Space-Filling Designs for High-Dimensional Mixture Experiments with Multiple Constraints

1. Space-Filling Designs for High-Dimensional Mixture Experiments with Multiple Constraints John J. Borkowski Montana State University Bozeman, MT ICAQM 2006 Conference Taipei, Taiwan June 10, 2006

2. OUTLINE • Motivation • Number-theoretic (NT) design generation in the hypercube • Number-theoretic mixture design (NTMD) generation. • The High-Dimensional Multiple-Component Constraint (MCC) Problem • An example: 8 components, 5 MCCs • Final Comments

3. Motivation Constrained mixture experiments: • q components (or ingredients) • xi is the proportion of the ithcomponent for i = 1, 2, … , q and Σxi = 1 • Single-component constraints (SCC) 0 ≤ Li ≤ xi ≤ Ui ≤ 1 • Multiple-component constraints (MCC) Ci ≤ Σ Aixi ≤ Di

4. Motivation (cont.) • The goal is to generate designs with points “scattered uniformly” throughout constrained mixture spaces defined by SCCs and MCCs. • The designs must contain ”boundary” and “interior” points, even for high-dimensional regions (e.g., 8 or more mixture components). • Today • Discuss several number-theoretic (NT) approaches for generating space-filling mixture designs (NTMDs) in highly-constrained regions

5. Notation • x = (x1, x2, … xs) • N = desired design size • Cs = [0,1]s(unit cube) • Ts = {x : ∑xi= 1, xi≥ 0} (simplex) • Ts(a,b) = {xTs: 0 ≤ ai ≤ xi ≤ bi ≤ 1} where a = (a1, …, as) and b = (b1, …, bs) (constrained subspace of simplex)

6. 2. NT-Design Point Generation in Cs • Lattice Point (LP) method • Square root sequence (SRS) method • Powers of the (s+1)st root (PR) method • Cyclotomic field (CF) method • Halton-set(H1)method • Hammersley-set (H2) method LP : form lattices from integers SRS, PR, CF : use the fractional part of a number H1, H2 : based on radical inverses of integers

7. NT-Design Point Generators in Cs(Fang and Wang 1994) Hk= (h1k, h2k, …, hsk) is the NTdesign point generator for the kth design point Xk (where hik depends on the NT-method used) . Forms of the generator Hk = (h1k, h2k, …, hsk) 1. Lattice-point (LP) method: Hk = (kn1, kn2, …, kns) mod N where (i) ni (ii) ni < N (iii) ni≠ njfor i≠j (iv) gcd(N,ni) = 1 2. Square root sequence (SRS) method: Hk = (k√p1, k√p2, … , k√ps ) where (p1, p2, … , ps) are unique primes

8. NT-Design Point Generators in Cs 2. Square root sequence (SRS) method: Hk = (k√p1, k√p2, … , k√ps ) where (p1, p2, … , ps) are unique primes 3. Powers of the (s+1)st root (PR) method: Hk= (kq, kq2, kq3, … , kqs) and q = p 1/(s+1)for some prime p 4. Cyclotomic field (CF) method: Hk = ( kc(p,1) , kc(p,2), … , kc(p,s) ) where c(p,i) = | 2 cos(2πi/p) | for some prime p ≥ 2s+3

9. NT-Design Point Generators in Cs Note: For k,m ,  b0, b1, …, br(<m) such that k = b0 +b1m + b2m2 + … +brmr Let y(k,m) = ∑ ( bi/ mi+1 ), which is called the radical inverse of kwith base m. 5. Halton-set(H1)method: Hk = ( y(k,p1), y(k,p2),…, y(k,ps) ) where the pi are distinct primes 6. Hammersly-set (H2) method: Hk = ( (2k-1)/2N , y(k,p2),…, y(k,ps) )

10. The kth row Xk of NT-design X(k = 1,…,N ) • LPmethod: Xk = ( 2Hk-11s ) / 2N • SRS, PR, and CFmethods Xk = ({Hk}) = ({kn1},{kn2}, … , {kns}) where {kni} is the fractional part of kni • H1andH2methods: Xk = Hk

11. Example: LP-method (N =21, s=2)

12. Example: LP-method (N =21, s=2)

13. What is a “good” NTD generator? • We want the points generated by the design generator to be uniformly “scattered” in cube Cs . • To determine the degree of “uniformity of scatter”, we need an assessment criterion.

14. Two Assessment Criteria Let u1, u2, … , uR be a random sample of vectors from Cs. (evaluation set). Let d(x, xD) = the distance between any x Cs and its nearest design point xD. • Mean-squared distance: msd(X) = (1/R) ∑ d(ui, xD)2 • Maximum-distance: md(X) = maxi [d(ui, xD)] for i=1, 2,…, R Small msd(X) and md(X) imply points in Cs tend to be “close” to the design.

15. Example revisited: N =21 , s=2(LP-method NT-designs, R=15000 points)

16. 3. Number-theoretic mixture designs (NTMDs) • Fang and Yang (2000) provide a mapping G of points in Cq-1 into Tq(a,b). • Reconsider the 21-point, 2-factor NT-designs. • Suppose we want to generate a 21-point NTMD such that .1 ≤ x1 ≤ .7 0≤ x2 ≤ .8 .1 ≤ x3 ≤ .6 • After applying G to the NT- design points in C2 , we get three-component NTMDs in T3(a,b)

17. Application of map G (C2 into T3(a,b))

18. Application of map G (C2 into T3(a,b))

19. Generating NTMDs in Tq(a,b)(no multiple component constraints) • Generate NT-designs in Cq-1 from a set of generators • Apply map G to each NT-design X to generate a NTMD T in Tq(a,b) • Using an evaluation set in Tq(a,b), calculate msd(T) or md(T) for each T • Select the NTMD with the smallest criterion value

20. 4.The High-Dimensional Multiple-Component Constraint (MCC) Problem • For high-dimensional mixture problems, there are often MCCs: Ci≤∑ Aixi ≤ Di • Many points in a NTMD will not satisfy all multiple-component constraints • We need a method to generate NTMDs of specified size N that satisfies all constraints

21. Generating N -point NTMDs with MCCs • Generate NTMDs in Tq(a,b) of size N * > N using one or more of the six NT-methods. • Remove points that do not satisfy the MCCs yielding a NTMD T* • Consider only those NTMD T* designs that contain exactly N points • Generate an evaluation set satisfying all constraints • Calculate msd(T*) and/or md(T*) for each modified NTMD and compare

22. Generation of NTMD T* with MCC: x1- x2 ≥ 0

23. Generation of NTMD T* with MCC: x1- x2 ≥ 0

24. 5. Eight Component MCC Example: Koons (Technometrics 1989) x1: Earthy hematite ore 0≤ x1 ≤ .45 x2: Specular hematite ore 0≤ x2 ≤ .90 x3: Flue dust 0≤ x3 ≤ .35 x4: BOF slag 0≤ x4 ≤ .20 x5: Mill scale 0 ≤ x5 ≤ .30 x6: Dolomite .04 ≤x6≤ .08 x7: Limestone .06 ≤x7 ≤ .12 x8: Coke .029 ≤x8≤ .072

25. Eight Component MCC Example (cont.) 5 Multiple Component Constraints • 0 ≤ -x1 + .5x2 • x3 + x4 + x5 ≤ .35 • 0 ≤ x1 + x2 - x3 - x4 - x5 • .46 ≤ .6x1 + .6x2 + .35x3 + .2x4 + .7x5 • .043 ≤ .17x3 + .85x8 ≤ .085

26. 20-Point, 8-component NTMDs Number of designs evaluated • LP method: 1630 • SRS method: 120 • PR method: 168 • CF method: 133 • H1 method: 120 • H2 method: 84 Evaluation set : 100,000 points

27. √msd values for the 20 Best DesignsMethod: 1=LP 2=SRS 3=PR 4=CF 5=H1 6=H2

28. √msd values (enlarged plot)Method: 1=LP 2=SRS 3=PR 5=H1

29. md values for the 20 Best DesignsMethod: 1=LP 2=SRS 3=PR 4=CF 5=H1 6=H2

30. md values (enlarged plot)Method: 1=LP 2=SRS 3=PR 5=H1

31. Best Designs for Each Method

32. 6. Final Comments • NTMD approach can provide designs in high-dimensional constrained regions with MCCs. • Other assessment criteria may be developed for the mixture problem. • May be able to tweak NTMD points to improve msd(T*) or md(T*).

33. Appendix: From Fang and Yang (2000) • Let G(u,d,Φ,Δ,k) = Δ { 1 - [ u(1-Φ)k + (1-u)(1-d)k ]1/k } • Let Δk = 1 – ( uk+1 + … + uq ) and Δq = 1 dk = max{ak /Δk , 1- (b1+…+bk-1)/Δk} Φk= max{bk /Δk , 1- (a1+…+ak-1)/Δk} where a=(a1, …, aq) and b=(b1, …, bq) are the lower and upper component limits • If u2,…uq is (q -1)-tuple of Unif(0,1) deviates, then ( y1, y2, … , yq ) is a random sample from the uniform distribution on Tq(a,b) where yk = G(uk, dk, Φk, Δk, k-1) k=q, q -1,…,2 y1 = 1 – ( y2 + … + yq ).

34. Selected References: • Fang, K.-T. & Wang Y. (1994) Number Theoretic Methods in Statistics, Chapman and Hall, London. • Fang, K.-T. & Yang, Z.-H. (2000) “On Uniform Design of Experiments with Restricted Mixtures and Generation of Uniform Distribution on Some Domains.”, Stat. and Prob. Letters, 46: 113-120.

35. Website This PowerPoint presentation can be found at my website: www.math.montana.edu/~jobo/ppt/index.html