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 Research supported by NSF DMS Grant

Content 1.Notations and Definitions on Nonregular Designs 2.Basics on Quaternary Codes 3.Motivation of using Quaternary Codes. 4.Previous Result I: Design Construction with Resolution New Application I: Design Construction with Resolution Previous Result II: Fast Search Results on Optimum 2 n-2 Designs 7.New Application II: Fast Search Results on Optimum 2 n-4 Designs 8.Future Extensions 9.Conclusion

J-characteristics (J k ) of D: Let a subset of k columns of D. Then When D is a regular design, Notations and Definitions Let D be a design of N runs and n factors. Generalized Resolution of D: Suppose that r is the smallest integer such that. Then we define the generalized resolution of D.

Basics on Quaternary (Z 4 ) Codes Algorithms of using Z 4 -codes to generate nonregular designs: 1.Let G (generator matrix) be an r x n full-rank matrix over Z 4. 2.A 4-level design C is formed based on G. 3.Applying, a 2-level design D is formed based on C. Z4Z ↓↓↓↓ Gray map Bijection Transformation of Gray Map : Z4Z ↓↓↓↓ Z22Z Z4Z4 0123

Basics on Quaternary (Z 4 ) 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. Z4Z ↓↓↓↓ Z22Z 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 Z 4 ?

Previous Result I: Design Construction with Resolution 3.5 Example - 16 runs design: Result: 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 Generalized Resolution = 3.5 [Xu and Wong, accepted by Statistica (2005)]

New Applications I: Design Construction with Resolution 4.0 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 [Phoa, contributed talk in JRC Tennessee, USA (2006)]

New Applications I: Design Construction with Resolution 4.0 Maximum Columns of Design with GR=4.0: # Rows in G# Columns in GDimension of DResolution 2416 x x x :::: N4 n-1 4 n x 4 n /24.0 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 2 n-2 Designs Structure for Optimum (over Z 4 ) 2 n-2 Designs: Notation: G = I k (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 [Phoa, contributed talk in JRC Tennessee, USA (2006)] Fast Calculation on GR of 2 n-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 2 n-2 Designs Search Results on Optimum Designs of different runs: k #col Run (a,b)(1,1) (2,1) (2,2)(3,1)(3,2) (4,2) (4,3) GR 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 2 n-4 Designs Structure for Optimum (over Z 4 ) 2 n-4 Even Designs: Dimension of G = k x (k+2). Dimension of D = 4 k x 2(k+2) where k = size of Identity matrix v 1 = First 4-level column in G v 2 = Second 4-level column in G We count the number of combination of each row of v 1 and v 2 using a where c ab = number of row combination with v 1i = a and v 2i = b a, b = {0,1,2,3} counting matrix cwith the following notation:

New Applications II: Fast Search Results on Optimum 2 n-4 Designs Fast Calculation on GR of 2 n-4 Design: GR = min{ where GRv12,GRv1,GRv2} + decimals

New Applications II: Fast Search Results on Optimum 2 n-4 Designs Fast Calculation on GR of 2 n-4 Even Design: The decimal depends on which equation the GR comes from: If it comes from the 3rd equation of either GRv 12, GRv 1 or GRv 2, If it comes from the 1st or 2nd equation of either GRv 12, GRv 1 or GRv 2, where if GR=GRv 12 if GR=GRv 1 if GR=GRv 2

New Applications II: Fast Search Results on Optimum 2 n-4 Designs Example – 256 run Design: Maximum Resolution of Regular Design with 256 run  6.0

New Applications II: Fast Search Results on Optimum 2 n-4 Designs Search Results on Optimum Designs of different runs: k #col Run k1048k GR Advantages of this Fast Search Method: 1.It saves tremendous amount of computer search time. 2.It is possible to search for design with super large run size. 3.The generalized resolution of our best results beats the regular design with same run size.

Future Extensions 1. Construction of Designs with higher Resolutions. GR = 4.5  Algorithm on GR = 4.5 Construction  Designs Constructions with properties similar to Nordstrom-Robinson Design. 2. Extensions of Fast Search Method. 2 n-6 Design  Optimality of 2 n-6 Design. Odd Design  Analog to the Odd Design. Ultimate Goal 3. The Ultimate Goal: Develop a Generalization on current method of calculating brand new, fast and clean GR of any given designs, which is brand new, fast and clean for both Optimum Search and Construction purpose.

Conclusion Nordstrom and Robinson code 1.Motivation: Nordstrom and Robinson code shows the potential of using quaternary code in designs of experiment. new construction method GR=4.0 2.A new construction method generates the design with GR=4.0, an extension from the previous GR=3.5 new search method optimum 2 n-4 designs 3.A new search method provides some new results to fast search for the optimum 2 n-4 designs, a generalization of its previous special case with 2 n-2 runs. extensions 4.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 1.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 – Hedayat, A.S., Sloane, N.J.A. and Stufken, J. (1999) Orthogonal Arrays: Theory and Applications, Springer, New York. 3.Xu, H. (2005) Some nonregular designs from the Nordstrom and Robinson code and their statistical properties, Biometrika, 92, 385 – Xu, H. and Wong, A. (2005) Two-level Nonregular designs from quaternary linear codes, accepted by Statistica Sinica. 5.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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