A class of mixed orthogonal arrays obtained from projection matrix inequalities

Transcription

1 Pang et al Journal of Inequalities and Applications :241 DOI /s R E S E A R C H Open Access A class of mixed orthogonal arrays obtained from projection matrix inequalities Shanqi Pang *, Yan Zhu and Yajuan Wang * Correspondence: pangshanqi@263net Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, School of Mathematics and Information Science, Henan Normal University, Xinxiang, , China Abstract By using the relationship between orthogonal arrays and decompositions of projection matrices and projection matrix inequalities, we present a method for constructing a class of new orthogonal arrays which have higher percent saturations As an application of the method, many new mixed-level orthogonal arrays of run sizes 108 and 144 are constructed MSC: 62K15; 05B15 Keywords: matrix inequality; orthogonal array; orthogonal projection matrix; matrix image 1 Introduction An n m matrix A, havingk i columns with p i levels, i = 1,2,,r, m = r k i, p i p j for i j, is called an orthogonal array OA of strength d and size n if each n d submatrix of A contains all possible 1 d row vectors with the same frequency Unless stated otherwise, we use the notation L n p k 1 1 pk r r for an OA of strength 2 An orthogonal array is said to have mixed levels if r 2 Orthogonal arrays have been used extensively in statistical design of experiments, computer science and cryptography Constructions of OAs have been studied extensively in the literature; see Hedayet et al [1, 2], Zhang et al [3, 7], Pang et al [5], Pang [6], Zhang et al [7], Chen et al [8], Du et al [9], etc Zhang et al [3] presentamethodofconstructionoforthogonalarraysofstrengthtwobyusing a relationship between orthogonal arrays and decompositions of projection matrices In the construction of new mixed orthogonal arrays, two goals should be kept in mind, first, we want the orthogonal array to be as close to a saturated main-effect plan as possible so that there will be a large number of factors and second, we want the p i,thenumber of levels, to be as large as possible so that the design has a high degree of flexibility see Mandeli [10] In this paper by using projection matrix inequalities and further exploring the relationship, matrix images of a class of orthogonal arrays can be found Therefore we constructa classof new orthogonalarrays If, asin Mandeli [10], we still define the percent saturation of an OA L n p k 1 1 pk r r tobe r k ip i 1/n 1 100%, then the orthogonal arrays constructed in this paper have higher percent saturations 2 Basic concepts and main theorems The following definitions, notations, and results are needed in the sequel 2015 Pang et al This article is distributed under the terms of the Creative Commons Attribution 40 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made

2 Pang et al Journal of Inequalities and Applications :241 Page 2 of 9 Definition 1 AmatrixA is said to be an orthogonal projection matrix if it is idempotent A 2 = A and symmetric A T = A Definition 2 Suppose that an experiment is carried out according to an array A = a ij n m =a 1,,a m, and Y =Y 1,,Y n T is the experimental data vector In the analysis of variance Sj 2, the sum of squares of the jth factor, is defined as p j 1 Sj n = Y s 1 2 Y s, 1 I i=0 ij n s I ij s=1 where I ij = {s : a sj = i} and I ij is the number of elements in I ij From1, S 2 j is a quadratic form in Y and there exists a unique symmetric matrix A j such that S 2 j = Y T A j Y Thematrix A j is called the matrix image MI of the jth column a j of A, denotedbyma j =A j The MI of a subarray of A is defined as the sum of the MIs of all its columns In particular, we denote the MI of A by ma Let 1 r be the r 1vectorof1 sthenm1 r =P r where P r = 1 r 1 r1 T r Let r=0,,r 1 T, e i r=0 0 i T 1 r and I r be the identity matrix of order r Then mr = τ r where τ r = I r P r The following permutation matrices are very useful: N r = e 1 re T 2 r+ + e r 1re T r r+e rre T 1 r and Kp, q= p q e i pe T j q e j qe T i p, j=1 where is the usual Kronecker product in the theory of matrices Sometimes, it is necessary and easy to use the following properties of these two permutation matrices to obtain the orthogonal arrays needed: N r r=1 r +r, mod r, Kp, q 1 q p =p 1 q, Kp, q q p =p q and Kp, qp q τ p K T p, q=τ p P q, Kp, qτ q τ p K T p, q=τ p τ q Lemma 1 For any permutation matrix S and any array L, msl 1 r = SmL P r S T, and ms1 r L = SP r mls T Lemma 2 Let A be an OA of strength 1, ie, A =a 1,,a m = S 1 1r1 p 1,,S m 1rm p m, where r j p j = nands i is a permutation matrix, for i =1,,m The following statements are equivalent

3 Pang et al Journal of Inequalities and Applications :241 Page 3 of 9 1 A is an OA of strength 2 2 ma is a projection matrix 3 ma i ma j =0i j 4 The projection matrix τ n can be decomposed as τ n = ma ma m +, where rk =n 1 m j=1 p j 1istherankofthematrix Lemma 3 Suppose L and H are OAs satisfying mlmh=0then K =L, H is also an OA Lemma 4 Let L, H and K be OAs of run size n Then K, H is also an OA if mk ml, where A B means that the difference B A is nonnegative Lemmas 1, 2, 3,and4 can be found in Zhang et al [3] Now we state the following theorems Theorem 1 If L =[r 1 p, H] is an OA, then mh I r τ p If L =[1 p r, K] is an OA, then mk τ p I r Proof From Lemma 2, wehaveml=τ r P p + mh Since ml τ rp, τ rp = τ r P p + I r τ p and mr 1 p =τ r P p, it follows that mh I r τ p Similarly, we can prove that mk τ p I r Corollary If p is a prime and Dr, m; p is a p-level difference matrix, then both Dr, m; p p and p Dr, m; p are OAs, and mdr, m; p p I r τ p and mp Dr, m; p τ p I r Proof From Bose and Bush [11], we see that L = [ r 1 p, Dr, m; p p ] is an OA From Theorem 1, it follows that mdr, m; p p I r τ p Similarly, we can prove that p Dr, m; pisanoaandmp Dr, m; p τ p I r Theorem 2 If p is a prime and Dr, m; p is a p-level difference matrix, then Dr, m; p 0 q p is an OA, and mdr, m; p 0 q p I r P q τ p Proof From Lemma 1 and Theorem 1,wehavemDr, m; p p 0 q I r τ p P q, m Dr, m; p 0 q p = m I r Kq, p Dr, m; p p 0 q I r Kq, p I r τ p P q I r Kq, p T = I r P q τ p Theorem 3 If τ p τ q = k S iτ p P q S T i is an orthogonal decomposition of τ p τ q, then I r τ p τ q = k I r S i I r τ p P q I r S i T is an orthogonal decomposition of I r τ p τ q If there exists an OA H such that mh I r τ p, then L =[I r S 1 H 1 q,,i r S k H 1 q ] is also an OA

4 Pang et al Journal of Inequalities and Applications :241 Page 4 of 9 Proof Since τ p τ q = k S iτ p P q S T i,wehave k I r τ p τ q = I r S i τ p P q Si T = k I r I r I r S i τ p P q Si T Using the property ABC DEF=A DB EC F, we obtain I r τ p τ q = k I r S i I r τ p P q I r S i T Because of the orthogonality in each step, the above decomposition of I r τ p τ q is orthogonal By Lemma 1,wehave m I r S i H 1 q =I r S i H 1 q I r S i T I r S i I r τ p P q I r S i T By Lemmas 3 and 4,weseethatI r S i H 1 q andi r S j H 1 q isorthogonali j It follows from Lemma 2 that L is an OA 3 Someexamples These matrix inequalities in Theorems 1, 2, and 3 are very useful for construction of orthogonal arrays We illustrate their applications with some examples We begin with OAs L andl andtheirproperties: L = 3 1 3,1 3 3, a, b = For this OA L 9 3 4, there exists a 9 9permutationT 2 as follows such that 3 1 3,1 3 3 = T 2 a, b From Lemma 2, wehaveml = τ 9, ma, b=τ 9 τ 3 P 3 P 3 τ 3 = τ 3 τ 3 and m3 1 3,1 3 3 = T 2 τ 3 τ 3 T T 2 Hence T τ 9 = T i τ 3 τ 3 Ti T, where T 1 = I 9 and T 2 =

5 Pang et al Journal of Inequalities and Applications :241 Page 5 of 9 On the other hand, we have τ 9 τ 3 P 3 P 3 τ 3 = τ 3 τ 3 Also,τ 3 τ 3 = 2 Q iτ 3 P 3 Q T i,whereq 1 = diagi 3, N 3, N3 2K3, 3 and Q 2 = diagi 3, N3 2, N 3K3, 3 Consider the orthogonal array L : L = 4 1 4,1 4 4, c, d, f T = From Lemma 2,wehaveτ 16 = 2 S iτ 4 P 4 Si T +τ 4 τ 4,whereS 1 = I 16 and S 2 = K4, 4 Moreover, we have τ 4 τ 4 = 5 i=3 S iτ 4 P 4 Si T,whereS 3 = diagq 1, Q 2, Q 3, Q 4 K4, 4, S 4 = diagq 1, Q 3, Q 4, Q 2 K4, 4, S 5 = diagq 1, Q 4, Q 2, Q 3 K4, 4, Q 1 = I 4, Q 2 = I 2 N 2, Q 3 = N 2 I 2 and Q 4 = N 2 N 2 Example 1 Construction of orthogonal arrays of run size 108 Orthogonally decompose the projection matrix τ 108 as follows: τ 108 = I 12 τ 9 + τ 12 P 9 = I 12 τ 3 τ 3 + I 12 τ 3 P 3 + I 12 P 3 τ 3 +P 6 τ 2 + P 3 τ 2 τ 2 + τ 3 I 4 + P 3 τ 2 P 2 P 9 = I 12 τ 3 τ 3 + I 12 τ 3 P 3 +P 6 τ 2 + P 3 τ 2 τ 2 P 9 + I 12 P 3 τ 3 +τ 3 I 4 + P 3 τ 2 P 2 P 9 = I 12 τ 3 τ 3 + I 12 τ 3 +P 6 τ 2 + P 3 τ 2 τ 2 P 3 P3 + Q I 12 τ 3 +τ 3 I 4 + P 3 τ 2 P 2 P 3 P3 Q T, where Q = I 12 K3, 3 Now we want to find OAs H 1, H 2 and H 3 such that mh 1 I 12 τ 3 τ 3,andmH 2 I 12 τ 3 +P 6 τ 2 + P 3 τ 2 τ 2 P 3,andmH 3 I 12 τ 3 +τ 3 I 4 + P 3 τ 2 P 2 P 3 Let be the Kronecker sum mod 3 in ordinary matrix theory and D12, 12, 3 be a difference matrix as follows: D12, 12, 3 =

6 Pang et al Journal of Inequalities and Applications :241 Page 6 of 9 It follows from the corollary of Theorem 1 that H = L =D12, 12, 3 3 is an OA and mh I 12 τ 3 Fromthepropertyτ 3 τ 3 = 2 Q iτ 3 P 3 Q T i and Theorem 3,we have an orthogonal array H 1 : H 1 = L = [ I r Q 1 H 1 3, I r Q 2 H 1 3 ] and mh 1 I 12 τ 3 τ 3 From the orthogonal array L inzhanget al [4], we can get an orthogonal array L =K12, 3L =[L , L ] satisfying ml τ 36 = I 12 τ 3 +τ 12 P 3 Deleting the orthogonal array L from L , we obtain an OA H 2 = L whose MI satisfies mh 2 I 12 τ 3 +P 6 τ 2 + P 3 τ 2 τ 2 P 3 By the definition of an OA, there exists a permutation matrix T such that TL contains the two columns and Deleting these two columns from TL , we obtain an OA H 3 = L satisfyingmh 3 I 12 τ 3 +τ 3 I 4 + P 3 τ 2 P 2 P 3 Hence we can construct a new OA L as follows: L = [ H 1, H 2 1 3, QH ] 2 The percent saturation for this OA is 842% Also, similarly constructing of H 2 and by use of the orthogonal arrays L , L , L andl inzhanget al [3], L inhedayetet al [2], L andl inzhanget al [4],wecanobtainsevenOAsL , L , L , L , L , L andl whosemisare less than or equal to I 12 τ 3 +P 6 τ 2 + P 3 τ 2 τ 2 P 3 On the other hand, by the definition of an OA, any OA of run size 36 with two factors having two levels can contain the two columns and through row permutations For example, there exists OA L in Example 2, L , L , L , L , L , L , L , L inzhang et al [3], L inhedayetet al [2], L , L , L , L in Zhang et al [4], and L , L , L inxu[12] Deleting these two columns and from these arrays, we can obtain 17 OAs L , L , L , L , L , L , L , L , L , L , L , L , L , L , L , L , and L whose MIs are less than or equal to I 12 τ 3 + τ 3 I 4 + P 3 τ 2 P 2 P 3 Replacing H 2 in 2 by those seven OAs and H 3 in 2bythese17OAs,respectively,we can construct = 143 OAs such as L , etc The orthogonal arrays constructed not only are new but also have higher percent saturations Example 2 Construction of orthogonal arrays of run size 144 From the above properties of OAs L andl , we have τ 9 = T i τ 3 τ 3 Ti T,

7 Pang et al Journal of Inequalities and Applications :241 Page 7 of 9 where T 1 = I 9 and T 2 is the above permutation matrix and τ 16 = S i τ 4 P 4 Si T + τ 4 τ 4 By using the properties I 9 = I 9 I 9 I 9, T i I 9 T T i = I 9, S i P 16 S T i = P 16 i =1,2,ABC DEF= A DB EC F, we can orthogonally decompose τ 144 as follows: τ 144 = I 9 τ 16 + τ 9 P 16 = T i S i I 9 τ 4 + τ 3 τ 3 P 4 P 4 Ti S i T + I 9 τ 4 τ 4 From the L , we have τ 4 τ 4 = 5 S i τ 4 P 4 Si T i=3 Hence, τ 144 can be further decomposed as τ 144 = T i S i I 9 τ 4 + τ 3 τ 3 P 4 P 4 Ti S i T + 5 I 9 S i I 9 τ 4 P 4 I 9 S i T i=3 Now we want to find OAs H 1 and H 2 such that mh 1 I 9 τ 4 + τ 3 τ 3 P 4 and mh 2 I 9 τ 4 By the definition of an OA and through row permutation, any OA with two 3-level columns can contain the two columns , For example, from the result in Bose and Bush [11], we see that K 1 =[1 3 L 12 12, 3 D12, 12, 3] is an OA, where D12, 12, 3 is a difference matrix in Example 1 It is clear that K 1 contains two columns and ,anddiagI 3, N 2 3, N 3 I 4 [3 1 12, ] = [3 1 12, ] Set K 2 =diagi 3, N 2 3, N 3 I 4 K 1,wegetanOAK 2 = L , which contains two columns and Deleting these two columns from K 2,weobtainan OA H 1 = L ,whoseMIsatisfiesmH 1 =τ 36 τ 3 P 12 P 3 τ 3 P 4 = I 9 τ 4 + τ 3 τ 3 P 4 Now we need to find another OA H 2 such that mh 2 I 9 τ 4 Byusingrowpermutations of the orthogonal array L cf [13],wecanobtainanOAL as follows: L x x x y x y x y y z y y x x z y 1 x x x z z y x x z x z z x y z z 2 x y x y y y x x x y y z z x x y 3 x z y x x y x y z y y x y x x y = 4 x x x y y y z x y y y x y x x z, 5 x x z z x y x y y y y y x x x y 6 x x x x x y z x x y x z z y y z 7 x y y z z y z y z y x y z z z y 8 x x x y z y x z y x y y x x y y

8 Pang et al Journal of Inequalities and Applications :241 Page 8 of 9 where i =i, i, i, i T,fori =0,1,,8, x = 0, 0, 1, 1 T, x = 1, 1, 0, 0 T, y = 0, 1, 0, 1 T, y = 1,0,1,0 T, z = 0, 1, 1, 0 T and z = 1, 0, 0, 1 T Deleting the column from the L , we obtain an OA H 2 = L By using Theorem 1,weseethatmH 2 I 9 τ 4 From Theorem 3 the decomposition of τ 144,weconstructanewOAL as follows: L = [ T 1 S 1 H 1 1 4, T 2 S 2 H 1 1 4, I 9 S 3 H 2 1 4, I 9 S 4 H 2 1 4, I 9 S 5 H ] 3 The percent saturation for this OA is 764% In fact, the percent saturations for the arrays constructed by Wang and Wu [14] are between 497% and 538% and the highest percent saturation for the arrays with 144 runs constructed by Mandeli [10]is622% Also, by the definition of an OA, any OA of run size 36 with two factors having three levels can contain the two columns and through row permutations For example, there exist OAs L , L , L inzhanget al [3], L , L , L , L , L , L , L , L inzhanget al [4], L , L inxu[12], L in Finney [15], etc having and Deleting these two columns, we can obtain 14 OAs L , L , L , L , and so on Replacing H 1 in 3bythese OAs, respectively, we can construct 195 OAs such as L , etc The orthogonal arrays constructed not only are new but also have higher percent saturations Competing interests The authors declare that they have no competing interests Authors contributions The research and writing of this manuscript was a collaborative effort of all the authors All authors read and approved the final manuscript Acknowledgements The authors sincerely thank the referees for their many valuable comments which help us improving the paper This work is supported by the National Natural Science Foundation of China Grant No and IRTSTHN 14IRTSTHN023 Received: 5 January 2015 Accepted: 20 July 2015 References 1 Hedayet, AS, Pu, K, Stufken, J: On the construction of asymmetrical orthogonal arrays Ann Stat 20, Hedayat, AS, Sloane, NJA, Stufken, J: Orthogonal Arrays: Theory and Applications Springer, New York Zhang, Y, Lu, Y, Pang, S: Orthogonal arrays obtained by orthogonal decomposition of projection matrices Stat Sin 92, Zhang, Y, Pang, S, Wang, Y: Orthogonal arrays obtained by the generalized Hadamard product Discrete Math 238, Pang, S, Zhang, Y, Liu, S: Normal mixed difference matrix and the construction of orthogonal arrays Stat Probab Lett 69, Pang, S: Construction of a new class of orthogonal arrays J Syst Sci Complex 203, Zhang, Y, Li, W, Mao, S, Zheng, Z: Orthogonal arrays obtained by generalized difference matrices with g levels Sci China Math 541, Chen, G, Ji, L, Lei, J: The existence of mixed orthogonal arrays with four and five factors of strength two J Comb Des 228, Du, J, Wen, Q, Zhang, J, Liao, X: New construction and counting of systemic orthogonal array of strength tieice Trans Fundam Electron Commun Comput Sci E96-A9, Mandeli, JP: Construction of asymmetrical orthogonal arrays having factors with a large non-prime power number of levels J Stat Plan Inference 47, Bose, RC, Bush, KA: Orthogonal arrays of strength two and three Ann Math Stat 23,

9 Pang et al Journal of Inequalities and Applications :241 Page 9 of 9 12 Xu, H: An algorithm for constructing orthogonal and nearly orthogonal arrays with mixed levels and small runs Technometrics 44, Kuhfeld, W: Orthogonal arrays Wang, J, Wu, CFJ: An approach to the construction of asymmetrical orthogonal arrays J Am Stat Assoc 6, Finney, DJ: Some enumerations for the 6 6 Latin squares Util Math 21,