1 Introduce to the Count Function and Its Applications Chang-Yun Lin Institute of Statistics, NCHU
Outlines 2
Count Function
4 6/8 -2/8 2/8 -2/8 2/8 -2/8 -2/8 -2/8
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
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Construct a regular design
Aberration criterion
Non-regular design
Generalized word length pattern
Orthogonal array ( 1, 1): 4 (-1, 1): 4 ( 1,-1): 4 (-1,-1): 4
Orthogonal array
Projection
15 Isomorphic designs 123 I II III VI V IV
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17 Optimal design Is the minimum aberration design local optimal or global optimal ? Should we find it among all designs ?
18 Design enumeration Design generation Isomorphism examination
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20 Projection ? A (-2) A (-1) A (-3) A ?
21 Assembly method OA
22 3/4 1/4 -1/4 3/4 1/43/4
23 -1/43/4 1/43/4
24 1/4 3/4
25 -1/41/43/4
26 Incomplete count function
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30 Hierarchical structure OA(n, k=2, 2, d) OA(n, k=4, 2, d) OA(n, k=3, 2, d) … …
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34 Measure A Measure B Measure B Measure A Isomorphism examintion
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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 Row permutation Sign switch Column permutation A’ Measure (A) Measure (A’) = Row permutation Sign switch Column permutation
40 Row permutation =
41 Sign switch Positive split N vector of t=1 Negative split N vector of t=1
42 Sign switch =
43 Column permutation
44 Column permutation || t ||=1|| t ||=2|| t ||=3 Split-N matrix =
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46 Efficiency
47 Efficiency
48 Projection D (-1) DD’ D (-2) D (-3) D’ (-1) D’ (-2) D’ (-3)
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51 Simplified methods
52 Comparisons EX 6
53 EX 7
54 EX 8
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 Thank You
57 Summary
58 Group structure
59 Projection index set OA(n, k=2, 2, d) {1,1,4} Projection index set {1,1,4}{2,3,4} 4 {2,4,4}
60 Summary
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 Object
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