D-optimal composite designs

These added points keep symmetric properties, and can be considered as the star points which are rotated with rotation angles. Hence we employ Givens rotation matrix to represent them.

Example: two factors、three factors

From the above cases, the optimal added points are on the surface of the design space, k-ball with radius . Hence in the following cases of more than 3 factors, we assume all added points are on sphere, i.e. r = k, and the coordinates of these added support points can be obtained by multiplying the Given rotation matrices G_ij(θ_ij), i < j.

Definition 1. Let G_ij(θ_ij) be a k-dimensional Givens rotation matrix which represents a rotation in the plan spanned by the axes vector x_i and x_j , i < j. This matrix is an orthogonal matrix, and θ_ij is the rotation angles about the two axes x_i and x_j. The G_ij(θ_ij) is a correction of the identity matrix I_k with G_ij(θ_ij) (i ; i) = G_ij(θ_ij) (j ; j) = cosθ_ij and G_ij(θ_ij) (i ; j) = - G_ij(θ_ij) (j ; i)= sinθ_ij.

Then 2k symmetric support points can be represented by

where D_k(b) is a k × k diagonal matrix with the diagonal elements, b.

Example: four factors、five factors、six factors、seven factors、eight factors、nine factors

Table 1 displays the point efficiencies of the D-optimal composite designs, and these values are equal to or close to the point efficiencies of the corresponding CCDs. Therefore, we would conjecture that star points are D-optimal added points.

Factor	Parameter	N	Best Peff	Peff of CCD
2	6	9	0.6285	0.6285
3	10	15	0.7116	0.7116
4	15	25	0.7673	0.7673
5	21	27	0.8002	0.8002
6	28	45	0.8383	0.8384
7	36	79	0.8545	0.8547
8	45	81	0.8785	0.8787
9	55	147	0.7843	0.7845

Table 1. The P_eff value of optimal composite designs for k = 2, …, 9.