3.1.5 Central Composite Designs

These types of experimental design are frequently used together with response models of the second order. The design consists of three types of points:

axial points The $2 \cdot n$ axial points are created by a Screening Analysis (see Section 3.1.3).

cube points The 2ⁿ cube points come from a Full Factorial design (see Section 3.1.4)

center point A single point in the center is created by a Nominal design (see Section 3.1.2).

For the further description of the designs, the ranges -- minimum and maximum values -- of the control parameters are scaled to [-1,+1]. A graphic of a three dimensional Central Composite Circumscribed (CCC) design is shown in Figure 3.3. Here the axial points are located on a hyper-cube with the radius b_i. The cube build by the cube points has side-length of $2 \cdot b_i$.

Figure 3.3: The example points of a Central Composite Circumscribed design with three input parameters.

All three types of Central Composite designs, (Central Composite Circumscribed (CCC), Central Composite Inscribed (CCI), and Central Composite Face-centered (CCF)), have the same structure shown in Figure 3.3 but with other values for a_i and b_i. These values of the three Central Composite designs are listed in Table 3.2.

Table 3.2: Factors a and b for Central Composite designs with full factorial cube points.

	$\frac{b_i}{a_i}$	rangei
CCC	$\sqrt{n}$	2 ai
CCI	$\sqrt{n}$	2 bi
CCF	1	2 ai = 2 bi

Two of these designs -- CCC and CCI -- have a special characteristic; they are rotatable. A design is said to be rotatable, if upon rotating the design points about the center point the moments of the distribution of the design remain unchanged. For rotatable Central Composite designs the factor $\frac{b_i}{a_i}$ must be $\sqrt{n}$.

This table is only valid for a single center point^3.1. This is not a big restriction for numerical simulation because the the results of two simulations with the same input data have to be the same.

The rotatability and the small number of necessary experiments make CCC and CCI designs very well suited for estimating the coefficients in a second order model as will be explained in Section 3.2.

Footnotes

... point^3.1

For more than one center-point the factor of $\frac{b_i}{a_i}$ is given in [29].