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Introduce to the Count Function and Its Applications

This informative guide delves into the count function, its properties, coefficients, aberration, orthogonal arrays, and various applications. From design enumeration to isomorphism examination, learn how the count function plays a crucial role in statistical analysis. Discover the history and key concepts like orthogonal arrays, projection, and design optimization. Gain insights into implementing count functions for different design types and exploring versatile word length patterns. Uncover the significance of non-regular designs and the advantages of using orthogonal arrays. With in-depth examples and theoretical discussions, enhance your understanding of the count function's role in statistical analysis.

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Introduce to the Count Function and Its Applications

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  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 ?

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

  29. Measure B Measure A Isomorphism examintion Measure B Measure A

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

  31. Counting vector

  32. Theorem 4 : ? Theorem 5 :

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

  34. Row permutation =

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

  36. Sign switch =

  37. Column permutation

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

  39. Efficiency

  40. Efficiency

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

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