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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: fredphoa@stat.ucla.edu.

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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: fredphoa@stat.ucla.edu

Research supported by NSF DMS Grant

### Content

• 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

### Notations and Definitions

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.

### Basics on Quaternary (Z4) Codes

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:

### Basics on Quaternary (Z4) Codes

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)

### Motivation of using Quaternary Codes

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?

### Previous Result I:Design Construction with Resolution 3.5

[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 1’s

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

Result:

Generalized Resolution = 3.5

### New Applications I:Design Construction with Resolution 4.0

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

Result from Previous Steps:

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

Result:

Generalized Resolution = 4.0

### New Applications I:Design Construction with 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.

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

[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 1’s in the last column

b = number of 2’s in the last column

Fast Calculation on GR of 2n-2 Design:

1. If b ≥ 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.

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

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

 Optimality of exceptionally large design is possible

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

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}

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

Fast Calculation on GR of 2n-4 Design:

GR = min{

GRv12,

GRv1,

GRv2

} + decimals

where

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

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

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

Example – 256 run Design:

Maximum Resolution of Regular Design with 256 run

 6.0

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

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.

### Future Extensions

1. Construction of Designs with higher Resolutions.

 Algorithm on GR = 4.5 Construction

 Designs Constructions with properties similar

to Nordstrom-Robinson Design.

2. Extensions of Fast Search Method.

 Optimality of 2n-6 Design.

 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.

### Conclusion

• 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.

### Acknowledgements

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 – 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 – 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!

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