Myers, R.H., Montgomery, D.C. & Anderson-Cook, C.M. (2009) Response Surface Methodology, Third Edition. Chapter 7. Experimantal Designs for Fitting Response Surfaces - I. (response surface methodology) ( ) ( ) ( ) ( ) 1 2 3 Myers-Montgomery 7... 2013/04/09 RSM ( ). : ( ) 1
1 : 1. 2. 3. 4. 5. ( ) 6. 7. 8. * 1 9. 10. V ar[ŷ(x)]/σ 2 * 2 *1 4.10 *2 3.4 2
2 6 y x 1, x 2,..., x k E(y) = f(x, θ) f R(x) R (region of interest) R O(x) (region of operability) R O Figure 7.1 R current best guess ŷ(x) f(x, θ) 2.1 E(y) = X 1 β 1 E(y) = X 1 β 1 + X 2 β 2 β 1 b 1 : E(b 1 ) = β 1 + Aβ 2 A = (X 1X 1 ) 1 X 1X 2 A (alias matrix) E(ŷ) E(y) = Rβ 2 3
R = X 1 A X 2 s 2 6 (lack-of-fit test) E(s 2 ) = σ 2 + β 2R Rβ 2 p 1 s 2 Rβ 2 β 2 y j E[ŷ j f(x j, β)] 2 E[ŷ j f(x j, β)] 2 = V ar(ŷ j ) [Bias(ŷ j )] 2 V ar(ŷ j ) [Bias(ŷ j )] 2 β 2R Rβ 2 = [Bias(ŷ j )] 2 p 1 j=1 V ar(ŷ j ) [Bias(ŷ j )] 2 4
3 x 1, x 2,..., x k y N 1 +1 : ŷ i = b 0 + b 1 x i1 + b 2 x i2 + + b k x ik V ar(b i ) 3.1 X X X *3 : N j = 1, 2,..., k X j [ 1, +1] j = 1, 2,..., k V ar(b i )/σ 2 x i i = 1, 2,..., k ±1. 3 8 2 3 1 ( III) *4 X X = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ±1 ŷ = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 3 2 3 3.2 *3 X 2 0 *4 2 3 (2 3 ) 2 3 2 3 1 5
. 5 *5 : 5 y i = β 0 + β i x i + 5 β ij x i x j + ϵ i=1 i<j=2 2 5 2 5 1 ( V) X : 16 16 (saturated) R 2 100% *6 ( 0 ) *5 y i = β 0 + β 1 x 1 + (3 ) + β 5 x 5 + β 12 x 1 x 2 + (8 ) + β 45 x 4 + x 5 + ϵ X *6 1 6
? 3 : y i = β 0 + β 1 x i1 + β 2 x i2 + β 3 x i3 + ϵ i (i = 1, 2,..., 8) Figure 7.2 3 1 3 σ 2 /8 2 σ 2 /8, σ 2 /4 1 2 1 3 3 3.3 - k N = k + 1 k 2 θ cosθ = 1 k k = 2, N = 3 Figure 7.3 k = 3, N = 4 Figure 7.4 2 3 1 ( III) k 7
3.4 - : P V (x) = V ar[ŷ(x)] = σ 2 x (m) (X X) 1 x (m) x (m) x (1) = [1, x 1, x 2,..., x k ] k = 2 x (1) = [1, x 1, x 2, x 1 x 2 ] P V (x) 8
SP V (x) = N V ar[ŷ(x)] σ 2 x ρ x = Nx (m) (X X) 1 x (m) SP V (x) = 1 + ρ 2 x SPV 9
4 (1 ) 2 (fractions) *7 *8 k k y = β 0 + β i x i + β ii x 2 i + k β ij x i x j + ϵ i=1 i=1 i<j=2 1 + 2k + k(k 1)/2 6 : 1. 3 2. k 1 + 2k + k(k 1)/2 *9 10 8 y ŷ(x) 4.1 (central composite designs; CCDs) 3 k 2 V * 10 2 2k α α 0 *7 5 *8 2 y i = β 0 + β 1 x 1 + β 2 x 2 + β 11 x 2 1 + +β 22x 2 2 + β 12x 1 x 2 + ϵ *9 2 6, 3 10 *10 V 2 k = 2 k = 3 10
n c pure error * 11 Figure 7.5 k = 2, α = 2 Figure 7.6 k = 3, α = 3 α n c 4.2 (design moments) X = 1 x 11 x 21... x k1 1 x 12 x 22... x k1....... 1 x 1N x 2N... x kn *11 2 11
* 12 : (first moments): [i] = 1 N N u=1 x iu (second pure moments): [ii] = 1 N N u=1 x2 iu (second mixed moments): [ij] = 1 N N u=1 x iux ju i j (odd moments) (even moments) (moment matrix) M = X X N 2 k : 1 0... 0 M = X X N = 0 1.... = I k 0 1 2 k M M ( ) *12 12
3.4 : SP V (x) = Nx (m) (X X) 1 x (m) ( X ) 1 X SP V (x) = x (m) x (m) = x (m) M 1 x (m) N (design rotatability) 2 N V ar[ŷ(x)]/σ 2 N V ar[ŷ(x)]/σ 2 ( α n c ) ( ) Appendix 1. 0 [i] = 0 (i = 1, 2,..., k) [ij] = 0 (i j, i, j = 1, 2,..., k) [ii] = λ 2 (i = 1, 2,..., k) λ 2 III 2 ±1 λ 2 = 1 x ρ x SP V (x) = 1 + ρ 2 x 13
0 [iiii]/[iijj] (i j) 3 4.3 0 [iiii]/[iijj] F ( F = 2 k ) [iiii] [iijj] = F + 2α4 F 3 α = 4 F k = 2 α = 2 2/4 = 2 = 1.414, k = 3 α = 2 3/4 = 1.682 * 13 k = 4 ( ) Figure 7.8, Figure 7.9 k = 2 n c = 1 n c = 5 N N V ar[ŷ(x)]/σ 2 3.5 k 8 ( ) 3 5 α = k, 3 5 8 *13 k = 3, α = 3... k = 3, α = 3 12 6 24 24 12... 14
4.4 3.4 : P V (x) = V ar[ŷ(x)] = σ 2 x (m) (X X) 1 x (m) σ 2 SP V (x) = Nx (m) (X X) 1 x (m) 15
4.2 SPV M = X X N ( ) N N SPV UP V (x) = V ar[ŷ(x)]/σ 2 = x (m) (X X) 1 x (m) SPV UPV σ 2 (MSE) UPV MSE EP V = MSE x (m) (X X) 1 x (m) 4.5 (cuboidal design region) α = 1 (face-centered cube, FCD) Figure 7.11 16
1 2 Figure 7.12 0 Figure 7.13 1 Figure 7.14 3 SPV pure error 4.6 α α 4.7 2 (α = k) 5 ( ) 3 5 (α = 1) 3 1 2 3 k k = 2 k = 3 27 k > 3 4.8 Box & Behnken (1960) 3 (bakanced incomplete block design, BIBD) 3, 3 BIBD 1 2 ( 1) 3 17
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x 1 x 2 2 2 ( ±1 ) x 3 (0) x 1 x 2 x 2 x 3 k = 3 (Box-Behnken design) ( ) k = 4 k = 2 k = 3, k = 4 12 + n c, 24 + n c 14 + n c, 24 + n c (k=4,7 ) Figure 7.17 3 5 19
4.9 2 (equiradial designs)... Figure 7.20 2 20
2 4 4.10 2 2 I = AB 2 1/2 * 14 y i = β 0 + β 1 x i1 + β 2 x i2 + δ 1 z i1 + δ 2 z i2 + ϵ i δ 1, δ 2 z i1, z i2 y i 1 z i1 = 1, z i1 = 0 X z 2 z 1 y i = β 0 + β 1 x i1 + β 2 x i2 + δ 1 (z i1 z 1 ) + ϵ i X *14 4 2 +1 1 2 21
4 (z u1 z 1 )x u1 = 0 u=1 4 (z u1 z 1 )x u2 = 0 u=1 * 15 k b k k y u = β 0 + β i x ui + β ii x 2 ui + k b β ij x ui x uj + δ m (z um z m ), i=1 i=1 i<j=2 m=1 u = 1, 2,..., N z mu u m 1 N x ui (z um z m ) = 0, i = 1, 2,..., k, m = 1, 2,..., b u=1 N x ui x uj (z um z m ) = 0, i j, m = 1, 2,..., b u=1 N x 2 ui(z um z m ) = 0, i = 1, 2,..., k, m = 1, 2,..., b u=1 [i] = 0, [ij] = 0 ( 2 ) x ui = 0 x ui x uj = 0(i j) 3 m N u=1 x2 ui z um = ẑ m N u=1 x2 ui *15 x x 1u, x 2u 22
k = 2 2 2 (α = 2) k = 3 3 α = 3 2 2 3 2 ( ) 1 2, 3, 5, 9, 17,... k = 2 2, k = 3 3 k = 2 (α = 1) 23
k = 3 k = 4 k = 4, k = 5 1 3 5, 2 4 6 24
* 16 *16 25