SpaceFilling Designs for HighDimensional Mixture Experiments with Multiple Constraints

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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

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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

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Motivation

• Constrained mixture experiments
• q components (or ingredients)
• xi is the proportion of the ith component
• for i 1, 2, , q and S xi 1
• Single-component constraints (SCC)
• 0 Li xi Ui 1
• Multiple-component constraints (MCC)
• Ci S Aixi Di

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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

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Notation

• x (x1, x2, xs)
• N desired design size
• Cs 0,1s (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)

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2. NT-Design Point Generation in Cs

• Lattice Point (LP) method
• Square root sequence (SRS) method
• Powers of the (s1)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

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NT-Design Point Generators in Cs (Fang and Wang
1994)

• Hk (h1k, h2k, , hsk) is the NT design 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
  lt N
• (iii) ni ? nj for i?j (iv)
  gcd(N,ni) 1
• 2. Square root sequence (SRS) method
• Hk (kvp1, kvp2, , kvps )
• where (p1, p2, , ps) are unique primes

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NT-Design Point Generators in Cs

• 2. Square root sequence (SRS) method
• Hk (kvp1, kvp2, , kvps )
• where (p1, p2, , ps) are unique primes
• 3. Powers of the (s1)st root (PR) method
• Hk (kq, kq2, kq3, , kqs )
• and q p 1/(s1) 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(2pi/p) for some
  prime p 2s3

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NT-Design Point Generators in Cs

• Note For k,m ? ?, ? b0, b1 , , br (ltm) such
  that
• k b0 b1m b2m2 brmr
• Let y(k,m) ? ( bi / mi1 ), which is called
  the
• radical inverse of k with 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) )

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The kth row Xk of NT-design X (k 1,,N )

• LP method
• Xk ( 2Hk-11? s ) / 2N
• SRS, PR, and CF methods
• Xk (Hk) (kn1, kn2, , kns)
  
• where kni is the fractional part of kni
• H1 and H2 methods Xk Hk

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Example LP-method (N 21, s2)
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Example LP-method (N 21, s2)
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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.

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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 i1, 2,,
  R
• Small msd(X) and md(X) imply points in Cs tend
  to be close to the design.

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Example revisited N 21 , s2(LP-method
NT-designs, R15000 points)
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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)

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

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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

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Generating N -point NTMDs with MCCs

• Generate NTMDs in Tq(a,b) of size N gt 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

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Generation of NTMD T with MCC x1- x2 0
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Generation of NTMD T with MCC x1- x2 0
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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

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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 .17 x3 .85x8 .085

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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

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vmsd values for the 20 Best DesignsMethod 1LP
2SRS 3PR 4CF 5H1 6H2
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vmsd values (enlarged plot)Method 1LP 2SRS
3PR 5H1
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md values for the 20 Best DesignsMethod 1LP
2SRS 3PR 4CF 5H1 6H2
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md values (enlarged plot)Method 1LP 2SRS
3PR 5H1
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Best Designs for Each Method
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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).

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Appendix From Fang and Yang (2000)

• Let G(u,d,F,?,k) ? 1 - u(1-F)k
  (1-u)(1-d)k 1/k
• Let ?k 1 ( uk1 uq ) and ?q 1
• dk maxak /?k , 1- (b1bk-1)/?k
• Fk maxbk /?k , 1- (a1ak-1)/?k
• 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, Fk, ?k, k-1) k q, q
  -1,,2
• y1 1 ( y2 yq ).

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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.

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Website

• This PowerPoint presentation can be found at my
  website
• www.math.montana.edu/jobo/ppt/index.html