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

2. Outlines 2

3. Count Function

4. 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. 1001011010010110 4/8 0 0 0 0 0 0 4/8

7. Construct a regular design

9. Aberration criterion

10. Non-regular design

11. Generalized word length pattern

12. Orthogonal array ( 1, 1): 4 (-1, 1): 4 ( 1,-1): 4 (-1,-1): 4

13. Orthogonal array

14. Projection

15. 15 Isomorphic designs 123 I II III VI V IV

16. 16

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

18. 18 Design enumeration Design generation Isomorphism examination

19. 19

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

21. 21 Assembly method OA

22. 22 3/4 1/4 -1/4 3/4 1/43/4

23. 23 -1/43/4 1/43/4

24. 24 1/4 3/4

25. 25 -1/41/43/4

26. 26 Incomplete count function

27. 27 102 102 ?

28. 28

29. 29

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

31. 31

32. 32

33. 33

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

35. 35

36. 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. 37 Counting vector

38. 38 ? Theorem 4 : Theorem 5 :

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

40. 40 Row permutation =

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

42. 42 Sign switch =

43. 43 Column permutation

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

45. 45

46. 46 Efficiency

47. 47 Efficiency

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

49. 49

50. 50

51. 51 Simplified methods

52. 52 Comparisons EX 6

53. 53 EX 7

54. 54 EX 8

55. Summary Count function Coefficients; aberration; orthogonal array; projection; isomorphism Design enumeration Assembly method: generates a design from the LOO projections Hierarchical structure: sequentially generates designs through the assembly method Isomorphism examination 55

56. 56 Thank You

57. 57 Summary

58. 58 Group structure

59. 59 Projection index set 1 2 1 3 2 2 4 3 1 4 3 4 2 OA(n, k=2, 2, d) 1 1 4 {1,1,4} Projection index set 11 34 4 4 2 2 {1,1,4}{2,3,4} 4 {2,4,4}

60. 60 Summary

61. 61 Future work Design enumeration Design generation 2 levels 3 levels Factorial design Block design Projection More efficient assembly method Isomorphism examination Classification method Split-N matrix: 2 levels 3 levels Initial screening method Complete classification method => 3 factor designs => Block design => more efficient measure

62. 62 Object

63. 63