Comparison with Spherical CCD

0.0447 −0.0068 0.0183 −1.8387 2.1370 −0.2239 −0.0006 −0.0069

−0.4135 0.0187 −0.0283 −0.6461 −0.8190 −2.5961 0.0016 −0.0118 0.5810 −0.7941 0.3617 1.8885 1.4977 −1.0446 −0.0084 −0.0136

−2.6638 0.3541 0.4499 0.5339 0.5253 0.1233 −0.0120 0.0140

−0.5402 −1.1908 −2.5037 0.0863 0.1083 0.0491 −0.0054 0.0003

−0.3187 −2.4136 1.1819 −0.5854 −0.4815 0.3180 −0.0100 0.0123

−0.0086 −0.0105 0.0014 0.0061 0.0075 −0.0032 2.7846 0.4957 0.0176 0.0070 −0.0058 0.0008 0.0073 −0.0181 −0.4956 2.7845



8. These points are different from the original star points of the spherical CCD clearly.

• n₀ = 4

When n₀ = 4, the sixteen locally D-optimal star points are

±

−0.1154 0.0969 0.1144 −0.1490 −0.0345 2.8179 −0.0006 −0.0069 1.1404 −0.6068 1.0294 −1.9710 1.1759 −0.0641 0.0016 −0.0118 2.1037 −1.2997 −0.0935 1.0461 −0.8659 0.1793 −0.0084 −0.0136 0.7290 0.4968 −2.5103 −0.9406 0.1742 0.0671 −0.0120 0.0140 0.8122 1.8104 0.7427 −0.7243 −1.7241 −0.1185 −0.0054 0.0003

−1.0341 −1.5520 −0.2541 −1.2607 −1.6921 −0.0660 −0.0100 0.0123 0.0090 −0.0041 −0.0079 −0.0045 −0.0103 0.0022 2.7896 0.4669 0.0141 −0.0039 0.0191 0.0075 0.0091 0.0071 −0.4668 2.7895



8. These points are different from the original star points of the spherical CCD clearly.

4.3 Comparison with Spherical CCD

We employ the relative locally D-efficiency in (1) to compare our conditionally locally D-optimal central composite designs and the spherical CCD’s. The relative locally D-efficiency of our conditionally locally D-optimal central composite design

and spherical CCD is defined as det¡

M_b⁻¹(ξ_D^∗)¢ det¡

M_b⁻¹(ξ_CCD)¢,

where ξ_D^∗ is our conditionally locally D-optimal central composite design with respect to b⁽⁰⁾ and ξCCD is the spherical CCD . If the relative locally D-efficiency is less than 1, our conditionally locally D-optimal central composite design is better than the corresponding spherical CCD. That is the lower relative locally D-efficiency is, the better our conditionally locally D-optimal central composite design we get. Based on the different first-order design with n0 center points, the relative locally D-efficiencies are shown as follows. For each k, we have tried the cases of different b⁽⁰⁾ by choosing values of R and angels φj, j = 1, · · · , k − 1. Given a first-order design (included the factorial design and n₀ center points) and R, the best and the worst values of the relative locally D-efficiencies are shown in Tables 2 to 9. The results based on full factorial designs are in Tables 2 to 5, and the results in Tables 6 to 9 are the cases that we choose fractional factorial designs in the first-order designs.

R\k 2 3 4 5 6 7 8

0.25√

k 0.9920 0.9847 0.9777 0.9698 0.9595 0.9546 0.9516 (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) 0.5√

k 0.9423 0.9070 0.8870 0.8727 0.8686 0.8657 0.8689 (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) 0.75√

k 0.8935 0.8436 0.8241 0.8137 0.8152 0.8230 0.8357 (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000)

√k 0.8618 0.8072 0.7903 0.7839 0.7888 0.8032 0.8215 (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) Table 2. The relative locally D-efficiencies of our conditionally locally D-optimal cen-tral composite designs and the spherical CCD’s for n₀ = 1 and 2^kfull factorial design as the first-order design. In parentheses, the values, 1.0000, which indicate the highest rel-ative locally D-efficiencies show our conditionally locally D-optimal central composite designs never work worse than the spherical CCD’s.

R\k 2 3 4 5 6 7 8 0.25√

k 0.9965 0.9919 0.9874 0.9829 0.9775 0.9721 0.9681 (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) 0.5√

k 0.9668 0.9439 0.9275 0.9148 0.9068 0.9011 0.9047 (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) 0.75√

k 0.9314 0.8979 0.8782 0.8684 0.8624 0.8589 0.8748 (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000)

√k 0.9066 0.8685 0.8503 0.8432 0.8392 0.8336 0.8598 (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) Table 3. The relative locally D-efficiencies of our conditionally locally D-optimal cen-tral composite designs and the spherical CCD’s for n0 = 2 and 2^kfull factorial design as the first-order design. In parentheses, the values, 1.0000, which indicate the highest rel-ative locally D-efficiencies show our conditionally locally D-optimal central composite designs never work worse than the spherical CCD’s.

R\k 2 3 4 5 6 7 8

0.25√

k 0.9977 0.9948 0.9920 0.9888 0.9848 0.9799 0.9703 (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) 0.5√

k 0.9767 0.9581 0.9447 0.9321 0.9266 0.9203 0.9194 (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) 0.75√

k 0.9493 0.9225 0.9044 0.8907 0.8851 0.8802 0.8910 (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000)

√k 0.9288 0.8981 0.8800 0.8695 0.8631 0.8565 0.8773 (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) Table 4. The relative locally D-efficiencies of our conditionally locally D-optimal cen-tral composite designs and the spherical CCD’s for n0 = 3 and 2^kfull factorial design as the first-order design. In parentheses, the values, 1.0000, which indicate the highest rel-ative locally D-efficiencies show our conditionally locally D-optimal central composite designs never work worse than the spherical CCD’s.

R\k 2 3 4 5 6 7 8 0.25√

k 0.9978 0.9960 0.9934 0.9908 0.9884 0.9840 0.9752 (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) 0.5√

k 0.9768 0.9661 0.9453 0.9420 0.9377 0.9307 0.9292 (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) 0.75√

k 0.9591 0.9349 0.9198 0.9044 0.9033 0.8985 0.9012 (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000)

√k 0.9417 0.9151 0.8978 0.8859 0.8842 0.8720 0.8892 (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) (1.0000) Table 5. The relative locally D-efficiencies of our conditionally locally D-optimal cen-tral composite designs and the spherical CCD’s for n0 = 4 and 2^kfull factorial design as the first-order design. In parentheses, the values, 1.0000, which indicate the highest rel-ative locally D-efficiencies show our conditionally locally D-optimal central composite designs never work worse than the spherical CCD’s.

R\k 5(2⁵⁻¹_V ) 6(2⁶⁻¹_{V I} ) 7(2⁷⁻¹_{V II}) 8(2⁸⁻²_V ) 0.25√

k 0.9695 0.9610 0.9531 0.9484 (1.0000) (1.0000) (1.0000) (1.0000) 0.5√

k 0.8682 0.8567 0.8459 0.8330 (1.0000) (1.0000) (1.0000) (1.0000) 0.75√

k 0.8043 0.7957 0.7914 0.7806 (1.0000) (1.0000) (1.0000) (1.0000)

√k 0.7706 0.7645 0.7639 0.7524 (1.0000) (1.0000) (1.0000) (1.0000)

Table 6. The relative locally D-efficiencies of our conditionally locally D-optimal cen-tral composite designs and the spherical CCD’s for n0 = 1 and fractional factorial design as the first-order design. In parentheses, the values, 1.0000, which indicate the highest relative locally D-efficiencies show our conditionally locally D-optimal central composite designs never work worse than the spherical CCD’s.

R\k 5(2⁵⁻¹_V ) 6(2⁶⁻¹_{V I} ) 7(2⁷⁻¹_{V II}) 8(2⁸⁻²_V ) 0.25√

k 0.9844 0.9814 0.9747 0.9717 (1.0000) (1.0000) (1.0000) (1.0000) 0.5√

k 0.9106 0.9071 0.8945 0.8880 (1.0000) (1.0000) (1.0000) (1.0000) 0.75√

k 0.8616 0.8584 0.8488 0.8427 (1.0000) (1.0000) (1.0000) (1.0000)

√k 0.8371 0.8333 0.8278 0.8198 (1.0000) (1.0000) (1.0000) (1.0000)

Table 7. The relative locally D-efficiencies of our conditionally locally D-optimal cen-tral composite designs and the spherical CCD’s for n0 = 2 and fractional factorial design as the first-order design. In parentheses, the values, 1.0000, which indicate the highest relative locally D-efficiencies show our conditionally locally D-optimal central composite designs never work worse than the spherical CCD’s.

R\k 5(2⁵⁻¹_V ) 6(2⁶⁻¹_{V I} ) 7(2⁷⁻¹_{V II}) 8(2⁸⁻²_V ) 0.25√

k 0.9910 0.9872 0.9822 0.9803 (1.0000) (1.0000) (1.0000) (1.0000) 0.5√

k 0.9429 0.9293 0.9175 0.9130 (1.0000) (1.0000) (1.0000) (1.0000) 0.75√

k 0.9007 0.8884 0.8771 0.8736 (1.0000) (1.0000) (1.0000) (1.0000)

√k 0.8779 0.8666 0.8569 0.8535 (1.0000) (1.0000) (1.0000) (1.0000)

Table 8. The relative locally D-efficiencies of our conditionally locally D-optimal cen-tral composite designs and the spherical CCD’s for n0 = 3 and fractional factorial design as the first-order design. In parentheses, the values, 1.0000, which indicate the highest relative locally D-efficiencies show our conditionally locally D-optimal central composite designs never work worse than the spherical CCD’s.

R\k 5(2⁵⁻¹_V ) 6(2⁶⁻¹_{V I} ) 7(2⁷⁻¹_{V II}) 8(2⁸⁻²_V ) 0.25√

k 0.9932 0.9902 0.9861 0.9844 (1.0000) (1.0000) (1.0000) (1.0000) 0.5√

k 0.9536 0.9417 0.9303 0.9270 (1.0000) (1.0000) (1.0000) (1.0000) 0.75√

k 0.9173 0.9054 0.8944 0.8914 (1.0000) (1.0000) (1.0000) (1.0000)

√k 0.8915 0.8861 0.8763 0.8739

(1.0000) (1.0000) (1.0000) (1.0000)

Table 9. The relative locally D-efficiencies of our conditionally locally D-optimal cen-tral composite designs and the spherical CCD’s for n₀ = 4 and fractional factorial design as the first-order design. In parentheses, the values, 1.0000, which indicate the highest relative locally D-efficiencies show our conditionally locally D-optimal central composite designs never work worse than the spherical CCD’s.

From above tables, our composite designs usually perform better than spherical CCD’s.

We also can identify that when φ_j = 0, j = 1, · · · , k − 1, i.e. the extremum point, b⁽⁰⁾ = (0, · · · , 0, R), is the axial point, spherical CCD’s are also the best designs we are looking for. When R increases, the relative locally D-efficiencies decrease. Roughly the relative locally D-efficiencies decrease when k increases. Based on the same R and k, roughly the relative locally D-efficiencies of the fractional factorial designs (Tables 6 to 9) in the first-order design are lower than the full factorial designs (Tables 2 to 5) in the first-order design. The relative locally D-efficiencies are lower for the cases with fewer center points. Thus given small number of center points and a possible extremum point, locally D-optimal criterion should be adopted to search the possible additional star points to improve the coefficient estimation.

5 Conclusion

In this thesis, we study that which 2k additional experimental points, i.e. star points, should be added when the first-order design and center points are given. Fol-lowing the same structure of CCD, we find the symmetric and orthogonal 2k additional star points according to the locally D-optimal criterion. Because the design space can be described as a k-ball, we employ the polar coordinates to represent all experimental points. Then we rewrite the star points in terms of Givens rotation matrix. Hence our goal is transferred into an optimization problem. When k is larger, a simulated annealing algorithm, the Best Angles and Radius sampler, is employed to find the 2k star points numerically. In Section 4, based on the different first-order designs, the details are discussed and given a comparison with the spherical CCD’s.

References

[1] Box, G. E. P. and Hunter, J. S. (1957). Multi-factor Experimental Designs for Exploring Response Surfaces. The Annals of Mathematical Statistics, 28, 195-241.

[2] Box, G. E. P. and Wilson, K. B. (1951). On the Experimental Attainment of Optimum Conditions. Journal of the Royal Statistical Society, Ser. B, 13, 1-45.

[3] Chen, R.-B. and Lin, D. K. J. (2009). Conditionally Optimal Star Points in Central Composite Designs for Response Surface Methodology. Submitted.

[4] Cheng, R. C. H., Melas, V. B., and Pepelyshev, A. N. (2000). Optimal design for evaluation of an extremum point. In: Atkinson, A., Bogacka, B., and Zhigljavsky, A. (eds), Optimum Design 2000, Kluwer, 15-24.

[5] Melas, V. B., Pepelyshev, A. N., and Cheng, R. C. H. (2003). Designs for estimat-ing an extremal point of quadratic regression models in a hyperball. Metrika, 58.

193-208.

A Appendix: The Proof of Theorem

Suppose the first-order design contains a 2^k full factorial design and n₀ center points. Let b⁽⁰⁾ be the given extremum point and s^∗₁, · · · , s^∗_2k be the star points of the locally D-optimal design, ξ^∗_D, for b⁽⁰⁾. There exist a vector fb⁽⁰⁾ = (bf⁽⁰⁾₁ ,bf⁽⁰⁾₂ , · · · ,bf⁽⁰⁾_k ) with bf⁽⁰⁾_i ≥ 0, i = 1, 2, · · · , k, and the corresponding fs_m, m = 1, 2, · · · , 2k, such that ³

bf⁽⁰⁾´_T

= G¡ b⁽⁰⁾¢_T

and (fs_m)^T = G (s^∗_m)^T, where G = G(ϑ), ϑ = (ϑ₁₂, · · · , ϑ_1k, ϑ₂₃, · · · , ϑ_2k, · · · , ϑ_(k−1)k), is the rotation matrix and can be represented as the prod-uct of the Givens rotation matrices, Q

1≤i<j≤kG_ij(ϑ_ij), with ϑ_ij ∈ (^π₂)n, n ∈ Z. Then e

s₁, · · · , fs_2k are the additional star points of the locally D-optimal design for fb⁽⁰⁾.

<proof>

In our design, we fix the supports of the first-order design on particular positions before adding star points. If we want to keep the supports of the first-order design after rotation, ϑ_ij should be suggested as a value which belongs to a multiple of ^π₂.

Let

ξ_D^∗ =





0 , c1 , · · · , c₂^k , s^∗₁ , · · · , s^∗_2k n₀/n , 1/n , · · · , 1/n , 1/n , · · · , 1/n



and

ξ =e





0 , ec1 , · · · , ec₂^k , es^∗₁ , · · · , es^∗_2k n₀/n , 1/n , · · · , 1/n , 1/n , · · · , 1/n



,

where ec_i = Gc_i, i = 1, · · · , 2^k and the set {c₁, · · · , c₂^k} is equal to the set {ec₁, · · · , ec₂^k}.

According to ξ_D^∗ is the conditionally locally D-optimal design for b⁽⁰⁾, we want to show ξ is the conditionally locally D-optimal design for fe b⁽⁰⁾. First, consider the original notation,

f_b(0)(x) = (f_b^T(0)(1)(x), f_b^T(0)(2)(x))^T, where

f_b^T(0)(1)(x) = ((b⁽⁰⁾₁ − x₁), · · · , (b⁽⁰⁾_k − x_k)),

f_b^T(0)(2)(x) = ((b⁽⁰⁾₁ − x₁)², · · · , (b⁽⁰⁾_k − x_k)², 2(b⁽⁰⁾₁ − x₁)(b⁽⁰⁾₂ − x₂), · · · , 2(b⁽⁰⁾₁ − x₁)(b⁽⁰⁾_k − x_k), 2(b⁽⁰⁾₂ − x₂)(b⁽⁰⁾₃ − x₃), · · · , 2(b⁽⁰⁾₂ − x₂)(b⁽⁰⁾_k − x_k), · · · , 2(b⁽⁰⁾_k−1− x_k−1)(b⁽⁰⁾_k − x_k), 1),

and

M₁(ξ_D^∗) = R

f_b(0)(1)(x)f_b^T(0)(1)(x)ξ^∗_D(dx), M₂(ξ_D^∗) = R

f_b⁽⁰⁾₍₂₎(x)f_b^T(0)(1)(x)ξ^∗_D(dx), M₃(ξ_D^∗) = R

f_b(0)(2)(x)f_b^T(0)(2)(x)ξ^∗_D(dx),

M_b⁽⁰⁾(ξ_D^∗) = M₁(ξ_D^∗) − M₂^T(ξ_D^∗)M₃⁻¹(ξ^∗_D)M₂(ξ^∗_D).

The number of supports, n0+ 2^k+ 2k, is large than estimated parameters, ^(k+2)(k+1)₂ . Thus M is a full rank matrix. So the inverse, M₃⁻¹(ξ_D^∗), exists surely.

Let

f_bg(0)(ex) = (f_g^T

b⁽⁰⁾(1)(ex), f_g^T

b⁽⁰⁾(2)(ex))^T, where

f_g^T

b⁽⁰⁾(1)(ex) = (( fb⁽⁰⁾1 − ex1), · · · , ( fb⁽⁰⁾k− exk)), f_g^T

b⁽⁰⁾(2)(ex) = (( fb⁽⁰⁾₁ − ex₁)², · · · , fb⁽⁰⁾_k− ex_k)², 2( fb⁽⁰⁾₁− ex₁)( fb⁽⁰⁾₂− ex₂), · · · , 2( fb⁽⁰⁾₁− ex₁)( fb⁽⁰⁾_k− ex_k), 2( fb⁽⁰⁾₂− ex₂)( fb⁽⁰⁾₃− ex₃), · · · ,

2( fb⁽⁰⁾2− ex2)( fb⁽⁰⁾k− exk), · · · , 2( fb⁽⁰⁾k−1− exk−1)( fb⁽⁰⁾k− exk), 1), f_bg(0)(1)(ex) = L₁(ϑ)f_b⁽⁰⁾₍₁₎(x),

f_bg(0)(2)(ex) = L2(ϑ)f_b⁽⁰⁾₍₂₎(x), L₁(ϑ) =Q

1≤i<j≤kG_ij(ϑ_ij) with ϑ_ij ∈ (^π₂)n, n ∈ Z, is a k × k rotation matrix, L₂(ϑ) is a ^k2+k+2₂ × ^k2+k+2₂ transformation matrix and a function of L₁(ϑ).

Here L1(ϑ) and L2(ϑ) are also invertible. The proof we show in final part, (**).

Then

M₁(eξ) = Z

f_bg(0)(1)(ex)f_g^T

b⁽⁰⁾(1)(ex)eξ(dex)

= Z

L1(ϑ)f_b⁽⁰⁾₍₁₎(x)¡

L1(ϑ)f_b⁽⁰⁾₍₁₎(x)¢_T

ξ_D^∗(dx)

= L₁(ϑ) Z

f_b(0)(1)(x)f_b^T(0)(1)(x)ξ_D^∗(dx)L^T₁(ϑ)

= L₁(ϑ)M₁(ξ_D^∗)L^T₁(ϑ), M₂(eξ) =

Z

f_bg(0)(2)(ex)f_g^T

b⁽⁰⁾(1)(ex)eξ(dex)

= Z

L₂(ϑ)f_b⁽⁰⁾₍₂₎(x)¡

L₁(ϑ)f_b⁽⁰⁾₍₁₎(x)¢_T

ξ_D^∗(dx)

= L₂(ϑ) Z

f_b⁽⁰⁾₍₂₎(x)f_b^T(0)(1)(x)ξ_D^∗(dx)L^T₁(ϑ)

= L₂(ϑ)M₂(ξ_D^∗)L^T₁(ϑ),

M3(eξ) = first-order terms, L₁(ϑ).

Because the determinant of any Givens rotation is 1, det

In other words, eξ is the conditionally locally D-optimal design for fb⁽⁰⁾. Thus es₁, · · · , fs_2k are the additional star points of the conditionally locally D-optimal design for fb⁽⁰⁾. ¤

(**) Without loss of generality, the general forms are

L₁(ϑ) = Y

Q₄=













g₁₁g₂₂+ g₁₂g₂₁ · · · g_1pg_2q+ g_1qg_2p · · · g_1(k−1)g_2k+ g_1kg_2(k−1)

... . .. ... . .. ...

g_p1g_q2+ g_p2g_q1 · · · g_ppg_qq+ g_pqg_qp · · · g_p(k−1)g_qk+ g_pkg_q(k−1)

... . .. ... . .. ...

g_(k−1)1g_k2+ g_(k−1)2g_k1 · · · g_(k−1)pg_kq+ g_(k−1)qg_kp · · · g_{(k−1)(k−1)}g_kk+ g_(k−1)kg_k(k−1)











 ,

is a ^k(k−1)₂ × ^k(k−1)₂ matrix, 1 ≤ n ≤ k and 1 ≤ p < q ≤ k.

Because any Givens rotation is invertible and orthogonal, the product,Q

1≤i<j≤kGij(ϑij), is invertible and orthogonal, too. Consider the given condition, ϑ_ij ∈ (^π₂)n, n ∈ Z, Gij(ϑij) becomes a ”signed permutation matrix”, i.e. in a square matrix, there is ex-actly one nonzero entry in each row and each column and 0’s elsewhere, where the nonzero entries are ±1. It means any product of two different entries in each row or each column is ”zero”. Then the product, Q

1≤i<j≤kG_ij(ϑ_ij), is a signed permutation matrix i.e. L₁(ϑ) is a signed permutation matrix. Thus we get that Q₁ is a permuta-tion matrix, Q₂ and Q₃ are zero matrices with different dimension. Now we focus on Q₄. In Q₄, each row vector is related to each two different row vectors in L₁(ϑ). The property of permutation matrix can help us to get a result that there is exactly one nonzero entry, ±1, in each row of Q₄ and 0’s elsewhere and each row is different. It means Q₄ is a signed permutation matrix. As a result, L₂(ϑ) is a signed permutation matrix, too. Because a signed permutation matrix is invertible, L₁(ϑ) and L₂(ϑ) are invertible.

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