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_at_stat.ucla.edu

Research supported by NSF DMS Grant
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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-2 Designs
• New Application II
• Fast Search Results on Optimum 2n-4 Designs
• Future Extensions
• Conclusion

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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.
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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
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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)
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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?
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Previous Result IDesign 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
1s
Step 2 Delete columns whose first non-zero and
non-two entry are 3s.
Result
Generalized Resolution 3.5
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New Applications IDesign Construction with
Resolution 4.0
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
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New Applications IDesign Construction with
Resolution 4.0

• Maximum Columns of Design with GR4.0

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
Ngt256. This construction generalizes the case for
higher run sizes.
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Previous Result II Fast Search Results on
Optimum 2n-2 Designs
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 1s
in the last column b number of 2s in the
last column
Fast Calculation on GR of 2n-2 Design
1. If b a, then GR 2(a1).
2. If b lt a, then for even design, GR
min2ba1 , 2(a1) decimals.
for odd design, GR min2ba , 2(a1)
decimals.
3.
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Previous Result II Fast Search Results on
Optimum 2n-2 Designs

• 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

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

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

Dimension of G k x (k2). Dimension of D 4k x
2(k2) 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
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New Applications II Fast Search Results on
Optimum 2n-4 Designs
Fast Calculation on GR of 2n-4 Design
GR min
GRv12,
GRv1,
GRv2
decimals
where
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New Applications II Fast Search Results on
Optimum 2n-4 Designs
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 GRGRv12
if GRGRv1
if GRGRv2
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New Applications II Fast Search Results on
Optimum 2n-4 Designs
Example 256 run Design
Maximum Resolution of Regular Design with 256
run ? 6.0
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New Applications II Fast Search Results on
Optimum 2n-4 Designs

• 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.

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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 is brand new,
fast and clean for both Optimum Search and
Construction purpose.
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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 GR4.0, an extension from the previous
  GR3.5
• 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.

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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

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Thank you very much for your attention!
Question?
How many times I appear in this talk?