Three-dimensional response surface plot

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.

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. 

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. 

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... 

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. 

Figure 11.4 is a pseudo-three-dimensional representation of a response surface showing a system response, yt, plotted against the two system factors, xx and x2. 

Fig. 3 Application of the Doehlert experimental design to optimize a MIP for propranolol with respect to the type of cross-linker (EDMA or TRIM) and the degree of cross-linking, (a) Three-dimensional representation of response surfaces for the percentage of bound [3H]propanolol to the molecularly imprinted polymer (MIP) and the corresponding non-imprinted control polymer (NIP), (b) Contour plot of the function describing binding of [3H]propanolol to MIPs relative to the degree and the kind (bi or trifunctional) cross-linking. The values were corrected for non-specific binding to the non-imprinted control polymer. Adapted from [31] with kind permission from Springer Science + Business Media...

Response surfaces in more than one dimension (more than one parameter) are hard to visualize. Two representations are common for two-dimensional optimization problems, where the response surface as a function of the two parameters forms a three-dimensional picture. Figure 5.2 shows a pseudo-isometric three-dimensional plot of such a surface (figure 5.2a) as well as a contour plot (figure 5.2b). 

A plot showing the shape of the fitted response surface model is shown in Fig. 3. Three-dimensional projections of response surface models make it easy to see how... 

Fig.3.8 Three-dimensional plot of the estimated response surface in the elimination reaction.

Fig.5.3 Three dimensional plots of the response surfaces yj, bomylamine unreacted...

The ANOVA table indicates that the variation of and produces a systematic variation ofy], and that this variation is probably significantly above the noise level given by the residual sum of squares.. A three-dimensional plot of the response surface is shown in Fig. 6.4. The twist of the plane caused by the interaction is clearly seen. 

Fig.12.12 Contour plots of a response surface when all eigenvalues have the same sign. The three-dimensional isocontours describe a score of concentric ellipsoids.

The contour plots are shown in Figures 10 and 11. The response surface was a three-dimensional hypersurface in a four-dimensional space, which was hardly able to be plotted. Instead, the response surface for spray rate and air temperature—the two dominating factors—was sliced at three levels of the third factor, atomizing air pressure. These were the low, mid, and high factorial levels of 2.3, 2.9, and 3.5 atmospheres (bar) for air pressure. 

Graphical inspection of response surfaces is restricted to three-dimensional plots. How do you plot response surfaces if more than two factors are included in the study ... 

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. 

A response surface is a representation of the output or response of a chemical system or instrument expressed as a function of the relevant independent variables. When possible, the response surface is depicted as a two-dimensional or three-dimensional plot. It is a representation of the instrumental or systemic response as the independent variables are varied, and this representation is widely used in optimization. Much of experimentation can be considered as an exercise in the exploration and exploitation of the relevant response surfaces. 

Data systems for comprehensive GC require special tools extending the available features of most current GC-MS software suites. Because of the 3D matrix of multiple chromatograms and conventional or MS detection, special tools are required to display and evaluate comprehensive GC separations. The time/response data streams are converted to a matrix format for the two-dimensional contour plot providing a colour code of peak intensities (Figure 2.137), or the three-dimensional surface plot generation (Figure 2.138), for visualization. 

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