Uniform Designs and Their Constructions Yu Tang Soochow University - - PowerPoint PPT Presentation

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Uniform Designs and Their Constructions

Yu Tang

Soochow University

• Apr. 23, 2015

Yu Tang (Soochow University) @ Shanghai Jiaotong University

• Apr. 23, 2015

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Content

Brief introduction to uniform design

Background Measure of uniformity — discrepancy

Combinatorial properties of uniform designs

Three examples

Construction methods

Via combinatorial configuration By optimization approach Using level permutation

Conclusion

Yu Tang (Soochow University) @ Shanghai Jiaotong University

• Apr. 23, 2015

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Background

§ Motivated by a system engineering project (1978) ♠ 6 factors, each with at least 12 levels ♠ the number of runs cannot exceed 50 ♠ no orthogonal array is available § Uniform design ♠ proposed by Professor Yuan Wang and Professor Kai-Tai Fang ♠ solve the above problem using 31 runs § Widely applied in many fields manufacturing system engineering pharmaceutics natural sciences · · · · · ·

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• Apr. 23, 2015

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Achievements

Number of papers on uniform designs published The second prize of the 2008 National Natural Science Award

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• Apr. 23, 2015

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Three cases with 16 points over a unit square domain

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛

△

❛ ❛ ❛ ❛ ❛ ❛ ❛

△

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛

×

❛

△ ×

• ne time

two times three times *from Fang and Ma (2001)

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Star discrepancy

Number theory (quasi-Monte Carlo) method Star discrepancy (Weyl (1916)) D(Pn) = max x∈C s

• N(Pn, [0, x))

n − Vol([0, x))

• ,

[0, x) denotes the hypercube [0, x1) × · · · × [0, xm) N(Pn, [0, x)) represents the number of points of Pn falling in [0, x) Vol(A) means the volume of A

D1 = D2 = D3 = 0.23438.

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Modified Lp-discrepancy

Modified Lp-discrepancy

Wrap-around L2-discrepancy (Hickernell (1998a))

WD2

2 =

• u=∅
• C2u

N(Pu ∩ Jw(x′

u, xu))

n − Vol(Jw(x′

u, xu))

2 dx′

udxu,

Centered L2-discrepancy (Hickernell (1998b))

CD2

2 =

• u=∅
• Cu

N(Pu ∩ Jw(xu)) n − Vol(Jw(xu)) 2 dxu

· · · · · ·

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Closed form

§ Wrap-around L2-discrepancy (Hickernell 1998a)

(WD2(P))2 = − 4

3

m +

1 n2 n

• k=1

n

• j=1

m

• i=1

3 2 − |xki − xji| (1 − |xki − xji|)

• .

§ Centered L2-discrepancy (Hickernell 1998b)

(CD2(P))2 =

1 n2 n

• k=1

n

• j=1

m

• i=1
• 1 + 1

2

• xki − 1

2

• + 1

2

• xji − 1

2

• − 1

2 |xki − xji|

• − 1

n n

• k=1

m

• i=1
• 1 − 1

2

• xki − 1

2

• − 1

2

• xki − 1

2

• 2

+ 13 12 m .

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Numerical results

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛

△

❛ ❛ ❛ ❛ ❛ ❛ ❛

△

❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛

×

❛

△ ×

• ne time

two times three times W1 = 0.02789 < W2 = W3 = 0.03423. C1 = 0.01138 < C2 = 0.01236 < C3 = 0.01797.

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• Apr. 23, 2015

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Content

Brief introduction to uniform design

Background Measure of uniformity — discrepancy

Combinatorial properties of uniform designs

Three examples

Construction methods

Via combinatorial configuration By optimization approach Using level permutation

Conclusion

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• Apr. 23, 2015

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Example 1

dij = ♯{(xik, xjk) : xik = xjk, k = 1, · · · , m} row 1 2 3 4 5 1 1 1 1 1 1 2 2 2 2 2 1 3 1 1 2 2 2 4 2 2 1 1 2 5 1 2 1 2 3 6 2 1 2 1 3 7 1 2 2 1 4 8 2 1 1 2 4 = ⇒ d12 = 1 d34 = 1 d56 = 1 d78 = 1 d13 = 2 d14 = 2 d23 = 2 · · · · · ·

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Uniform designs under the discrete discrepancy

Theorem 1

(Fang, Lin and Liu (2003); Fang, Ge, Liu and Qin (2004); Fang, Lu, Tang and Yin (2004)) Let X be a U-type design U(n; q1 × · · · × qm). Denote ˜ γ = m

i=1n/qi − m

n − 1 and γ = ⌊˜ γ⌋ where ⌊x⌋ denotes the integer part of x. Then D2(X; a, b) = −

m

• j=1
• a+(qj−1)b

qj

• + am

n + 2bm n2 n−1

• i=1

n

• j=i+1

a

b

dij , (1)

n−1

• i=1

n

• j=i+1

a

b

dij ≥ n(n − 1)[(γ + 1 − ˜ γ) a

b

γ + (˜ γ − γ) a

b

γ+1], (2) and the lower bound on the right hand side of (2) can be achieved if and only if all dijs (i = j) take the same value γ, or take only two values γ and γ + 1.

*discrete discrepancy — proposed in Hickernell and Liu (2002)

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Example 2

d0

ij = ♯{(xik, xjk) : xik = xjk, k = 1, · · · , m};

d1

ij = ♯{(xik, xjk) : |xik − xjk| = 1 or q − 1, k = 1, · · · , m};

· · · · · · dq/2

ij

= ♯{(xik, xjk) : |xik − xjk| = q/2, k = 1, · · · , m} row 1 2 3 4 5 6 7 1 4 2 3 3 1 2 1 2 1 4 2 3 3 1 2 3 2 1 4 2 3 3 1 4 1 2 1 4 2 3 3 5 3 1 2 1 4 2 3 6 3 3 1 2 1 4 2 7 2 3 3 1 2 1 4 8 4 4 4 4 4 4 4 = ⇒ d0

12 = 1

d0

13 = 1

d0

23 = 1

· · · · · · d1

12 = 4

d1

13 = 4

d1

23 = 4

· · · · · · d2

12 = 2

d2

13 = 2

d2

23 = 2

· · · · · · Fij = (d0

ij, d1 ij, . . . , dq/2 ij

)

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Uniform designs under the wrap-around L2 discrepancy

Theorem 2

(Fang, Tang and Yin (2005) A lower bound of the wrap-around L2-discrepancy on U(n; qm) with even q and odd q is given by ∆ + n−1

n

3

2

m(n−q)

q(n−1) 5

4

• mn

q(n−1)

3 2 − 2(2q−2) 4q2

• 2mn

q(n−1) · · ·

• 3

2 − (q−2)(q+2) 4q2

• 2mn

q(n−1) ;

∆ + n−1

n

3

2

m(n−q)

q(n−1)

3 2 − 2(2q−2) 4q2

• 2mn

q(n−1) · · ·

• 3

2 − (q−1)(q+1) 4q2

• 2mn

q(n−1) ,

respectively, where ∆ = − 4 3 m + 1 n 3 2 m . A U-type design U(n; qm) is a uniform design under the wrap-around L2-discrepancy, if its all Fij distributions, i = j, are the same. In this case, the WD2-value of this design achieves the above lower bound.

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Example 3

di = ♯{xik = 2, k = 1, · · · , m}; dij = ♯{(xik, xjk) : xik = xjk = 2, k = 1, · · · , m}

row 1 2 3 4 5 6 1 2 3 1 3 3 2 2 1 1 1 2 1 2 3 3 2 2 1 3 1 4 3 1 3 2 2 3 5 1 3 2 1 2 3 6 2 2 3 3 1 1 = ⇒ d1 = 4 d2 = 4 d3 = 4 d4 = 4 d5 = 4 d6 = 4 d12 = 1 d13 = 1 d14 = 0 · · · · · ·

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Uniform designs under the centered L2 discrepancy

Theorem 3

(Fang, Maringer, Tang and Winker (2006)) For a U-type design U(n, 3m), let µ = 2 3m

• , γ =

2m(n − 3) 9(n − 1)

• , and denote nµ = (µ + 1)n − 2mn

3 and nγ = (γ + 1)n(n − 1) 2 − mn(n − 3) 9 . If f (µ) ≥ f (0), then

CD2(P)2 ≥ ( 13 12 )

m

− 2 n

• nµ(

10 9 )

µ

+ (n − nµ)( 10 9 )µ+1

• +

1 n2

• nµ(

4 3 )

µ

+ (n − nµ)( 4 3 )µ+1

• +

2 n2

• nγ(

4 3 )

γ

+ ( n(n − 1) 2 − nγ)( 4 3 )γ+1

• .

(3)

The lower bound can be obtained if and only if nµ dis take the value µ, n − nµ dis take the values µ + 1 and nγ dijs take the value γ, n(n−1)

2

− nγ dijs take the value γ + 1, for all i = j. f (x) = 1 3 4 3 x − 2n 9 10 9 x

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Content

Brief introduction to uniform design

Background Measure of uniformity — discrepancy

Combinatorial properties of uniform designs

Three examples

Construction methods

Via combinatorial configuration By optimization approach Using level permutation

Conclusion

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• Apr. 23, 2015

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Constructions of uniform designs

Previous methods

good lattice method (Wang and Fang (1981) Latin square method (Fang, Shiu and Pan (1999)) expending orthogonal design method (Fang (1995))

Combined with combinatorial properties

constructions by combinatorial configurations combinatorial optimization level permutation

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Combinatorial approach

Combinatorial configurations have been proved to be very useful in the constructions of uniform designs Resolvable pairwise balanced designs (including resolvable balanced incomplete block designs), resolvable packing designs and resolvable covering designs have been extensively used to construct uniform designs under the discrete discrepancy

• M. Q. Liu and R. C. Zhang (2000)
• X. Lu, W. B. Hu and Y. Zheng (2003)
• K. T. Fang, G. N. Ge and M. Q. Liu (2002)
• K. T. Fang, G. N. Ge, M. Q. Liu and H. Qin (2003)
• K. T. Fang, X. Lu, Y. Tang and J. X. Yin (2004)

Perfect resolvable balanced incomplete block design (perfect PBIBD) to construct uniform designs under the wrap-around L2 discrepancy (Fang, Tang and Yin (2005))

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An example of RBIBD

A perfect (12, 3, 2; 6)-RBIBD can be formed by the following 11 parallel classes (point set V = {0, 1, . . . , 10, ∞}):

P1 = {0, 1,3}, {2, 6, 8}, {4, 5, 9}, {7, 10, ∞} P2 = {1,2, 4}, {3, 7, 9}, {5, 6, 10}, {8, 0,∞} P3 = {2, 3, 5}, {4, 8, 10}, {6, 7, 0}, {9, 1,∞} P4 = {3, 4, 6}, {5, 9, 0}, {7, 8, 1}, {10, 2, ∞} P5 = {4, 5, 7}, {6, 10, 1}, {8, 9, 2}, {0,3, ∞} P6 = {5, 6, 8}, {7, 0,2}, {9, 10, 3}, {1,4, ∞} P7 = {6, 7, 9}, {8, 1,3}, {10, 0,4}, {2, 5, ∞} P8 = {7, 8, 10}, {9, 2, 4}, {0, 1,5}, {3, 6, ∞} P9 = {8, 9, 0}, {10, 3, 5}, {1,2, 6}, {4, 7, ∞} P10 = {9, 10, 1}, {0,4, 6}, {2, 3, 7}, {5, 8, ∞} P11 = {10, 0,2}, {1,5, 7}, {3, 4, 8}, {6, 9, ∞} For t = 1, 2, . . . , q 2

• , each pair x, y ∈ V is t-apart in the same number of

parallel classes

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Uniform design constructed by perfect RBIBD

U(12; 411) derived from the perfect (12, 3, 2; 6)-RBIBD

row 1 2 3 4 5 6 7 8 9 0 1 1 4 3 2 4 2 3 3 1 2 1 1 1 1 4 3 2 4 2 3 3 1 2 2 2 1 1 4 3 2 4 2 3 3 1 3 1 2 1 1 4 3 2 4 2 3 3 4 3 1 2 1 1 4 3 2 4 2 3 5 3 3 1 2 1 1 4 3 2 4 2 6 2 3 3 1 2 1 1 4 3 2 4 7 4 2 3 3 1 2 1 1 4 3 2 8 2 4 2 3 3 1 2 1 1 4 3 9 3 2 4 2 3 3 1 2 1 1 4 10 4 3 2 4 2 3 3 1 2 1 1 ∞ 4 4 4 4 4 4 4 4 4 4 4

⇐ =

{0, 1, 3}, {2, 6, 8}, {4, 5, 9}, {7, 10, ∞} {1,2, 4}, {3, 7, 9}, {5, 6, 10}, {8, 0,∞} {2, 3, 5}, {4, 8, 10}, {6, 7, 0}, {9, 1,∞} {3, 4, 6}, {5, 9, 0}, {7, 8, 1}, {10, 2, ∞} {4, 5, 7}, {6, 10, 1}, {8, 9, 2}, {0, 3, ∞} {5, 6, 8}, {7, 0, 2}, {9, 10, 3}, {1, 4, ∞} {6, 7, 9}, {8, 1,3}, {10, 0,4}, {2, 5, ∞} {7, 8, 10}, {9, 2, 4}, {0, 1, 5}, {3, 6, ∞} {8, 9, 0}, {10, 3, 5}, {1, 2, 6}, {4, 7, ∞} {9, 10, 1}, {0,4, 6}, {2, 3, 7}, {5, 8, ∞} {10, 0,2}, {1,5, 7}, {3, 4, 8}, {6, 9, ∞}

dt

ij = ♯{(xik, xjk) : |xik − xjk| = t or q − t, k = 1, · · · , m}

Fij = (d0

ij, d1 ij, . . . , dq/2 ij

)

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Existence results

Theorem 4

For any prime n = q1q2 − 1, there exists a (n + 1, q1, (q1 − 1)(n − 1); 2q1(n − 1))-PRBIBD, hence we can obtain a uniform design U(n + 1; qn(n−1)

2

) under WD2.

Theorem 5

For any prime n = q1q2 − 1 ≡ 3 (mod 4), there exists a (n + 1, q1, (q1 − 1) n−1

2 ; q1(n − 1))-PRBIBD, hence we can obtain a

uniform design U(n + 1; q

n(n−1) 2

2

) under WD2.

Theorem 6

For any odd prime q, there exists a (q, 1, 0; 2)-PRBIBD, hence we can

• btain a uniform design U(q; qq−1) under WD2.

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Optimization approach

Many existing uniform designs including designs listed on the uniform design homepage are obtained by implementing a threshold accepting (TA) algorithm The TA algorithm utilizes a random optimization approach, thus the result of a better design will depend more on luck and the iteration times for refinement We combine the basic optimization framework with combinatorial requirements for specific discrepancies and propose a new combinatorial optimization algorithm, called balance-pursuit heuristic (Fang, Maringer, Tang and Winker (2006)) This new algorithm will be fast and more efficient than the existing threshold accepting heuristic, as it can set up a final best status for the searching goal according to the inner combinatorial structure

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Algorithm using optimization approach Searching low-discrepancy designs under CD2

1 Initialize nR, nSr and the sequence of thresholds τr, r = 1, 2, . . . , nR 2 Generate starting design U0 ∈ U(n, qm) 3 for r = 1 to nR do 4 for i = 1 to nSr do 5 Randomly generate U1 ∈ N(U0) 6 if CD2(U1) < CD2(U0) × (1 + τr) then 7 U0 = U1 8 end if 9 end for 10 end for

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Optimization methods for uniform designs

· · · ⇛ 3 2 3 3 2 1 3 2 3 1 2 2 2 2 3 1 1 2 2 3 1 3 3 2 2 3 2 1 3 1 2 1 3 3 1 2 2 3 1 3 1 2 1 1 1 1 1 3 ⇛ 3 2 3 3 1 1 3 2 3 1 2 2 2 2 3 1 1 2 2 3 1 3 3 2 2 3 2 1 3 1 2 1 3 3 1 2 2 3 1 3 1 2 1 1 2 1 1 3 ⇛ · · · · · · ⇛ CD2

2 = 0.051835

⇛ CD2

2 = 0.051460

⇛ · · ·

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Construction using optimization approach

Threshold accepting algorithm

Winker and Fang (1998) Fang and Ma (2001a,b) Fang, Ma and Winker (2002) Fang, Lu and Winker (2003)

Balance pursuit algorithm (combinatorial optimization)

• K. T. Fang, Y. Tang and J. X. Yin,

Journal of Complexity, 2005.

• K. T. Fang, D. Maringer, Y. Tang and P. Winker,

Mathematics of Computation, 2006.

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Illustration of the balance-pursuit algorithm

3 2 3 3 14.000000 2 1 3 2 13.000000 3 1 2 2 13.111111 2 2 3 1 13.000000 1 2 2 3 13.111111 1 3 3 2 14.000000 2 3 2 1 13.000000 3 1 2 1 14.111111 3 3 1 2 14.111111 2 3 1 3 14.222222 1 2 1 1 14.111111 1 1 1 3 15.333333 ⇛ 3 2 3 3 14.000000 1 1 3 2 14.111111 3 1 2 2 13.111111 2 2 3 1 13.000000 1 2 2 3 13.000000 1 3 3 2 14.111111 2 3 2 1 13.000000 3 1 2 1 14.111111 3 3 1 2 14.111111 2 3 1 3 14.222222 1 2 1 1 14.000000 2 1 1 3 14.111111 σi =

• j=i

(4 3)dij

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Content

Brief introduction to uniform design

Background Measure of uniformity — discrepancy

Combinatorial properties of uniform designs

Three examples

Construction methods

Via combinatorial configuration By optimization approach Using level permutation

Conclusion

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Combinatorial isomorphic designs

Table: Three regular 33−1 designs

D1 D2 D3 1 2 1 1 1 2 1 2 2 2 2 1 1 1 1 2 1 1 1 2 1 1 1 1 1 1 2 1 2 1 1 2 2 2 2 2 2 1 2 1 2 1 1 2 1 2 2 2 1 2 2 2 2 2

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Graph

(D1) (D2) (D3)

Figure: Points of three 33−1 designs

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Geometrically isomorphic

Geometrically isomorphic: if one can be obtained from the other by variable exchange and/or reversing the level order of one or more factors Geometrically isomorphic ⇒ same centered L2-discrepancy φ(D) = 1 N2

N

• i=1

N

• j=1

n

• k=1
• 1 + 1

2

• uik − 1

2

• + 1

2

• ujk − 1

2

• − 1

2 |uik − ujk|

• − 2

N

N

• i=1

n

• k=1
• 1 + 1

2

• uik − 1

2

• − 1

2

• uik − 1

2

• 2

+ 13 12 n , where uik = (2xik + 1)/(2s).

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Level permutation

Efficient way to construct uniform fractional factorial designs

Start with a minimum aberration design Linearly permute its levels Choose the design with the smallest CD value

Good uniformity, in addition to minimum aberration Suitable for studying both nominal and quantitative factors

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Why it works?

Existing regular minimum aberration designs (Xu (2005)) Keep MA property when levels being permuted MA designs have low discrepancies (on average)

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Average centered L2-discrepancy

For an (N, sn)-design D, apply all s! level permutations to each column and obtain (s!)n combinatorially isomorphic designs Denote the set of these designs as P(D) Compute the CD value for each design and define ¯ φ(D) as the average CD value of all designs in P(D), i.e., ¯ φ(D) = 1 (s!)n

• D′∈P(D)

φ(D′).

Theorem 7

For an (N, 3n)-design D, ¯ φ(D)= 13 12 n − 2 29 27 n + 29 27 n

n

• i=0

2 29 i Ai(D).

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27-run designs

Table: Comparison of designs with 27-run designs

Minimum aberration designs permuted Designs on UD homepage n Ave φ Min φ Max φ A2 A3 4 0.046549 0.046547⋄ 0.046553 5 0.063818 0.063689 0.063878 2 6 0.083786 0.083475⋄ 0.083923 4 7 0.108701 0.108061∗ 0.109118 10 8 0.137749∗ 0.136644∗ 0.138483∗ 16 9 0.172783∗ 0.170996∗ 0.174090∗ 24 10 0.218927∗ 0.213994∗ 0.221241 42 11 0.273255 0.264549∗ 0.276195 60 12 0.338698 0.325027∗ 0.343084 80 13 0.418900 0.397890∗ 0.425576 104 φ A2 A3 0.046547 0.063525 2.6667 0.083475 5.3333 0.108698 0.0988 12.1728 0.138657 0.3457 18.4444 0.175343 0.6914 30.0494 0.219131 1.3580 40.9877 0.272383 2 56 0.336401 2.3210 75.4568 0.414783 3.5309 96.1975 “⋄”: the same as the CD value of the best existing design; “∗”: smaller than the CD value of the best existing design Yu Tang (Soochow University) @ Shanghai Jiaotong University

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81-run designs

Table: Level permutations of 81-run designs

n Ave φ Min φ Best level permutations 5 0.062691 0.062690 6 0.081294 0.081290 0 1 7 0.102528 0.102515 0 2 1 8 0.126795 0.126764 0 2 1 0 9 0.154565 0.154497 0 2 1 0 1 10 0.186393 0.186255 0 2 1 0 1 0 · · · · · · · · · · · · · · · · · · · · · 16 0.540883 0.534813 0 0 1 1 2 2 2 0 2 2 2 2 17 0.640085 0.631437 0 0 1 1 2 2 2 0 2 2 2 2 0 18 0.755854 0.743782 0 0 1 1 2 2 2 0 2 2 2 2 0 1 19 0.898270 0.883749 0 0 1 1 2 2 2 0 2 2 2 2 0 1 2 20 1.066298 1.048120 0 0 1 1 2 2 2 0 2 2 2 2 0 1 2 1

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Content

Brief introduction to uniform design

Background Measure of uniformity — discrepancy

Combinatorial properties of uniform designs

Three examples

Construction methods

Via combinatorial configuration By optimization approach Using level permutation

Conclusion

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Conclusion

Combinatorial properties can provide a clear direction for a U-type design to be a uniform design When the distributions of certain distances between distinct rows (dijs) as well as characters of single row (dis) become as even as possible, then the corresponding discrepancy of the design can be expected to achieve the minimum Based on this idea, three effective construction methods for uniform designs are explored

Constructing by combinatorial configurations Searching with optimization algorithm Using level permutation

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References on combinatorial approach

• Y. Tang, M. Y. Ai, G. N. Ge and K. T. Fang (2007), Optimal

mixed-level supersaturated designs and a new class of combinatorial designs, Journal of Statistical Planning and Inference, 137, 2294–2301.

• K. T. Fang, Y. Tang and J. X. Yin (2006), Resolvable partially

pairwise balanced designs and their applications in computer experiments, Utilitas Mathematica, 70, 141–157.

• K. T. Fang, X. Lu, Y. Tang and J. X. Yin (2004), Constructions of

uniform designs by using resolvable packings and coverings, Discrete Mathematics, 274, 25–40.

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References mainly on optimization approach

• K. T. Fang, D. Maringer, Y. Tang and P. Winker (2006), Lower

bounds and stochastic optimization algorithms for uniform designs with three or four levels, Mathematics of Computation, 75, 859–878.

• K. T. Fang, Y. Tang and J. X. Yin (2005), Lower bounds for

wrap-around L2-discrepancy and constructions of symmetrical uniform designs, Journal of Complexity, 21, 757–771.

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References on level permutation

• G. Xu, J. Zhang and Y. Tang (2014), Level permutation method for

constructing uniform designs under the wrap-around L-2-discrepancy, Journal of Complexity, 30, 46–53.

• Y. Tang and H. Xu (2013), An effective construction method for

multi-level uniform designs, Journal of Statistical Planning and Inference, 143, 1583–1589.

• Y. Tang, H. Xu and D. K. J. Lin (2012), Uniform fractional factorial

designs The Annals of Statistics, 40, 891–907.

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Thanks

§ Conference organization committee § Professor Ellichi Bannai and Professor Yaokun Wu for the invitation § NNSF of China with grant No. 11271279 § NSF of Jiangsu Province with grant No. BK2012612

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Thank you for your time and attention! ytang@suda.edu.cn

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