Three-dimensional response surface

Univariate optimization is a common way of optimizing simple processes, which are affected by a series of mutually independent parameters. For two parameters such a simple problem is illustrated in figure 5.3a. In this figure a contour plot corresponding to the three-dimensional response surface is shown. The independence of the parameters leads to circular contour lines. If the value of x is first optimized at some constant value of y (line 1) and if y is subsequently optimized at the optimum value observed for x, the true optimum is found in a straightforward way, regardless of the initial choice for the constant value of y. For this kind of optimization problem univariate optimization clearly is an attractive method. 

FIGURE 11 Three-dimensional response surface plot of release rate as function of composition of coating solution. 

Figure 5 An additive model three-dimensional response surface plot for fines from the milling study.

The predicted data were processed by Minitab Release 14 Statistical Software (Minitab Inc., Pennsylvania, USA) to produce a three-dimensional response surface using the grid layout described in section 7.2.5. 

Fig. 6A.11. Three-dimensional response surface representation of the diuron degradation as a function of potassium ferrioxalate and hydrogen peroxide concentrations.

Three-dimensional response surface plots were generated for each quality parameter. Calculation of optimal synthesis conditions for optimum IB strength and FE was performed using a multiple response method designated as desirability [22-24]. This optimization method incorporates desired values and priorities for each variable. 

For three-dimensional optimization processes (leading to four-dimensional response surfaces) graphical presentation in a single two-dimensional figure is no longer possible. Any solution to this problem is bound to be difficult to construct and to interpret. A series... 

The response surface plot is a three-dimensional space surface which is formed by the response value of interaction of test factors. The effects of test factors on the response value can be found by analyzing the response surface. The effect of interaction on DS is shown in Fig. 5.2. In the plot, the interaction between reaction time and reaction temperature had a major effect on the DS for the quick drop of the response surface and the serried contour hne. Dialysis time had less effect on the DS. The design point was gained under the reaction time 15 h, reaction temperature 30°C, dialysis time 8 h and the DS was 4.0. Figure 5.3 shows the effects of interaction on yield of HP-/3-CDs. The contour line of dialysis... 

Scatter plots are the most common type of graph used to show relationships between a dependent variable (responses) and independent variables (factors). Judiciously selecting and sizing the symbols in scatter plots allows one to communicate trends and the measured values associated error. Frequently, to show the effect of more than one independent variable, different symbol types or colors can be used. Three-dimensional plots, surface plots, and contour plots are becoming more common to illustrate the effect of two or more factors on the dependent variable. Bar charts (and pie charts) are popular for presentations, and histograms are useful to compare the distribution of populations. Ternary plots are used in thermodynamics and to demonstrate explosion limits of different compositions of gases. 

Fig. 43 Three-dimensional contour surfaces of the valence MOs responsible for the cyclic delocalization of the election density in the Uj and U5 rings of the cyc/o-U X ( = 3, 4 X = OH, NH) clusters (figures in italics are the orbital energies, in eV). Reprinted with permission from [185]. Copyright ACS Journal Archives...

Example of a two-factor response surface displayed as (a) a pseudo-three-dimensional graph and (b) a contour plot. Contour lines are shown for intervals of 0.5 response units. 

Note These equations are from Doming, S. N. Morgan, S. L. Experimental Design A Chemometric Approach. Elsevier Amsterdam, 1987, and pseudo-three-dimensional plots of the response surfaces can be found in their figures 11.4, 11.5, and 11.14. The response surface for problem (a) also is shown in Color Plate 13. 

A receptor is a surface membrane component, usually a protein, which regulates some biological event in response to reversible binding of a relatively small molecule40 . The precise three-dimensional structures of the binding sites of receptors still remain unknown today. Thus, this section mainly describes the correlation of shape similarity between the molecules which would bind to a given receptor with their biological activity. 

Figure 1.6 Impact frustration (a) the Chicxulub crater, seen as a three-dimensional gravity map, thought to be responsible for the extinction of the dinosaurs (b) the cratered surface of the Moon. (Reproduced from photos by courtesy of NASA)...

As for the mechanical response of thin lipid films, surface pressure(fl)-surface area(A) characteristics of lipid monolayer at air/water interface have been well studied under quasi-static conditions. It has been established that different phases are observed for the ensemble of lipid molecules in a two-dimensional arrangement, similarly to the gas, liquid, and solid phases and some other intermediate phases as in three-dimensional molecular assemblies. 

Attaching the catalyst molecules to the electrode surface presents an obvious advantage for synthetic and sensor applications. Catalysis can then be viewed as a supported molecular catalysis. It is the object of the next section. A distinction is made between monolayer and multilayer coatings. In the former, only chemical catalysis may take place, whereas both types of catalysis are possible with multilayer coatings, thanks to their three-dimensional structure. Besides substrate transport in the bathing solution, the catalytic responses are then under the control of three main phenomena electron hopping conduction, substrate diffusion, and catalytic reaction. While several systems have been described in which electron transport and catalysis are carried out by the same redox centers, particularly interesting systems are those in which these two functions are completed by two different molecular systems. 

In Chapter 2 it was seen that a response surface for a one-factor system can be represented by a line, either straight or curved, existing in the plane of two-dimensional experiment space (one factor dimension and one response dimension). In two-factor systems, a response surface can be represented by a true surface, either flat or curved, existing in the volume of three-dimensional experiment space (two factor dimensions and one response dimension). By extension, a response surface associated with three- or higher-dimensional factor space can be thought of as a hypersurface existing in the hypervolume of four- or higher-dimensional experiment space. 

---