Surface models Parameter space

The results of three-component mixture designs are often presented as response surfaces over the triangular mixture space as shown in Figure 12.34. The Scheffe model parameters are seen to be equivalent to the responses at the vertexes. 

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-peri-odic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations. 

Here E ( y1 ) stands for the single-particle contribution to the total energy, allowing for molecule interaction with the surface <2 is the heat released in adsorption of molecules z on the /Lh site Fj the internal partition function for the z th molecules adsorbed on the /Lh site F j the internal partition function for the zth molecule in the gas phase the dissociation degree of the z th molecule, and zz the Henry local constant for adsorption of the zth molecule on the /Lh site. Lateral interaction is modeled by E2k([ylj ), and gj (r) allows for interaction between the z th and /Lh particles adsorbed on the /th and gth sites spaced r apart. In the lattice gas model, separations are conveniently measured in coordination-sphere numbers, 1 < r < R. For a homogeneous surface, molecular parameters zz and ej(r) are independent of the site nature, while for heterogeneous, they may depend on it. 

The third solution to the problem may be found in the use of more efficient computers, algorithms and computational methods. For instance, if segmentation of the parameter space (linear interpolation) is used, large parts of the retention surfaces and hence of the response surface may remain unaltered when a new data point is added to the existing set. The use of simple model equations instead of linear segmentation may also be more efficient from a computational point of view. However, such simple equations may only be used for the description of the retention behaviour in a limited number of cases and if the model equations become more complex the advantage quickly disappears. For example, d Agostino et al. used up to sixth order polynomial equations [537] and their procedure also led to excessive calculation times. 

Soft surface We assume in this model that the surface-layer energy shifts are proportional to the dilatation strain (3.37) caused by the missing interactions.1 With attractive forces of the type (3.39) in r- 5, the ratio p may be calculated for various values of the parameters a and d of the model (the spacing and size of the molecules , Fig. 3.23). It is concluded that p is practically independent of these parameters (from p = 25.2 for a = d = 0 it becomes 19.8 for a = d = 1). The typical values a = 6 and d = 10 lead to the ratio p = 20. On this point, we make the remark that only forces in r-5 are capable of yielding values of p compatible with the experimental value... 

By a carefull spacing of the settings of the variables it is possible to obtain estimates of the parameters so that they describe the following different features of the fitted response surface model and nothing else ... 

Figure 5-2 The plot of the misfit functional value as a function of model parameters tn. The vector of the steepest ascent, l(m ), shows the direction of "climbing on the hill" along the misfit functional surface. The intersection between the vertical plane P drawn through the direction of the steepest descent at point m and the misfit functional surface is shown by a solid parabola-type curve. The steepest descent step begins at a point 0(m ) and ends at a point 0(m +i) at the minimum of this curve. The second parabola-type curve (on the left) is drawn for one of the subsequent iteration points. Repeating the steepest descent iteration, we move along the set of mutually orthogonal segments, as shown by the solid arrows in the space M of the model parameters.

Figure 5-4 The plot of the misfit functional value as a function of model parameters m. In the framework of the Newton method one tries to solve the problem of minimization in one step. The direction of this step is shown by the arrows in the space M of model parameters and at the misfit surface.

Equations (12.55), sometime referred to as multiphonon transition rates for reasons that become clear below, are explicit expressions for the golden-rule transitions rates between two levels coupled to a boson field in the shifted parallel harmonic potential surfaces model. The rates are seen to depend on the level spacing 21, the normal mode spectrum mo,, the normal mode shift parameters Ao-, the temperature (through the boson populations ) and the nonadiabatic coupling... 

Parameter will henceforth be referred to as the asymmetry of the model mixture, where xb > 1 characterizes a binary mixture in which the formation of B-B pairs is energetically favored, whereas for XB < L this is the case for A-A pairs. For the special case xb 1 the asymmetric mixture degenerates to the symmetric case previously studied in Refs. [84] and [85]. In addition, we define the selectivity of the solid surfaces by specifying Xs in Eq. (4.125d) in a fashion similar to xb in Eq. (4.125c). Hence, the parameter space of our model is spanned by the set , aBi s, Xb> X s -... 

For higher values of. this simple model is not capable of describing the increasingly steep drop in ignition temperatures, since the assumption of a predominantly oxygen covered surface in the model breaks down at this point. This emphasizes that, while simplified chemical mechanisms can yield very good fits with experimental data if the chemistry is well understood, their applicability is typically limited to a rather narrow parameter space, and detailed reaction mechanisms are needed for a reliable description over a wide parameter range. 

The parametric surface must have points for all possible value combinations within the parameter ranges. This condition is fulfilled when any point in the parameter space is mapped to a point in the model space (Figure 3-34). The concept of parameter space is based on the recognition that parameters can be considered as local coordinates. The parameter space is a flat surface enclosed by a rectangle with dimensions equal to parameter ranges. 

Interpretive methods involve a given set of preplanned experiments from which chromatographic results are used for predicting retention surfaces of individual solutes according to given mathematical or statistical models. The model for the retention surfaces of individual solutes is then used to calculate the response surface over the parameter space explored, and to locate the optimum. 

The essence of the SM technique is approximation of functions g by simple algebraic expressions s within a subset H of parameter space 0. The approximating functions for the responses are obtained using the methodology of the response surface technique [17,19,27], by means of a relatively small number of computer simulations, referred as computer experiments. They are performed at pre-selected combinations of the parameter values and the entire set of these combinations is called a design of computer experiments. The computer experiments are performed using the complete dynamic model (3) and the functions obtained in this manner are referred as surrogate models. 

The proofs of Eqs. (9.59) and (9.60) explicitly rely on the linear character of the model. The above relations are thus only correct under the same conditions. Therefore, one speaks of the joint confidence region under linear assumptions. If the model is not linear in its parameters, the surface in the parameter space defined by Eq. (9.63) no longer is a contour of constant residual sums of squares. Although the probability level is correct, the contour itself has only been approximated. This property provides a qualitative measure of the degree of nonlinearity of the model it is rather simple to determine the coordinates of... 

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