University, Wuhan, China c College of Physical Science and Technology, Central China Normal. University, Wuhan, China Published online: 25 Apr 2014.

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1 This article was downloaded by: [ ] On: 0 April 01, At: 16:7 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, 7-1 Mortimer Street, London W1T JH, UK Communications in Statistics - Theory and Methods Publication details, including instructions for authors and subscription information: A Lower Bound for the Wrap-around L - discrepancy on Combined Designs of Mixed Two- and Three-level Factorials Zujun Ou ac, Hong Qin b & Xu Cai c a College of Mathematics and Statistics, Jishou University, Jishou, China b Faculty of Mathematics and Statistics, Central China Normal University, Wuhan, China c College of Physical Science and Technology, Central China Normal University, Wuhan, China Published online: 5 Apr 01. To cite this article: Zujun Ou, Hong Qin & Xu Cai (01) A Lower Bound for the Wrap-around L - discrepancy on Combined Designs of Mixed Two- and Three-level Factorials, Communications in Statistics - Theory and Methods, :10-1, 7-85 To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &

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3 Communications in Statistics Theory and Methods, : 7 85, 01 Copyright Taylor & Francis Group, LLC ISSN: print / 15-15X online DOI: / Regression and Experimental Designs A Lower Bound for the Wrap-around L -discrepancy on Combined Designs of Mixed Two- and Three-level Factorials ZUJUN OU, 1, HONG QIN, AND XU CAI Downloaded by [ ] at 16:7 0 April 01 1 College of Mathematics and Statistics, Jishou University, Jishou, China Faculty of Mathematics and Statistics, Central China Normal University, Wuhan, China College of Physical Science and Technology, Central China Normal University, Wuhan, China The objective of this article is to study the issue of employing the uniformity criterion measured by wrap-around L -discrepancy to assess the optimal foldover plans. For mixed two- and three-level fractional factorials as the original designs, general foldover plan and combined design under a foldover plan are defined, and the equivalence between any foldover plan and its complementary foldover plan is investigated. A lower bound of wrap-around L -discrepancy of combined designs under a general foldover plan is obtained, which can be used as a benchmark for searching optimal foldover plans. Moreover, it also provides a theoretical justification for optimal foldover plans in terms of uniformity criterion. Keywords Mixed level factorials; Wrap-around L -discrepancy; Combined design; Foldover plan; Lower bound. Mathematics Subject Classification 6K15; 6K Introduction The foldover is a useful technique in construction of two-level factorial designs. Many works on optimal foldover plans in terms of aberration criterion or clear effects criterion have been published. We can refer to Box et al. (1978), Montgomery and Runger (1996), Li and Mee (00), Li and Lin (00), Li et al. (00), Ye and Li (00), Li and Jacroux (007) and Wang et al. (009). Recently, there has been considerable interest in employing the discrepancy to assess the optimal foldover plans. The foldover plan such that the combined design (defined below) Received August 0, 01; Accepted February 11, 01. Address correspondence to Hong Qin, Faculty of Mathematics and Statistics, Central China Normal University, Wuhan 0079, China. qinhong@mail.ccnu.edu.cn 7

4 Lower Bound of Discrepancy on Combined Designs 75 Downloaded by [ ] at 16:7 0 April 01 has the smallest discrepancy value over all foldover plans is called the optimal foldover plan. Fang et al. (00) first used the uniformity criterion measured by centered L -discrepancy (Hickernell, 1998) to search the optimal foldover plan. Lei et al. (010, 01) obtained some lower bounds of centered L -discrepancy of combined designs when all Hamming distances between any distinct pair of runs of the initial design are equal. Subsequently, Ou et al. (011) obtained lower bounds of various discrepancies, including wrap-around L -discrepancy, of combined designs for general case. Qin et al. (01) extended the ones of Ou et al. (011) for symmetric designs to a set of asymmetric designs under a special case, where the last p-level column of foldover design is retained as the same in the original design. There are many measures to assess the uniformity of various designs. Among them, the wrap-around L -discrepancy has nice properties, such as it is invariant under reordering the runs, relabeling coordinates, and coordinate shift, we can refer to Fang et al. (005) and Zhou and Ning (008). In this article, we take the wrap-around L -discrepancy as the uniformity measure, and try to find a lower bound of wrap-around L -discrepancy of combined designs. To find a good lower bound of combined designs for a kind of discrepancy is an important issue, because this lower bound can serve as a benchmark for searching optimal foldover plans. While the work of Ou et al. (011) gave a lower bound of wrap-around L -discrepancy, of combined designs under a general foldover plan, the present paper aims at obtaining further results. We extend the findings in Ou et al. (011) from two-level symmetrical factorials to mixed two- and three-level fractional factorials since they are most popular among mixed factorials, and extend the ones in Qin et al. (01) from a kind of special foldover plan to the full foldover plan space if the last factor has three levels. First, the original design is a mixed two- and three-level factorial, which is different from that in Ou et al. (011). Second, the foldover plan considered in Qin et al. (01) is a special case of this article when the last factor has three levels. This article is organized as follows. Some notations and preliminaries are described in Sec.. Section gives the equivalence between any foldover plan and its complementary foldover plan, and a lower bound of wrap-around L -discrepancy of combined design under a general foldover plan also is presented in Sec.. Section gives some examples to illustrate our theoretical results and lends further support to our theoretical results.. Notations and Preliminaries Let U(n; q 1 q s ) denote an U-type design with n runs and s factors, which corresponds to an n s matrix d with entries 0, 1,...,q j 1inthejth column such that all the entries in this column appear equally often. The set of all such designs is denoted by U(n; q 1 q s ). When some q i s are equal, the notations U(n; qs 1 1 qs l l ) and U(n; q s 1 1 qs l l ) are used to express the U-type design and the corresponding set, where s 1 + +s l = s. For any design d U(n; q 1 q s ), note that the rows and columns of d are identified with the runs and factors, respectively. In this article, we mainly focus on the design in U(n; s 1 s ) since two- and three-level mixed factorials are most popular among mixed factorials. Consider a design in U(n; s 1 s ) with the design matrix d = (x 1,x,...,x s ), where x j is the jth column of d,j = 1,...,s. Define Ɣ ={γ = (γ 1,...,γ s1,γ s1 +1,...,γ s ) γ j1 =

5 76 Ou et al. 0, 1, 1 j 1 s 1 ; γ j = 0, 1,,s 1 j s}, then for any γ = (γ 1,...,γ s ) Ɣ,itdefines a foldover plan for design d U(n; s 1 s ), for 1 j 1 s 1,thej 1 th factor is mapped to x j 1 (γ j1 ) = (x j 1 γ j1 ) = (x 1j1 + γ j1,...,x nj1 + γ j1 ) (mod ) and for s 1 j s, thej th factor is mapped to x j (γ j ) = (x j γ j ) = (x 1j + γ j,...,x nj + γ j ) (mod ). Downloaded by [ ] at 16:7 0 April 01 For any design d U(n; s 1 s ) and a foldover plan γ Ɣ, the foldover design, denoted by d γ, is obtained by mapping the columns in d defined above according to the foldover plan γ. Thus, it is to be noted that each foldover design is generated by a foldover plan. The design obtained by augmenting the runs of the foldover design d γ to those of the design d is called the combined design, denoted by d(γ ), that is, d(γ ) = ( d ). d γ Let 1 s be the 1 s vector with all elements unity, for any γ = (γ 1,...,γ s1,γ s1 +1,...,γ s ) Ɣ, define γ = (γ (), 1 s γ () ), where γ () = (γ 1,...,γ s1 ), γ () = (γ s1 +1,...,γ s ) and 1 s γ () = ( γ s1 +1,..., γ s )(mod), and γ is called as the complementary foldover plan of γ in this article. Example.1. Consider a design d 1 U(1; ) with n = 1, s 1 = and s = 1asthe original design, these runs are shown in the first part of Table 1 (Run No. 1 1). If we take γ 1 = (1, 0, 1, 0, ) Ɣ, then we can obtain the foldover design d 1γ1 (Run No. 1 ) by the above foldover method. The original design d and the foldover design d 1γ1 forms the combined design d 1 (γ 1 ) (Run No. 1 ). Furthermore, the complementary foldover plan of γ 1 is γ 1 = (1, 0, 1, 0, 1). It is to be noted that any treatment combination (x 1,x,...,x s ) can be mapped to (u 1,u,...,u s ), where, u j1 = x j 1, 1 j 1 s 1, and u j = x j,s 1 j s. 6 In this article, we consider the wrap-around L -discrepancy, due to Hickernell (1998), given by [WD(d)] = ( ) s n n s i,j=1 l=1 [ ] u il u jl (1 u il u jl ). (1)

6 Lower Bound of Discrepancy on Combined Designs 77 Table 1 Combined design d 1 (γ 1 ) corresponding to γ 1 d 1 Run No. x 1 x x x x 5 Downloaded by [ ] at 16:7 0 April d 1γ1 Run No. γ 1 = ( ) Under a foldover plan γ, the wrap-around L -discrepancy value of the combined design d(γ ), denoted by WD(d(γ )), can be calculated by the following formula: [WD(d(γ ))] = ( ) s n n s l=s 1 +1 i,j=1 l=1 n s i,j=1 l=1 [ ] u il u jl (1 u il u jl ) ( n s1 ( ) ) u il + ( 1) γl+1 u jl γ l (1 u il + ( 1) γl+1 u jl γ l ) ( ) u il u (γ l) jl (1 u il u (γ l) jl ), ()

7 78 Ou et al. where u (γ l) jl = xl j (γ l)+1 and x l 6 j (γ l) = (x jl + γ l )(mod). For γ = (γ 1,γ,...,γ s ) Ɣ, lete 1 ={v γ v = 1,v = 1,...,s 1 } and E ={v γ v 0,v = s 1,...,s} of cardinality t 1 and t, respectively, let E 1 ={1,,...,s 1 } E 1 and E ={s 1,s 1 +,...,s} E.LetƔ t1 t be the set of all elements γ Ɣ such that γ exactly have t 1 and t non zero components among {γ 1,...,γ s1 } and {γ s1 +1,...,γ s }, respectively. Given d, s 1, s, t 1, and t,letf(n, s 1,s,t 1,t ) = min{[wd(d(γ ))] : γ Ɣ t1 t }. The optimal foldover plan in this paper is defined as the foldover plan γ such that [WD(d(γ )] = f (n, s 1,s,t 1,t ). The next section attempts to derive a lower bound to [WD(d(γ))].. Lower Bound to the Discrepancy Measure of the Combined Design Downloaded by [ ] at 16:7 0 April The Equivalence Between γ and γ in Terms of Uniformity Theorem.1. Let d be an U-type design in U(n; s 1 s ), for any γ Ɣ, γ is the complementary foldover plan of γ, then γ is equivalent to γ in terms of uniformity measured by wrap-around L -discrepancy, that is, [WD(d(γ ))] = [WD(d( γ ))]. Proof. For any γ Ɣ, leth ij be the Hamming distance between the ith and jth runs of d(γ ) in algebraic coding theory, h ij (γ ) be the Hamming distance between the ith run of d and jth run of the foldover design d γ.form = 0, 1,...,s, define H m (γ ) = 1 n {(i, j): h ij (γ ) = m, i, j = 1,...,n}, where denotes the cardinality of a set. It is to be note that H m (γ ) = H m ( γ )for m = 0, 1,...,s. Since the third term of the right side of () is decided by {H m (γ )} s m=0, and the first two terms of the right side of () is free from the foldover plan γ, this yields [WD(d(γ))] = [WD(d( γ ))], and γ is equivalent to γ, which completes the proof. Corollary.1. Let d be an U-type design in U(n; s 1 s ), γ Ɣ is the optimal foldover plan of d in the sense of uniformity measured by wrap-aroud L -discrepancy, then the complementary foldover plan γ of γ also is an optimal foldover plan of d. Remark.1. Theorem.1 is very useful to search the optimal foldover plans in the full foldover plan space Ɣ. Specifically, it only requires s 1 ( s )/ times search for a design d U(n; s 1 s ) based on Theorem.1, comparing to exhaustive search method which requires s 1 s times search... A Lower Bound of [WD(d(γ))] Based on the Row Distance It is to be noted that for l E, u il u (γ l) il (1 u il u (γ l) il ) = 9,

8 Lower Bound of Discrepancy on Combined Designs 79 then we can rewrite Eq. () as follows: [WD(d(γ ))] = ( ) s n n n 1 i j n l=1 1 i j n ( ) s n s ( l E 1 ( ) s1 t 1 ( 5 ) t1 ( [ ] u il u jl (1 u il u jl ) ) s t ( ) t 18 ( ) u il + u jl 1 (1 u il + u jl 1 ) Downloaded by [ ] at 16:7 0 April 01 l E 1 ( l E l E ( ) u il u jl (1 u il u jl ) ( ) u il u (γ l) jl (1 u il u (γ l) jl ) ( ) u il u jl (1 u il u jl ). () From the analytical expression of Eq. (), it is easy to see that [WD(d(γ))] is only a function of products of αij l u il u jl (1 u il u jl ), βij l u il+u jl 1 (1 u il +u jl 1 ) and λ l ij u il u (γ l) jl (1 u il u (γ l) jl ). However, for an U-type design, its α-, β-, and λ-values can only be limited to a specific set. More precisely, for an U-type design d U(n; s 1 s ), when 1 l s 1, both α- and β-values can only take possible values, i.e., 0, 1/, when s 1 l s, both α- and λ-values can only take two possible values, i.e., 0, /9. Table gives the distribution of α-values over the set {αij l :1 i j n} for l Ē i,i = 1,, 1 l s 1 and s 1 l s, respectively. Table gives the distribution of β-values over the set {βij l :1 i j n, l E 1} and the distribution of λ-values over the set {λ l ij :1 i j n, l E }. Note that given (n, s 1,s,t 1,t ), these distributions are the same for each design in d U(n; s 1 s ) and each foldover plan γ Ɣ t1 t. The following theorem provides a lower bound to [WD(d(γ))]. Table Distributions of α-values of a d U(n; s 1 s ) and a γ Ɣ t1 t l E 1 1 l s 1 l E s 1 l s α-values Number α-values Number α-values Number α-values Number 0 1 (s 1 t 1 )n(n ) 0 (s 1 t 1 )n 1 s 1 n(n ) 0 s 1 n 9 (s t )n(n ) 0 (s t )n 9 s n(n ) s n

9 80 Ou et al. Table Distributions of β-values and λ-values of a d U(n; s 1 s ) and a γ Ɣ t1 t l E 1 l E β-values Number λ-values Number 0 t 1 n t n 1 0 t 1 n(n ) 9 t n(n ) Theorem.. Consider a design d U(n; s 1 s ), for any 0 t i s i,i = 1,, and for any γ = (γ 1,...,γ s ) Ɣ t1 t, we have Downloaded by [ ] at 16:7 0 April 01 [WD(d(γ ))] ( ) s n ( ) s1n 5 (n 1) ( 18 ( ) 5 s1n t1 (n 1) ( ) s ( 1 + ( 5 6 ) sn (n 1) + n 1 n ( ) sn t (n 1). 18 ) t1 ( ) t ) + n 1 7 n ( ) s1n s1+t1 + sn s+t (n 1) (n 1) ( ) s1(n ) (n 1) + s(n ) (n 1) Moreover, [WD(d(γ))] achieves the above lower bound if all s l=1 ( αl ij ) and all l E 1 ( βl ij ) l E 1 E ( αl ij ) l E ( λl ij ) are the same for 1 i j n. Proof. By Eq. (), to minimize [WD(d(γ ))] over Ɣ for a given design d U(n; s 1 s ) is equivalent to minimizing with respect to α l ij and = 1 i j n l E 1 1 = s 1 i j n l=1 ( ) βl ij l E 1 E ( ) αl ij ( ) ( ) αl ij λl ij l E with respect to α l ij, βl ij, and λl ij. From Tables and, we know that for given (n, s 1,s,t 1,t ), the distributions of α-, β- and λ-values are the same, so both s 1 i j n l=1 ( ) αl ij and 1 i j n l E 1 ( ) βl ij l E 1 E ( ) ( ) αl ij λl ij l E are constants on Ɣ t1 t. Moreover, αl ij > 0, βl ij > 0 and λl ij > 0. Based on the geometric and arithmetic mean inequality, [WD(d(γ))] arrives at its minimum if all s l=1 ( αl ij ) and all l E 1 ( βl ij ) l E 1 E ( αl ij ) l E ( λl ij ) are the same for 1 i j n. The expression of the lower bound of [WD(d(γ))] is straightforward according to Table and Table for 1 and, which completes the proof.

10 Lower Bound of Discrepancy on Combined Designs 81 Remark.. If we ignore the foldover structure of d(γ ), then d(γ ) can be regarded as a design in U(n; s 1 s ). Following Zhou and Ning (008), a unify lower bound of [WD(d(γ))] for any t 1 = 0, 1,...,s 1 and t = 0, 1,...,s can be obtained as follows: where LB(n, s 1,s ) = [WD(d(γ ))] LB(n, s 1,s ), ( ) s n + n 1 n ( ( ) 1 n 1 ) s [ s 1 (n 1)+ s (n ) ] ( ) 5 s 1 n n 1 ( 18 ) s n (n 1). () Downloaded by [ ] at 16:7 0 April 01 Numerical examples (see Examples.1. in Sec. ) show that the lower bounds in Theorem. are tighter than LB(n, s 1,s ) for any t 1 = 0, 1,...,s 1 and t = 0, 1,...,s, since the lower bounds in Theorem. make full use of the foldover structure of d(γ ). Corollary.. If s 1 = 0 and consider a design d U(n; s ), for any 0 t s and for any γ Ɣ 0t, we have [WD(d(γ ))] ( ) s n + n 1 n ( ) sn s+t (n 1) ( ) s ( 1 + ( ) t ) + n 1 7 n ( ) s n t (n 1). 18 ( ) s(n ) (n 1) ( 18 ) sn (n 1) Remark.. If s = 1 and consider a design d U(n; s 1 ), for any 0 t 1 s 1, t = 0, 1, and for any γ Ɣ t1 t, one can obtain a lower bound of [WD(d(γ ))] by Theorem.. For any design d U(n; s 1 p)(p ), Qin et al. (01) also obtained a lower bound of centered L -discrepancy of combined design of d for any 0 t 1 s 1, t = 0, that is, for any γ Ɣ t1 0 which the levels of the last factor are retained as they are in the design d.. Numerical Examples In this section, we give some examples to show our theoretical results. For convenience, we denote the lower bound of [WD(d(γ))] which given in Theorem. as LB(t 1,t ) for any 0 t i s i, i = 1,, and γ Ɣ t1 t in the following Tables. One of the optimal foldover plan over the full foldover plan set Ɣ of d, the corresponding minimum squared wrap-around L -discrepancy value and its lower bound are marked by bold font. Given design d U(n; s 1 s ), one can find that the lower bounds LB(t 1,t ) in Theorem. are tighter than the lower bound LB(n, s 1,s ) given in Remark. for any t 1 = 0, 1,...,s 1 and t = 0, 1,...,s in the following four examples. Example.1. (Continued) Consider the design d 1 U(1; ) as shown in Example.1. Table shows the numerical results of d 1.From(),wehaveLB(1,, 1) =

11 8 Ou et al. Table Numerical results for d 1 t = 0 t = 1 t 1 [WD(d 1 (γ ))] LB(t 1,t ) γ [WD(d 1 (γ ))] LB(t 1,t ) γ (00000) (0 000) (10000) (0 0101) (11000) (1 010) (11010) (0 1111) (11110) (1 111) Downloaded by [ ] at 16:7 0 April 01 Example.. Consider a design d U(18; 7 ), given below, with n = 18,s 1 = 1,s = d = Table 5 shows the numerical results of d.from(),wehavelb(18, 1, 7) = 1.1. Table 5 Numerical results for d t 1 = 0 t 1 = 1 t [WD(d (γ ))] LB(t 1,t ) γ [WD(d (γ ))] LB(t 1,t ) γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0 000) ( ) ( ) (1 1 00) (0 011) (1 1 0) ( ) (1 1 11)

12 Lower Bound of Discrepancy on Combined Designs 8 Table 6 Numerical results for d t [WD(d (γ ))] LB(t 1,t ) γ (0000) (1000) (1100) (1110) (1111) Example.. Consider a design d U(9; ), given below, with n = 9,s 1 = 0,s =. Downloaded by [ ] at 16:7 0 April d = Table 6 shows the numerical results of d.from(),wehavelb(9, 0, ) = One can find that the lower bound LB(t 1,t ) given in Theorem. is tight when t = 0. Example. Consider a design d U(1; ), given below, with n = 1,s 1 = 11,s = d = All Hamming distances between any distinct pair of runs of d are equal to 1. Table 7 shows the partial numerical results of d only for t = 0 in order to save space. From (), we have LB(1, 11, 11) = 89.. One can find that the lower bounds LB(t 1,t )given in Theorem. are tight when (t 1,t ) = (0, 0) and (t 1,t ) = (11, 0).

13 8 Ou et al. Table 7 Partial numerical results for d t = 0 t 1 [WD(d (γ ))] LB(t 1,t ) γ Downloaded by [ ] at 16:7 0 April ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Acknowledgments The authors would like to thank the referee and Chief Editor for their valuable comments and suggestions that lead to improve the presentation of this article. Funding This work was partially supported by the National Natural Science Foundation of China (Grant Nos , ), SRFDP (No ), China Postdoctoral Science Foundation (No. 01M51716) and the Research Project Funded by Jishou University (No. jsdxxcfxbskyxm0111). References Box, G. E. P., Hunter, W. G., Hunter, J. S. (1978). Statistics for Experiments. New York: John Wiley and Sons. Chatterjee, K., Fang, K. T., Qin, H. (006). A lower bound for the centered L -discrepancy on asymmetric factorials and its application. Metrika 6: 55. Fang, K. T., Lin, D. K. J., Qin, H. (00). A note on optimal foldover design. Statist. Probab. Lett. 6:5 50. Fang, K. T., Lu, X., Winker, P. (00). Lower bounds for centered and wrap-around L -discrepancies and construction of uniform design by threshold accepting. J. Complex. 19: Fang, K. T., Tang, Y., Yin, J. X. (005). Lower bounds for wrap-around L -discrepancy and constructions of symmetrical uniform designs. J. Complex. 1: Hickernell, F. J. (1998). A generalized discrepancy and quadrature error bound. Math. Computat. 67:99. Lei, Y. J., Ou, Z. J., Qin, H., Zou, N. (01). A note on lower bound of centered L -discrepancy on combined designs. Acta Mathematica Sinica. 8(): Lei, Y. J., Qin, H., Zou, N. (010). Some lower bounds of centered L -discrepancy on foldover designs. Acta Math Sci. 0A(6):

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