Dimensional factors

The friction factor, f, fp/ in. must be obtained from Figure 10-138. Because it is not a dimensional factor, the Ap, relations take this into account. 

In Fig. 31.1a these scores are used as the coordinates of the four wind directions in 2-dimensional factor-space. From this so-called score plot one observes a large degree of association between the wind directions of 90, 180 and 270 degrees, while the one at 0 degrees stemds out from the others. 

These loadings have been used for the construction of the so-called loading plot in Fig. 31.1b which shows the positions of the three trace elements in 2-dimensional factor-space. The elements Na and Cl are clearly related, while Si takes a position of its own in this plot. 

Suppose you are given the task of preparing a ternary (three-component) solvent system such that the total volume be 1.00 liter. Write the equality constraint in terms of x X2, and Xj, the volumes of each of the three solvents. Sketch the three-dimensional factor space and clearly draw within it the planar, two-dimensional constrained feasible region. (Hint try a cube and a triangle after examining Figure 2.16.)... 

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. 

Figure 12.2 Location of a single experiment in two-dimensional factor space.

Figure 12.2 is a graphic representation of a portion of two-dimensional factor space associated with the system shown in Figure 12.1. In this illustration, the domain of factor (the horizontal axis ) lies between 0 and +10 similarly, the domain of factor X2 (the vertical axis ) lies between 0 and +10. The response axis is not shown in this representation, although it might be imagined to rise perpendicularly from the intersection of the factor axes (at jCj = 0, X2 = 0). Figure 12.2 shows the location in factor space of a single experiment at jc, = +3, X21 = +7.

Figure 12.3 is a pseudo-three-dimensional representation of a portion of the three-dimensional experiment space associated with the system shown in Figure 12.1. The two-dimensional factor subspace is shaded with a one-unit grid. The factor domains are again 0 < Xj < +10 and 0 < < +10. The response axis ranges from 0...

Figure 12.8 Factor combinations for a T factorial experimental design in three-dimensional factor space.

Figure 12.17 Upper left panel contours of constant response in two-dimensional factor space. Upper right panel a subset of the contours of constant response. Lower left panel canonical axes translated to stationary point of response surface. Lower right panel canonical axes rotated to coincide with principal axes of response surface.

Write a table similar to Table 12.3 for three-factor systems. What do the possible isoresponse contours look like in three-dimensional factor space [See, for example. Box (1954).]... 

In this chapter we investigate the interaction between experimental design and information quality in two-factor systems. However, instead of looking again at the uncertainty of parameter estimates, we will focus attention on uncertainty in the response surface itself. Although the examples are somewhat specific (i.e., limited to two factors and to full second-order polynomial models), the concepts are general and can be extended to other dimensional factor spaces and to other models. 

As we will see in Section 14.11, fractional factorial designs are often used to look for important factors. If it turns out that one or more factors is unimportant (say factor JCj), then a fractional factorial design can be collapsed into a less fractional design in a lower-dimensional factor space. Figure 14.6 shows an example of collapsing the design. 

The experimental design in two-dimensional factor space is shown in Figure 15.2. A simple model that would account for both [Na ] and time is... 

The optimum response is found within the factor space. Consider an n + 1 dimensional space in which the factor space is defined by n axes, and the final dimension (y in two dimensions, and z in three) is the response. Any combination of factor values in the n dimensional factor space has a response associated with it, which is plotted in the last dimension. In this space, if there are any optima, one optimum value of the response, called the global optimum, defines the goal of the optimization. In addition, there may be any number of responses, each of which is, within a sublocality of the factor space, better than any other response. Such a value is a local optimum. ... 

As explained in the previous section, truncatable here means that the excess free energy / depends only on K moment densities p,. Note that, in the first (ideal) term of (6), we have included a dimensional factor R(a) inside the logarithm. This is equivalent to subtracting T dap a) In R(a) from the free energy. Since this term is linear in densities, it has no effect on the exact thermodynamics it contributes harmless additive constants to the chemical potentials p a). However, in the projection route to a moment free energy, it will play a central role. 

Figure 11.2 is a graphic representation of a portion of two-dimensional factor space associated with the system shown in Figure 11.1. In this illustration, the...

Figure 11.3 is a pseudo-three-dimensional representation of a portion of the three-dimensional experiment space associated with the system shown in Figure 11.1. The two-dimensional factor subspace is shaded with a one-unit grid. The factor domains are again 0 < xl < +10 and 0 < x2 < +10. The response axis ranges from 0 to +8. The location in factor space of the single experiment at xn = +3, x2l = +7 is shown as a point in the plane of factor space. The response (yn = + 4.00) associated with this experiment is shown as a point above the plane of factor space, and is connected to the factor space by a dotted vertical line.

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