Central composite designs rotatability

The lower left panel in Figure 13.2 shows the central composite design in the two factors X, and X2. The factor domain extends from -5 to +5 in each factor dimension. The coordinate axes in this panel are rotated 45° to correspond to the orientation of the axes in the panel above. Each black dot represents a distinctly different factor combination, or design point. The pattern of dots shows a central composite design centered at (Xj = 0, Xj = 0). The factorial points are located 2 units from the center. The star points are located 4 units from the center. The three concentric circles indicate that the center point has been replicated a total of four times. The experimental design matrix is... 

Figure 13.3 Rotatable central composite design. Square points 2, star points 2 2, DF f = 3, DF = 3.

The rotatable central composite design in Figure 13.7 is related to the rotatable central composite design in Figure 13.3 through expansion by a factor of V2 the square points expand from 2 to 2 2 from the center the star points expand from 2 2 to 4 from the center. The experimental design matrix is... 

It can be shown that for a central composite design to be rotatable the... 

A possible disadvantage of the central composite design is that it requires five levels of each variable (0, 1, a). In some situations it might be necessary or preferable to have only three different settings of the variables. In this case a can be chosen to be 1 and the design is called a face-centered composite design. These designs are not rotatable. 

The rotatable feature of the central composite designs makes it possible to complete a balanced portion of the design, evaluate the results and possibly shift the design to another area in some of the variables. The shift in area of interest pivots on some of the runs already obtained and these become part of the new experimental design. 

Figure 3.13. Central composite designs for (a) two and (b) three factors. The design in panel a is rotatable.

Table 3.7. Levels for a three-factor, rotatable, central composite design.

Table 2.33 Position of the axial points for rotatability and orthogonality for central composite designs with varying number of replicates in the centre.

We examine the effect of the choice of a on the isovariance curves of a 2 factor central composite design with 3 centre points, shown in figure 5.8. The rotatability of the design with a = 1.414 can clearly be seen in figure 5.8c. 

It is identical to the 2-factor central composite design. The matrix is "almost orthogonal" if no extra centre-points are added. It is not rotatable, nor is its precision uniform in the experimental domain. 

Values of a for rotatable central composite designs. The total number of runs is... 

Three central composite designs that can be sequentially conducted in blocks and still preserve rotatability... 

A complete three-factor central composite design is depicted in Table 4.10 and Figure 4.12. The distance of the star points a from the center can be differently chosen. For a uniformly rotatable design,... 

A rotatable central composite design is often used in this case. This design consists of 2 " factorial design combined with a number of repeated tests in the central (zero) point plus 2k star points. In the central composite factorial design, the independent variables are varied at five levels that are called a zero level, +1 and -1 levels, and two star levels. The +1 and -1 levels provide an increase or decrease in the level of the factors, from its zero level, Xq, by one increment, 5j. Star levels provide an increase or decrease in the level of the factor from its zero level by one increment multiplied by a coefficient, a, which depends on the number of variables in the design. The value of a can be calculated as a = 2 (Cochran and Cox, 1957). An example of the two-variable rotatable central composite design is shown in Table 1.2. 

Table 1.2 Levels of variables in the two-variable rotatable central composite design...

The face-centered central composite design is not rotatable, but it does not require as many center points as the spherical CCD =2 3 is sufficient to provide a good variance... 

A rotatable central composite design is useful if it is desired to have a common variance for points equidistant from the centre. 

For the experimentation, a rotatable central composite design is selected. The process variables with their values... 

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