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

Uniform Designs and Their Constructions Yu Tang Soochow University Apr. 23, 2015 SoochowU.jpg Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 1 / 43

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 SoochowU.jpg Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 2 / 43

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 · · · · · · SoochowU.jpg Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 3 / 43

Achievements Number of papers on uniform designs published The second prize of the 2008 National Natural Science Award SoochowU.jpg Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 4 / 43

Three cases with 16 points over a unit square domain △ × ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ △ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ one time △ two times × three times ❛ *from Fang and Ma (2001) SoochowU.jpg Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 5 / 43

Star discrepancy Number theory (quasi-Monte Carlo) method Star discrepancy (Weyl (1916)) � N ( P n , [ 0 , x )) � � � D ( P n ) = max − Vol ([ 0 , x )) � , � � x ∈ C s n � [ 0 , x ) denotes the hypercube [0 , x 1 ) × · · · × [0 , x m ) N ( P n , [ 0 , x )) represents the number of points of P n falling in [ 0 , x ) Vol ( A ) means the volume of A D 1 = D 2 = D 3 = 0 . 23438. SoochowU.jpg Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 6 / 43

Modified L p -discrepancy Modified L p -discrepancy Wrap-around L 2 -discrepancy (Hickernell (1998a)) � 2 � N ( P u ∩ J w ( x ′ � u , x u )) WD 2 � − Vol ( J w ( x ′ d x ′ 2 = u , x u )) u d x u , n C 2 u u � = ∅ Centered L 2 -discrepancy (Hickernell (1998b)) � 2 � N ( P u ∩ J w ( x u )) � � CD 2 2 = − Vol ( J w ( x u )) d x u n C u u � = ∅ · · · · · · SoochowU.jpg Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 7 / 43

Closed form § Wrap-around L 2 -discrepancy (Hickernell 1998a) n n m � 3 � � 4 ( WD 2 ( P )) 2 = − � m � � � 1 + 2 − | x ki − x ji | (1 − | x ki − x ji | ) . n 2 3 k =1 j =1 i =1 § Centered L 2 -discrepancy (Hickernell 1998b) n n m � 1 + 1 � x ki − 1 � � � + 1 � x ji − 1 � � − 1 � � � � � ( CD 2 ( P )) 2 1 � � � � = 2 | x ki − x ji | � � � � n 2 2 2 2 2 k =1 j =1 i =1 n m � 2 � � m � � � � � 13 1 − 1 � x ki − 1 � − 1 � x ki − 1 � � − 1 � � � � + . � � � � n 2 2 2 2 12 � k =1 i =1 SoochowU.jpg Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 8 / 43

Numerical results △ × ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ △ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ ❛ one time △ two times × three times ❛ W 1 = 0 . 02789 < W 2 = W 3 = 0 . 03423. C 1 = 0 . 01138 < C 2 = 0 . 01236 < C 3 = 0 . 01797. SoochowU.jpg Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 9 / 43

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 SoochowU.jpg Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 10 / 43

Example 1 d ij = ♯ { ( x ik , x jk ) : x ik = x jk , k = 1 , · · · , m } row 1 2 3 4 5 1 1 1 1 1 1 2 2 2 2 2 1 d 12 = 1 d 13 = 2 3 1 1 2 2 2 d 34 = 1 d 14 = 2 4 2 2 1 1 2 = ⇒ d 56 = 1 d 23 = 2 5 1 2 1 2 3 · · · · · · d 78 = 1 6 2 1 2 1 3 7 1 2 2 1 4 8 2 1 1 2 4 SoochowU.jpg Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 11 / 43

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 ; q 1 × · · · × q m ) . Denote � m i =1 n / q i − m γ = ˜ and γ = ⌊ ˜ γ ⌋ where ⌊ x ⌋ denotes the integer part of x. Then n − 1 m n − 1 n � a � d ij , � � a +( q j − 1) b + a m n + 2 b m D 2 ( X ; a , b ) = − � � � (1) n 2 q j b j =1 i =1 j = i +1 n − 1 n � γ + (˜ � a � a � a � γ +1 ] , � d ij � � ≥ n ( n − 1)[( γ + 1 − ˜ γ ) γ − γ ) (2) b b b i =1 j = i +1 and the lower bound on the right hand side of (2) can be achieved if and only if all d ij s (i � = j) take the same value γ , or take only two values γ and γ + 1 . *discrete discrepancy — proposed in Hickernell and Liu (2002) SoochowU.jpg Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 12 / 43

Example 2 d 0 ij = ♯ { ( x ik , x jk ) : x ik = x jk , k = 1 , · · · , m } ; d 1 ij = ♯ { ( x ik , x jk ) : | x ik − x jk | = 1 or q − 1 , k = 1 , · · · , m } ; · · · · · · d q / 2 = ♯ { ( x ik , x jk ) : | x ik − x jk | = q / 2 , k = 1 , · · · , m } ij row 1 2 3 4 5 6 7 1 4 2 3 3 1 2 1 2 1 4 2 3 3 1 2 d 0 d 1 d 2 12 = 1 12 = 4 12 = 2 3 2 1 4 2 3 3 1 d 0 d 1 d 2 13 = 1 13 = 4 13 = 2 4 1 2 1 4 2 3 3 = ⇒ d 0 d 1 d 2 23 = 1 23 = 4 23 = 2 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 ij , . . . , d q / 2 F ij = ( d 0 ij , d 1 ) ij SoochowU.jpg Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 13 / 43

Uniform designs under the wrap-around L 2 discrepancy Theorem 2 (Fang, Tang and Yin (2005) A lower bound of the wrap-around L 2 -discrepancy on U ( n ; q m ) with even q and odd q is given by 2 mn 2 mn q ( n − 1) · · · � 3 � m ( n − q ) q ( n − 1) � 5 q ( n − 1) ; mn q ( n − 1) � � � � ∆ + n − 1 2 − 2(2 q − 2) 3 3 2 − ( q − 2)( q +2) � 4 q 2 4 q 2 n 2 4 2 mn 2 mn q ( n − 1) · · · q ( n − 1) , � 3 � m ( n − q ) q ( n − 1) � � � � 2 − 2(2 q − 2) 2 − ( q − 1)( q +1) ∆ + n − 1 3 3 n 2 4 q 2 4 q 2 � m � m � 4 + 1 � 3 . A U-type design U ( n ; q m ) is a respectively, where ∆ = − 3 n 2 uniform design under the wrap-around L 2 -discrepancy, if its all F ij distributions, i � = j, are the same. In this case, the WD 2 -value of this design achieves the above lower bound. SoochowU.jpg Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 14 / 43

Example 3 d i = ♯ { x ik � = 2 , k = 1 , · · · , m } ; d ij = ♯ { ( x ik , x jk ) : x ik = x jk � = 2 , k = 1 , · · · , m } row 1 2 3 4 5 6 d 1 = 4 1 2 3 1 3 3 2 d 2 = 4 d 12 = 1 2 1 1 1 2 1 2 d 3 = 4 d 13 = 1 3 3 2 2 1 3 1 = ⇒ d 4 = 4 d 14 = 0 4 3 1 3 2 2 3 · · · · · · d 5 = 4 5 1 3 2 1 2 3 d 6 = 4 6 2 2 3 3 1 1 SoochowU.jpg Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 15 / 43

Uniform designs under the centered L 2 discrepancy Theorem 3 (Fang, Maringer, Tang and Winker (2006)) For a U-type design U ( n , 3 m ) , � 2 � � 2 m ( n − 3) � , and denote n µ = ( µ + 1) n − 2 mn let µ = 3 m , γ = and 9( n − 1) 3 n γ = ( γ + 1) n ( n − 1) − mn ( n − 3) . If f ( µ ) ≥ f (0) , then 2 9 13 m 2 10 10 � µ � CD 2 ( P ) 2 ) µ +1 ≥ ( ) − n µ ( ) + ( n − n µ )( 12 9 9 n (3) 1 4 4 2 4 n ( n − 1) 4 � µ � � γ � ) µ +1 ) γ +1 + n µ ( ) + ( n − n µ )( + n γ ( ) + ( − n γ )( . n 2 n 2 3 3 3 2 3 The lower bound can be obtained if and only if n µ d i s take the value µ , n − n µ d i s take the values µ + 1 and n γ d ij s take the value γ , n ( n − 1) − n γ 2 d ij s take the value γ + 1 , for all i � = j. � x � x f ( x ) = 1 � 4 − 2 n � 10 SoochowU.jpg 3 3 9 9 Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 16 / 43

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 SoochowU.jpg Yu Tang (Soochow University) @ Shanghai Jiaotong University Apr. 23, 2015 17 / 43