Myers, Montgomery & Anderson-Cook (2009) Response Surface Methodology

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1 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) ( ) ( ) ( ) ( ) Myers-Montgomery /04/09 RSM ( ). : ( ) 1

2 1 : ( ) * V ar[ŷ(x)]/σ 2 * 2 * *

3 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

4 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

5 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 ± ( III) *4 X X = ±1 ŷ = b 0 + b 1 x 1 + b 2 x 2 + b 3 x *3 X 2 0 *4 2 3 (2 3 )

6 . 5 *5 : 5 y i = β 0 + β i x i + 5 β ij x i x j + ϵ i=1 i<j= ( V) X : (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

7 ? 3 : y i = β 0 + β 1 x i1 + β 2 x i2 + β 3 x i3 + ϵ i (i = 1, 2,..., 8) Figure σ 2 /8 2 σ 2 /8, σ 2 / k N = k + 1 k 2 θ cosθ = 1 k k = 2, N = 3 Figure 7.3 k = 3, N = 4 Figure ( III) k 7

8 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

9 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

10 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= k + k(k 1)/2 6 : k 1 + 2k + k(k 1)/2 * y ŷ(x) 4.1 (central composite designs; CCDs) 3 k 2 V * k α α 0 *7 5 *8 2 y i = β 0 + β 1 x 1 + β 2 x 2 + β 11 x β 22x β 12x 1 x 2 + ϵ *9 2 6, 3 10 *10 V 2 k = 2 k = 3 10

11 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 x k1 1 x 12 x x k x 1N x 2N... x kn *

12 * 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 : M = X X N = = I k k M M ( ) *12 12

13 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

14 0 [iiii]/[iijj] (i j) [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 = * 13 k = 4 ( ) Figure 7.8, Figure 7.9 k = 2 n c = 1 n c = 5 N N V ar[ŷ(x)]/σ k 8 ( ) 3 5 α = k, *13 k = 3, α = 3... k = 3, α =

15 : 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

16 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

17 1 2 Figure Figure Figure SPV pure error 4.6 α α (α = k) 5 ( ) 3 5 (α = 1) k k = 2 k = 3 27 k > Box & Behnken (1960) 3 (bakanced incomplete block design, BIBD) 3, 3 BIBD 1 2 ( 1) 3 17

18 18

19 x 1 x ( ±1 ) x 3 (0) x 1 x 2 x 2 x 3 k = 3 (Box-Behnken design) ( ) k = 4 k = 2 k = 3, k = n c, 24 + n c 14 + n c, 24 + n c (k=4,7 ) Figure

20 4.9 2 (equiradial designs)... Figure

21 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 *

22 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

23 k = (α = 2) k = 3 3 α = ( ) 1 2, 3, 5, 9, 17,... k = 2 2, k = 3 3 k = 2 (α = 1) 23

24 k = 3 k = 4 k = 4, k = ,

25 * 16 *16 25