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Introduce to the Count Function and Its

Applications

Chang-Yun Lin

Institute of Statistics, NCHU

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Outlines Count function:

Properties of the count function Coefficients (regular/non-regular designs) Aberration Orthogonal array Projection Isomorphism

Applications Design enumeration Isomorphism examination

Count Function

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𝐹 𝐴 (𝑥 )= 34𝐶𝜙−

14𝐶1+

14𝐶2−

14𝐶3+

14𝐶12−

14𝐶13−

14𝐶23−

14𝐶123

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

Coefficients (

Regular fractional factorial design

Example:

10010110

4/8 0 0 0 0 0 0 4/8

Construct a regular design

Design A , generators: , defining relation:

Count function of A

Word length pattern for

Design A , generators: , defining relation:

Word length pattern

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

𝑊𝐿𝑃=(0,0,0,3,0,0,0) 𝑊𝐿𝑃=(0,0,0 ,2 ,0 ,1 ,0) 𝑊𝐿𝑃=(0,0,0 ,1 ,2 ,0,0)

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

Generalized word length pattern

Regular design:

;

Non-regular design

;

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

Orthogonal array

for Example

Projection

Design A

Projection of A on factor j: Example:

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Isomorphic designs

1 2 3

I

II

III

VI

V

IV

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and are isomorphic if and only if there exist a permutation and a vector where ’s are either 0 or 1, such that

for all

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Optimal design

Is the minimum aberration design local optimal or global optimal ?

Should we find it among all designs ?

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Design enumeration

Design generation Isomorphism examination

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• Object: design enumeration for

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Projection

?A(-2)

A(-1)

A(-3)

A?

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Assembly method

OA

OA

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3/4

3/4

1/4

1/4

-1/4 -1/4

-1/4 -1/4

-1/43/4

1/43/4

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-1/43/41/43/4

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1/4 1/43/4

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-1/4 1/43/4

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Incomplete count function

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-1 10 2

-1 10 2

?

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29

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Hierarchical structure

OA(n, k=2, 2, d)

OA(n, k=4, 2, d)

OA(n, k=3, 2, d)

… …

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𝑶𝑨(𝒏 ,𝒅+𝟏 ,𝟐 ,𝒅)

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𝑶𝑨(𝒏 ,𝟓 ,𝟐 ,𝟐)

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𝑶𝑨(𝒏 ,𝒌 ,𝟐 ,𝒅)

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MeasureAMeasure

B

MeasureB

MeasureA

Isomorphism examintion

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Object

Propose a more efficient initial screening method

Measure development for initial screening Counting vector Split-N matrix

Efficiency comparison & enhancement Technique of projection

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Counting vector

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?Theorem 4 :

Theorem 5 :

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A

Row permutation

Sign switch

Column permutation

A’

Measure

Measure (A)

Measure

Measure (A’)=

Row permutation

Sign switch

Column permutation

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Row permutation

=

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Sign switch

1

3

5

7

2

4

6

8

Positive split N vector of t=1

Negative split N vector of t=1

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Sign switch

=

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Column permutation

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Column permutation

|| t ||=1 || t ||=2 || t ||=3

Split-N matrix=

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Efficiency

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Efficiency

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Projection

D(-1)

D D’D(-2)

D(-3)

D’(-1)

D’(-2)

D’(-3)

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Simplified methods

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ComparisonsEX 6

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EX 7

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EX 8

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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

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Thank You