Parameter-free models, optimization

Beam elements may easily be expanded forthe study of compensator systems (Li 2010). To all nodes we add 2 dof to the 6 dof of the beam node (Figure 5b). These 2 dof represent the displacement of the compensator s mass in the 2 plane directions of the floor. They are attached to the corresponding translational dof of the beam via the compensators stifihess and damping. The mass, stifihess and damping of the compensators are the free parameters for the optimization of a compensation system. At floors without compensators, small values of the compensators properties produce only negligible effects, so there is no need to use different element types for floors with and without compensator systems (Figure 5a). It is evident that such models allow for fast studies of the dynamic response of structures. 

Translating a known metabolic network into a dynamic model requires rate laws for all chemical reactions. The mathematical expressions depend on the underlying enzymatic mechanism they can become quite involved and may contain a large number of p>arameters. Rate laws and enzyme parameters are still unknown for most enzymes. Convenience kinetics is used to translate a biochemical network - manually into a dynamical model with plausible biological properties. It implements enzyme saturation and regulation by activators and inhibitors, covers all possible reaction stoichiometries, and can be specified by a small number of parameters. Its mathematical form makes it especially suitable for parameter estimation and optimization. In general, the convenience kinetics applies to arbitrary reaction stoichiometries and captures biologically relevant behavior such as saturation, activation, inhibition with a small number of free parameters. It represents a simple molecular reaction mechanism in which substrates bind rapadly and in random order to the enzyme. 

Random interface models for ternary systems share the feature with the Widom model and the one-order-parameter Ginzburg-Landau theory (19) that the density of amphiphiles is not allowed to fluctuate independently, but is entirely determined by the distribution of oil and water. However, in contrast to the Ginzburg-Landau approach, they concentrate on the amphiphilic sheets. Self-assembly of amphiphiles into monolayers of given optimal density is premised, and the free energy of the system is reduced to effective free energies of its internal interfaces. In the same spirit, random interface models for binary systems postulate self-assembly into bilayers and intro-... 

As seen in Chapter 2 a suitable measure of the discrepancy between a model and a set of data is the objective function, S(k), and hence, the parameter values are obtained by minimizing this function. Therefore, the estimation of the parameters can be viewed as an optimization problem whereby any of the available general purpose optimization methods can be utilized. In particular, it was found that the Gauss-Newton method is the most efficient method for estimating parameters in nonlinear models (Bard. 1970). As we strongly believe that this is indeed the best method to use for nonlinear regression problems, the Gauss-Newton method is presented in detail in this chapter. It is assumed that the parameters are free to take any values. 

We obtain an r.m.s. deviation of 0.84 kcal/mol with an optimal a of 0.181. One can also note the similarity between the a value of this model and that of the two-parameter model with a free a and /3. This suggests that the model is robust in the sense that the actual polar and non-polar free energy contributions are more or less invariant, as long as deviations from linear response are taken into account in a proper way. The FEP-derived model could be considered preferable to the two-parameter model since it contains only one free parameter, viz. oc. The results of adding a constant yto the new model was also investigated. Remarkably, the optimal value for such a y was found to be -0.02 kcal/mol, i.e. virtually zero. 

To summarize the attempts to refine the original LIE model, we found that an optimal equation for the binding free energy could be obtained with only one free parameter (a) and with the electrostatic coefficients (fi) derived from FEP simulations of some representative compounds in water. For the 18 compound training set that we used this model yielded a mean unsigned error of only 0.58 kcal/mol which seemed very promising. 

A single experiment consists of the measurement of each of the g observed variables for a given set of state variables (dependent, independent). Now if the independent state variables are error-free (explicit models), the optimization need only be performed in the parameter space, which is usually small. 

In the proposed technique, the user is free to pick the model formulation to suit his needs. The optimal set of parameters are found by solving the nonlinear optimization... 

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