed to be compntationally inexpensive and can be fitted directly to experimental data to hopefnlly yield immediately meaningfnl resnlts for complex systems. The challenge is that systematic improvement of empirical models beyond a certain point is extremely difficult and becomes more difficnlt as the basis for the model becomes less physical. 

Ca,= 1 and all other concentrations are zero, then Ca. (0 is the probability of finding the molecules in the i s state, Chizhov et al. s elegant solution is applied to the kinetic and spectral analysis of the bacteriorhodopsin photocycle. They discuss the problems involved in fitting data to such a complex system, as well as the difficulties of including reversibility of steps. 

Because of the assumed dual sorption mechanism present in glassy polymers, the explicit form of the time dependent diffusion equation in these polymers is much more complex than that for rubbery polymers (82-86). As a result exact analytical solutions for this equation can be found only in limiting cases (84,85,87). In all other cases numerical methods must be used to correlate the experimental results with theoretical estimates. Often the numerical procedures require a set of starting values for the parameters of the model. Usually these values are shroud guessed in a range where they are expected to lie for the particular penetrant polymer system. Starting from this set of arbitrary parameters, the numerical procedure adjusts the values until the best fit with the experimental data is obtained. The problem which may arise in such a procedure (88), is that the numerical procedures may lead to excellent fits with the experimental data for quite different starting sets of parameters. Of course the physical interpretation of such a result is difficult. 

Such complex information is known by experts in each analytical area. The problem for most practitioners is to obtain the needed information or knowledge and proceed with an analytical scheme. The extent to which this knowledge becomes available in computerized "expert systems" the easier and more efficient this task will become. This creates more freedom for the art, or the intuitive/oreative side of analysis. The critical role of the analyst is then the building of scientifically consistent mental picture, a concept or a model, to fit the accumulated data. 

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