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Applications of Using Quaternary Codes to Nonregular Designs

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Applications of Using Quaternary Codes to Nonregular Designs

Frederick K.H. Phoa

Department of Statistics

University of California at Los Angeles

07/12/2006 International Conference on

Design of Experiments and its Applications

Email: [email protected]

Research supported by NSF DMS Grant

- Notations and Definitions on Nonregular Designs
- Basics on Quaternary Codes
- Motivation of using Quaternary Codes.
- Previous Result I:
Design Construction with Resolution 3.5

- New Application I:
Design Construction with Resolution 4.0

- Previous Result II:
Fast Search Results on Optimum 2n-2Designs

- New Application II:
Fast Search Results on Optimum 2n-4Designs

- Future Extensions
- Conclusion

Let D be a design of N runs and n factors.

J-characteristics (Jk) of D:

Let a subset of k columns of D. Then

When D is a regular design,

Generalized Resolution of D:

Suppose that r is the smallest integer such that . Then we define

the generalized resolution of D.

Algorithms of using Z4-codes to generate nonregular designs:

- Let G (generator matrix) be an r x n full-rank matrix over Z4.
- A 4-level design C is formed based on G.
- Applying , a 2-level design D is formed based on C.

Gray map

Bijection Transformation of Gray Map:

Example:

1. Given a 2 x 4 generator matrix:

2. A 4-level design C is formed based on G.

3. A 2-level design D is formed based on C through

Gray map.

Generalized Resolution = 4.0

(GOOD DESIGN)

Application of Nordstrom and Robinson Code:

Hedayat, Sloane and Stufken (1999); Xu (2005)

It can form a 256 runs, 16 columns designs.

- Generalized Resolution = 6.5
- Minimum Runs to achieve this in
Regular Design = 512 runs

Why Z4?

[Xu and Wong, accepted by Statistica (2005)]

Example - 16 runs design:

From the list of all possible columns:

Step #1: Delete columns that do not contain any 1â€™s

Step #2: Delete columns whose first non-zero and non-two entry are 3â€™s.

Result:

Generalized Resolution = 3.5

[Phoa, contributed talk in JRC Tennessee, USA (2006)]

Additional Step on Previous Algorithm:

Result from Previous Steps:

Step #3: Delete columns whose sum of columns entries are even.

Result:

Generalized Resolution = 4.0

Maximum Columns of Design with GR=4.0:

This table suggests that the maximum columns of design with

GR =4.0 is N/2 with N runs.

Previous complete search algorithm breaks down when N>256.

This construction generalizes the case for higher run sizes.

[Phoa, contributed talk in JRC Tennessee, USA (2006)]

Structure for Optimum (over Z4) 2n-2 Designs:

Notation:

G = Ik(a,b) design

where

k = size of Identity matrix

a = number of 1â€™s in the last column

b = number of 2â€™s in the last column

Fast Calculation on GR of 2n-2 Design:

1. If b â‰¥ a, then GR = 2(a+1).

2. If b < a, then for even design, GR = min{2b+a+1 , 2(a+1)} + decimals.

for odd design, GR = min{2b+a , 2(a+1)} + decimals.

3.

Search Results on Optimum Designs of different runs:

- This search method bypasses the actual calculation of J-characteristics
- wordlength pattern
- Computer time on searching the optimum designs of particular run size
is much less than the traditional methods

ïƒ Optimality of exceptionally large design is possible

Structure for Optimum (over Z4) 2n-4 Even Designs:

Dimension of G = k x (k+2).

Dimension of D = 4k x 2(k+2)

where

k = size of Identity matrix

v1 = First 4-level column in G

v2 = Second 4-level column in G

We count the number of combination of each row of v1 and v2 using a

counting matrix c

with the following notation:

where

cab = number of row combination with

v1i = a and v2i = b

a, b = {0,1,2,3}

Fast Calculation on GR of 2n-4 Design:

GR = min{

GRv12,

GRv1,

GRv2

} + decimals

where

Fast Calculation on GR of 2n-4 Even Design:

The decimal depends on which equation the GR comes from:

If it comes from the 3rd equation of either GRv12, GRv1 or GRv2,

If it comes from the 1st or 2nd equation of either GRv12, GRv1 or GRv2,

where

if GR=GRv12

if GR=GRv1

if GR=GRv2

Example â€“ 256 run Design:

Maximum Resolution of Regular Design with 256 run

ïƒ 6.0

Search Results on Optimum Designs of different runs:

- Advantages of this Fast Search Method:
- It saves tremendous amount of computer search time.
- It is possible to search for design with super large run size.
- The generalized resolution of our best results beats the regular design with same run size.

1. Construction of Designs with higher Resolutions.

ïƒ Algorithm on GR = 4.5 Construction

ïƒ Designs Constructions with properties similar

to Nordstrom-Robinson Design.

2. Extensions of Fast Search Method.

ïƒ Optimality of 2n-6 Design.

ïƒ Analog to the Odd Design.

3. The Ultimate Goal:

Develop a Generalization on current method of calculating

GR of any given designs, which isbrand new, fast and clean

for both Optimum Search and Construction purpose.

- Motivation: Nordstrom and Robinson code shows the potential of using quaternary code in designs of experiment.
- A new construction method generates the design with GR=4.0, an extension from the previous GR=3.5
- A new search method provides some new results to fast search for the optimum 2n-4 designs, a generalization of its previous special case with 2n-2 runs.
- Future extensions on the design construction, fast search method are suggested.

Professor Hongquan Xu

UCLA Department of Statistics

for his valuable and fruitful comments and discussions.

References

- Deng, L.Y. and Tang, B (1999). Generalized resolution and minimum aberration criteria for Plackett-Burman and other nonregular factorial designs, Statistica Sinica, 9, 1071 â€“ 1082.
- Hedayat, A.S., Sloane, N.J.A. and Stufken, J. (1999) Orthogonal Arrays: Theory and Applications, Springer, New York.
- Xu, H. (2005) Some nonregular designs from the Nordstrom and Robinson code and their statistical properties, Biometrika, 92, 385 â€“ 397.
- Xu, H. and Wong, A. (2005) Two-level Nonregular designs from quaternary linear codes, accepted by Statistica Sinica.
- Phoa, F (2006) Applications of Quaternary Codes on Nonregular Design, contributed talk in JRC on Statistics in Quality, Industry and Technology, Knoxville, TN, USA

Thank you very much for your attention!

Question?

How many times I

appear in this talk?