Rowwise Complementary Designs Shao-Wei Cheng Institute of Statistical Science Academia Sinica Taiwan (based on a joint work with a Ph.D student Chien-Yu Peng)

Outline Introduction Columnwise and Rowwise Complementary Design Indicator Function Some Properties Isomorphism Generalized Wordlength Pattern and Resolution Minimum Aberration Orthogonality Moment Aberration Uniformity Generalization Summary

(Columnwise) Complementary Design Hadamard matrix number of factors Tang and Wu (1996), Chen and Hedayat (1996) useful when the number of factors in A is large

Rowwise Complementary Design (RCD) full factorial design with r replicates number of factors RCD r = the maximum number of appearances of design points in design run size

Indicator Function Fontana et al. (2000), Ye (2003), Cheng and Ye (2004), Cheng, Li, and Ye (2004), Tang (2001, J-characteristic) Definition Let D be a full factorial design of s factors and A be a subset of D. The indicator function of A is a function F A ( x ) defined on D such that where r x is the number of appearances of the point x in design A.

Let P be the collection of all subset of {1,2, …, s }, then the indicator function can be uniquely represented by where. cor( x I, x J ) = b I  J /b , where I  J=(I  J)\(I  J) (Note: x i 2 = 1)

 A 1 : (regular) FFD, 3=12 and 6=145 F A 1 = (1/4)(1+x 1 x 2 x 3 )(1+x 1 x 4 x 5 x 6 ) = (1/4)(1 + 1*x 1 x 2 x 3 + 1*x 1 x 4 x 5 x 6 + 1*x 2 x 3 x 4 x 5 x 6 ) I=123=1456=23456  A 2 : non-regular OA F A 2 = (1/4)[1 + (1/2)*x 2 x 4 x 6 + (  1/2)*x 2 x 5 x 6 + (1/2)*x 3 x 4 x 6 + (1/2)*x 3 x 5 x 6 + (1/2)*x 1 x 2 x 4 x 6 + (1/2)*x 1 x 2 x 5 x 6 + (  1/2)*x 1 x 3 x 4 x 6 + (1/2)*x 1 x 3 x 5 x 6 + (1)*x 1 x 2 x 3 x 4 x 5 ] b 0 = fractionaliasing (correlation)word Example wordlength = 4

full factorial design Let be the indicator function of, and be the RCD of, then the indicator function of is r = 1 r1r2r3...rkr1r2r3...rk r.rr.r rrr...rrrr...r r.rr.r general case r = max{r 1, …, r k } r  r 1 r  r 2 r  r 3. r  r k Indicator Function and RCD

Let A be a design the indicator function of its RCD is Example

Property 1: Isomorphism Suppose. If and are isomorphic, then and are also isomorphic.

Property 2: Generalized WLP and Resolution Generalized Wordlength Pattern (GWLP, Deng and Tang, 1999) where  i   (b I /b  ) 2 Generalized resolution (Tang and Deng, 1999) # I = i

Theorem Let and be the indicator functions of A and its RCD, then (i) GWLPs of A and its RCD are and, respectively. (ii) the design with larger run size has large generalized resolution and the difference between their generalized resolutions is.

Let A be a design with four factors. GWLP: (0, 1, 2, 0) its RCD GWLP: (0, 1, 2, 0)/9 Example

Property 3: Orthogonality A is an orthogonal array of strength t if and only if its RCD is an orthogonal array of strength t.

Example Cheng (1995) shows that 4-factor, 12-run designs there is only one OA with strength 2 the OA has 11 distinct runs (one run repeats twice) Q: why no OA with 12 distinct runs? 12 distinct runs  the indicator function of its RCD is 1  F A  its RCD has 4 runs, 4 factors the OA  the indicator function of its RCD is 2  F A  its RCD has 20 runs, 4 factors

The minimum aberration criterion is to sequentially minimize GWLP within a class of designs, called search class. Denote the search class of A by and the search class of its RCD by. example: if, then. Property 4: Minimum aberration

Theorem Design A is the MA design in the search class if and only if its RCD is the MA design in the search class S.

Property 5: Moment aberration Xu (2003), based on coding theory Let be the rowwise complementary design of, then its u- th power moment is where and are Stirling numbers of second kind. In particular, and.

Property 6: Uniformity Fang and his colleagues Symmetric L 2 -discrepancy ( SL 2 ) Centered L 2 -discrepancy ( CL 2 ) Wrap-around L 2 -discrepancy ( WL 2 )

Theorem Let A be an n  s two-level factorial design, then the discrepancies of its RCD can be expressed in terms of the discrepancies and GWLP of A as follows

Type I designs full factorial design number of factors Generalization Hadamard matrix

Type II designs full factorial design number of factors Hadamard matrix

Blocked factorial designs blocked indicator function (Cheng, Li, and Ye, 2004) If is a blocked indicator function which has q block factors (i.e., blocks and block size = runs), then the can be regarded as a blocked indicator function with blocks and block size = runs.

Summary propose RCD, useful when run size is large indicator function is a useful framework for studying RCD various relations between a design and its RCD are explored methods that combine the techniques of CCD and RCD can generate more designs