Introduce to the Count Function and Its Applications

1. Introduce to the Count Function and Its Applications Chang-Yun Lin Institute of Statistics, NCHU

2. Outlines • Count function: • Properties of the count function • Coefficients (regular/non-regular designs) • Aberration • Orthogonal array • Projection • Isomorphism • Applications • Design enumeration • Isomorphism examination

3. Count Function

4. 6/8 -2/8 2/8 -2/8 2/8 -2/8 -2/8 -2/8

5. History • Fontana, Pistone and Rogantin (2000) • Indicator function (no replicates) • Ye (2003) • Count function for two levels • Cheng and Ye (2004) • Count function for any levels

6. Coefficients ( • Regular fractional factorialdesign • Example: 1 0 0 1 0 1 1 0 4/8 0 0 0 0 0 0 4/8

7. Construct a regular design • Design A • , generators: , • defining relation: • Count function of A

8. Word length pattern for • Design A • , generators: , • defining relation: • Word length pattern

9. Aberration criterion • For any two designs and , • the smallest integer s.t.. • has less aberration than • if • has minimum aberration • If there is no design with less aberration than

10. Non-regular design • Any two effects (Placket-Burman design) • cannot be estimated independently of each other • not fully aliased • Advantages • Run size economy • Flexibility • Example

11. Generalized word length pattern • Regular design: • ; • Non-regular design • ;

12. Orthogonal array • n runs; k factors; s levels • strength d: • for any d columns, all possible combinations of symbols appear equally often in the matrix • Example: ( 1, 1): 4 (-1, 1):4 ( 1,-1):4 (-1,-1):4

13. Orthogonal array • for • Example

14. Projection • Design A • Projection of A on factor j: • Example:

15. Isomorphic designs 1 2 3 I II III IV V VI

16. and are isomorphic if and only if there exist a permutation and a vector where ’s are either 0 or 1, such thatfor all

17. Optimal design Is the minimum aberration design local optimal or globaloptimal? Should we find it among all designs? Q1. Q2.

18. Design enumeration • Design generation • Isomorphism examination

19. Object: design enumeration for

20. Projection A(-1) ? A(-2) A ? A(-3)

21. Assembly method OA OA

22. 3/4 1/4 -1/4 -1/4 3/4 1/4 3/4 -1/4 3/4 1/4 -1/4 -1/4

23. 3/4 1/4 3/4 -1/4

24. 3/4 1/4 1/4

25. 3/4 -1/4 1/4

26. Incomplete count function

27. -1 -1 0 0 1 1 2 2 ?

30. Hierarchical structure • OA(n, k=2, 2, d) … … • OA(n, k=4, 2, d) • OA(n, k=3, 2, d)

34. Measure B Measure A Isomorphism examintion Measure B Measure A

36. Object • Propose a more efficient initial screening method • Measure development for initial screening • Counting vector • Split-N matrix • Efficiency comparison & enhancement • Technique of projection

37. Counting vector

38. Theorem 4 : ? Theorem 5 :

39. A A’ Row permutation Sign switch Column permutation Measure Measure Row permutation Sign switch Column permutation Measure (A’) Measure (A) =

40. Row permutation =

41. Sign switch 1 2 3 4 5 6 7 8 Positive split N vector of t=1 Negative split N vector of t=1

42. Sign switch =

43. Column permutation

44. Column permutation = Split-N matrix || t ||=1 || t ||=2 || t ||=3

46. Efficiency

47. Efficiency

48. Projection D’(-1) D(-1) D(-2) D’(-2) D D’ D(-3) D’(-3)