Response contour lines

We can obtain a two-dimensional representation of the modeled surface by drawing its response contour lines, that is, lines of constant response. The contour lines of a plane are line segments. For example, if we let y = 70 in Eq. (6.3) we arrive at the expression... 

Fig. 6.3. Response contour lines for the plane described by Eq. (6.3). The arrow indicates the steepest ascent path starting at the center of the design. The values in parentheses are the experimentally determined responses.

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. 

Figure 24.5 (a) Streamlines at a bend are asymmetrically distributed across the river, (b) Contour lines of equal velocity in a cross section indicate that flow velocities strongly vary laterally as well as from the water surface to the bottom. Such variations are responsible for longitudinal dispersion. 

Point M in the figure is the optimum that by one definition of the research problem objective should be determined. Each intersection line in the plane is a line of constant response values and is called contour lines-contour diagram. 

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 5.7 Illustration of a two-dimensional Simplex optimization. Dotted lines are contour lines figures represent the response. Figure adapted from ref. [505]. Reprinted with permission.

Figure 14 A model calculation of the 2D-IR spectra of a idealized system of two coupled vibrators. The frequencies of these transitions were chosen as 1615 cm-1 and 1650 cm-1, the anharmonicity as A = 16 cm the coupling as = 7 cm and the homogeneous dephasing rate as T2 = 2 ps. The direction of both transitions as well as the polarization of the pump and the probe pulse were set perpendicular. The spectral width of the pump pulses was assumed 5 cm-1. The figure shows (a) the linear absorption spectrum and (b) the nonlinear 2D spectrum. In the 2D spectra, light gray colors and solid contour lines symbolize regions with a positive response, while negative signals are depicted in dark gray colors and with dashed contour lines.

Figure 16 Absorption spectrum of the cycfo-Mamb-Abu-Arg-Gly-Asp in D2O. The dashed line shows a representative spectrum of the pump pulses (width 12 cm-1) utilized to generate the 2D-IR spectra. (b,c) 2D pump-probe spectra of the same sample measured with the polarization of the probe pulse perpendicular and parallel to the polarization of the pump pulse, respectively. The dashed contour lines mark regions where the difference signal is negative (bleach and stimulated emission), while the solid contours lines mark regions where the response is positive (excited state absorption). The most prominent off-diagonal bands are marked by arrows. (d,e,f) A global least-squares fit of the experimental data, used to refine the coupling Hamiltonian in Equation (29c). (From Ref. 42.)...

The coefficients estimated from these 9 experiments are given in column B of table 10.11, where they may be compared with the values estimated from the entire data set of 42. The solid contour lines on figure 10.8 represent the predictions of the reduced design, dotted lines the predictions of the full data set. The two response surfaces are very close to one another. 

RSM for independent variables and mixture components 2D contour plots with overlay of contour lines for different responses, and limits of the domain. 3D plot of response surface... 

Figure 4.1 Response versus factors surface). By using search methods, plot In the case of response surface the response is measured along a methods (RSMs), the response is search path, here along a simplex described by a mathematical model path (cf. "Analytical Performance dotted contour lines of the response Characteristics" Section).

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... 

Contour plots allow the relationship between significant variables and responses to be visualised. These plots resemble topographical maps in that contour lines are drawn on a two dimensional plane to represent the surface of a response variable. This allows a highly visual, easily interpreted picture to be used to understand the process or system being studied. Thus, in the example given in Reference 31, it is very clear that the lowest particle size is achieved using intermediate bead dimensions and that it is essentially independent of pump rate. 

Instead of using a response surface graph, one often uses a contour plot. This translates the response surface in the same way as a geographical map of a mountainous area. The isoresponse lines can be viewed as the contour lines on the map. 

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