Construction of Supersaturated Design with Large Number of Factors by the Complementary Design Method

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1 Acta Mathematicae Appicatae Sinica, Engish Series Vo. 29, No. 2 (2013) DOI: /s & Acta Mathema cae Appicatae Sinica, Engish Series The Editoria Office of AMAS & Springer-Verag Berin Heideberg 2013 Construction of Supersaturated Design with Large Number of Factors by the Compementary Design Method Yan LIU 1, Min-Qian LIU 2 1 Schoo of Mathematica Sciences, Tianjin Norma University, Tianjin , China 2 Department of Statistics, Schoo of Mathematica Sciences and LPMC, Nankai University, Tianjin , China (E-mai: mqiu@nankai.edu.cn) Abstract Supersaturated designs (SSDs) have been widey used in factor screening experiments. The present paper aims to prove that the maxima baanced designs are a kind of specia optima SSDs under the E(f NOD ) criterion. We aso propose a new method, caed the compementary design method, for constructing E(f NOD ) optima SSDs. The basic principe of this method is that for any existing E(f NOD ) optima SSD whose E(f NOD ) vaue reaches its ower bound, its compementary design in the corresponding maxima baanced design is aso E(f NOD ) optima. This method appies to both symmetrica and asymmetrica (mixed-eve) cases. It provides a convenient and efficient way to construct many new designs with reativey arge numbers of factors. Some newy constructed designs are given as exampes. Keywords baanced design; coincidence number; compementary design; equidistant design; weak equidistant design 2000 MR Subject Cassification 62K15; 62K05 1 Introduction Supersaturated designs (SSDs) are a kind of factoria designs whose run sizes are ess than the numbers of their main effects. They are mainy used in the factor screening stage of an industria or scientific experiment to seect sparse and dominant effects. The anaysis of SSDs reies on the assumption of effect sparsity, which says that ony a few dominant factors actuay affect the response. The probem of constructing SSDs was first proposed by Satterthwaite [28]. Then Booth, Cox [2] examined two-eve SSDs systematicay. After then, such designs were not studied further unti the appearance of the work by Lin [18],Wu [31]. SSDs have received much interest in recent years because of their abiities of screening factors with ow cost. Some most recent studies on symmetrica SSDs incude Fang, Ge, Liu [8], Fang, Ge, Liu, Qin [10], Xu, Wu [33], Georgiou, Koukouvinos [14], Georgiou, Koukouvinos and Mantas [15], and Liu, Liu, Zhang [26]. For asymmetrica SSDs, Fang, Lin and Liu [11,12] proposed the E(f NOD ) criterion, and Yamada and Matsui [37] and Yamada, Lin [35] used the χ 2 criterion. Recent studies incude Li, Liu, Zhang [17], Fang, Ge, Liu, Qin [9], Koukouvinos, Mantas [16], Liu, Fang, Hickerne [20], Yamada, Matsui, Matsui, Lin, Tahashi [36],Ai,Fang,He [1], Zhang, Zhang, Liu [40], Tang, Ai, Ge, Fang [30], Chen, Liu [3,4], Liu, Lin [21],Liu,Cai [19], Liu, Zhang [22], Sun, Lin, Liu [29], Liu, Liu [24,25], Huang, Lin, Liu [13], and Yin, Zhang, Liu [38]. In particuar, Sun, Lin, Liu [29] provided a comprehensive Manuscript received June 23, Revised Marh 14, Supported by the Nationa Natura Science Foundation of China (Nos and ), the 131 Taents Program of Tianjin, the Fundamenta Research Funds for the Centra Universities (No ), and the Doctor Foundation of Tianjin Norma University (No. 52XB1205). 2 Corresponding author.

2 254 Y. LIU, M.Q. LIU ist of eary works on SSDs and an extensive review of the existing methods for mixed-eve SSDs. So far, most studies have focused on the construction and anaysis of SSDs with reativey sma numbers of factors. However, much practica experience indicates that SSDs with reativey arge numbers of factors aso have wider appications. This paper provides a new method, caed the compementary design method, for constructing E(f NOD ) optima SSDs with reativey arge numbers of factors. Some definitions and notations are introduced in the rest of this section. Section 2 reviews the E(f NOD ) criterion and its reated properties. Section 3 proves that a symmetrica maxima baanced design is an E(f NOD ) optima design. Section 4 introduces the new method for constructing symmetrica E(f NOD ) optima designs. Section 5 extends the method to the asymmetrica case and provides some discussion. Let D(n; q 1,,q m ) denote an asymmetrica (or mixed-eve) design of n runs and m factors with eves q 1,,q m,wherethetth factor takes vaues from a set of q t symbos {0,,q t 1}. This design can aso be written as an n m matrix D =(d it ), where the ith row and the tth coumn are denoted by d i and d t, respectivey. A D(n; q 1,,q m ) is caed an orthogona array (OA) of strength two, denoted by L n (q 1,,q m ), if a possibe eve-combinations for any two factors appear equay often. When m (q t 1) = n 1, the D(n; q 1,,q m ) design is caed t=1 saturated; and when m (q t 1) >n 1, the design is caed a supersaturated design (SSD). t=1 When some q t s are equa, we use notations D(n; q r1 1 qr )andl n (q r1 1 qr ) respectivey, where r k = m. Whenatheq t s are equa, we use D(n; q m )andl n (q m ) respectivey, and ca them symmetrica designs. Two coumns are caed orthogona if their eve-combinations appear equay often and caed fuy aiased if one can be obtained from the other by permuting eves. Fuy aiased coumns shoud be avoided in a design as they cannot accommodate two different factors independenty. A design is caed baanced if the eves appear equay often in each coumn of the design. Throughout this paper, we wi focus on baanced designs ony. 2 E(f NOD ) Criterion and Its Reated Properties For a D(n; q r1 1 qr ) design with r k = m, we suppose the tth and sth coumns d t and d s have q t and q s eves, respectivey. The E(f NOD ) criterion is defined as E(f NOD )= 1 t<s m / ( ) f NOD (d t,d s m ), 2 where f NOD (d t,d s )= q t 1 u=0 q s 1 v=0 ( n (ts) uv n q t q s ) 2, and n (ts) uv is the number of (u, v)-pairs in (d t,d s ). Let λ ij denote the coincidence number between the ith and jth rows. Fang, Lin, Liu [11,12] studied the reation between E(f NOD )andλ ij s and obtained a ower bound of E(f NOD ). Later the ower bound was improved by Fang, Ge, Liu, Qin [9]. Some of their concusions are presented in the foowing emma.

3 SSD with Large Number of Factors 255 Lemma 1. λ = ( n For any D(n; q r1 1 qr ) design D with n λ 2 ij i,j=1,i j E(f NOD )= m(m 1) + C f r k = m, n(n 1) [ ( λ +1 λ)(λ λ )+λ 2 ] + C f, (1) m(m 1) where C f = nm m 1 n 2 ( r k r k (r k 1) r h r ) k + m(m 1) q k qk 2 +, q h q k k,h=1,h k r k /q k m)/(n 1), and λ denotes the integer part of λ. The ower bound in (1) can be achieved if and ony if a the vaues of λ ij (i j) take at most two different vaues λ and λ +1. Remark 1. Note that m λ ij is caed the Hamming distance between the ith and jth rows, a design with equa Hamming distances, m λ, is caed an equidistant design, andadesign with Hamming distances m λ and m λ 1 is caed a weak equidistant design [39]. It is obvious that a E(f NOD ) optima designs whose E(f NOD ) vaues reach their ower bounds in (1) may be divided into two kinds: equidistant and weak equidistant designs. There are aso severa other criteria for evauating SSDs, such as the maximum f NOD (d t,d s ) (see [3,8,16]), χ 2 (see [34,37]), discrete discrepancy [6,11,12] and minimum moment aberration [32]. The χ 2 and E(f NOD ) are equivaent for the symmetric case. In this paper, we use E(f NOD )to evauate SSDs as [3,19,25] did recenty. 3 Symmetrica Maxima Baanced Design For any given run size n and eve numbers q 1,,q, we ca the design with the argest number of coumns that are not fuy aiased a maxima baanced design of n runs and q 1,,q eves. If a the eves are equa, we ca it a symmetrica maxima baanced design. Now et us discuss some properties of such designs. Theorem 1. For any positive integers n and q, wheren is divisibe by q, the symmetrica maxima baanced design of n runs and q eves is an equidistant D(n; q M ) design with M = n! q![(. n q )!]q Proof. Finding out the number of coumns of a symmetrica maxima baanced design with n runs and q eves corresponds to figuring out a possibe ways to put n different bas into q identica boxes, where each box has n/q bas. It may be easiy cacuated that there are M ways. So the symmetrica maxima baanced design of n runs and q eves is a D(n; q M )design. Now et us investigate how many t s satisfy d it = d jt for any two runs d i =(d i1,,d im ) and d j =(d j1,,d jm )ofthed(n; q M ) design, where i j. It corresponds to finding out how many ways there exist to put n 2 different bas into q boxes, where the first q 1boxes are identica, each of which has n/q bas, and the ast box has n/q 2 bas. It can be easiy cacuated that there are (n 2)! B = ( n q 2) (q 1)! ways, which does not depend on i and j andthusisaconstantfori j. Hence the D(n; q M ) design is an equidistant design.

4 256 Y. LIU, M.Q. LIU Remark 2. It is vauabe to note that the D(n; q M ) design contains a possibe baanced coumns that are not fuy aiased. This means that for any other baanced D(n; q m )design which aso has no fuy aiased coumns, a its coumns can be found in D(n; q M ). Then deeting a the coumns of D(n; q m )fromd(n; q M ), the resuting D(n; q M m )designformed by the remaining coumns is regarded as the compementary design of D(n; q m ). D(n; q m )and D(n; q M m ) may be viewed as a pair of compementary designs of D(n; q M ). This point pays an important roe in our compementary design method. Some exampes of symmetrica maxima baanced designs are given beow. Exampe 1. For n =6andq =2, 3, the maxima baanced designs are D(6; 2 10 )andd(6; 3 15 ) designs, respectivey, which areshownintabes1and2. Tabe 1 D(6; 2 10 ) Tabe 2 D(6; 3 15 ) Exampe 2. For n = 8 and q = 2, 4, the maxima baanced designs are D(8; 2 35 ) and D(8; ) designs, which are shown in Tabes 3 and 6. Tabe 3 D(8; 2 35 ) From Theorem 1 and Lemma 1, we know the maxima baanced designs are E(f NOD )optima designs without fuy aiased coumns. The next section provides a new method to construct optima SSDs with smaer numbers of coumns based on the maxima designs.

5 SSD with Large Number of Factors The Compementary Design Method From Theorem 1, we immediatey have the foowing resut. Theorem 2. Let D(n; q m ) and D(n; q M m ) denote a pair of compementary designs of the maxima baanced design D(n; q M ), then the D(n; q m ) design is equidistant (or weak equidistant) if and ony if the D(n; q M m ) design is equidistant (or weak equidistant). This theorem provides a convenient and efficient way to find E(f NOD ) optima symmetrica SSDs from existing equidistant or weak equidistant designs without fuy aiased coumns. Given any such design, the corresponding compementary design must aso be equidistant or weak equidistant without fuy aiased coumns. Some exampes are isted in the rest of this section to iustrate this method. Exampe 3 [8]. obtained an equidistant D(6; 3 5 ) design as shown in Tabe 4. Two new weak equidistant D(6; 3 4 )andd(6; 3 6 ) designs may be constructed from it by deeting one of its coumn or adding a new coumn, e.g. (0, 1, 2, 0, 1, 2) to it, respectivey. Deete a the coumns of D(6; 3 5 ), D(6; 3 4 )andd(6; 3 6 )fromthed(6; 3 15 ) design in Tabe 2, respectivey, then we get their compementary D(6; 3 10 ), D(6; 3 11 )andd(6; 3 9 ) designs, where the D(6; 3 10 )design is shown in Tabe 5 and it is an equidistant design, whie the D(6; 3 11 )andd(6; 3 9 )designsare two new weak equidistant designs. A these three designs have no fuy aiased coumns. Tabe 4 D(6; 3 5 ) Tabe 5 D(6; 3 10 ) Exampe 4. Given equidistant D(8; 4 7k )designswithk =1,, 6 (see [5,14]), deete a the coumns of each of them from the maxima baanced D(8; ) design in Tabe 6, then we get equidistant D(8; k )designswithk =1,, 6. A these six designs are new E(f NOD ) optima designs without fuy aiased coumns. The existing D(8; 4 42 ) (see [14]) design and its corresponding compementary D(8; 4 63 ) design are shown in Tabes 7 and 8. Other designs are eft out here.

6 258 Y. LIU, M.Q. LIU Tabe 6 D(8; ) Tabe 6 (continued) D(8; ) Tabe 7 D(8; 4 42 ) Tabe 8 D(8; 4 63 )

7 SSD with Large Number of Factors Extension and Discussion We now extend the compementary design method to the asymmetric case. For this case, we have Theorem 3. (1) For any integers n, q 1,,q satisfying the conditions that (i) n is divisibe by q k,fork =1,, and (ii) q k q h,fork h, the asymmetrica maxima baanced design with n runs and eve numbers q 1,,q is an equidistant D(n; q M1 1 q M ) design, where M k = n! q k![( n q )!] for k =1,,. q k k (2) Let D(n; q r1 1 qr ) and D(n; qm1 r1 1 q M r ) denote a pair of compementary designs of the D(n; q M1 1 q M ) design, then D(n; q r1 1 qr ) is equidistant (or weak equidistant) if and ony if D(n; q M1 r1 1 q M r ) is equidistant (or weak equidistant). This theorem may be easiy deduced from Theorems 1 and 2 and thus the proof is omitted here. Let us see an exampe that iustrates how to construct new designs based on Theorem 3. Exampe 5. For q 1 =2, q 2 =3andn = 6, the maxima baanced design is a D(6; ) design as shown in Tabe 9. From the FSOA method of [12], we construct an equidistant D(6; ) design as shown in Tabe 10. Two weak equidistant D(6; )andd(6; )designs can then be obtained by adding coumns (0, 1, 0, 1, 0, 1) and (0, 0, 1, 1, 2, 2) to this D(6; ) design, respectivey. Deete a the coumns of D(6; ), D(6; )andd(6; )fromthe D(6; ) design, respectivey, then we get their compementary D(6; ), D(6; )and D(6; ) designs. The D(6; ) design is shown in Tabe 11, which is equidistant, and the D(6; )andd(6; ) designs are weak equidistant. A of them are new E(f NOD )optima designs without fuy aiased coumns. Tabe 9 D(6; ) Tabe 10 D(6; )

8 260 Y. LIU, M.Q. LIU Tabe 11 D(6; ) Given any equidistant (or weak equidistant) design, we construct a new equidistant (or weak equidistant) design by the compementary design method. Existing papers have provided penty of equidistant and weak equidistant designs, such as Chen, Liu [3], Fang, Ge, Liu [5,6,8], Fang, Ge, Liu, Qin [7,9,10], Fang, Lin, Liu [12], Georgiou, Koukouvinos [14], Georgiou, Koukouvinos, Mantas [15], Koukouvinos, Mantas [16],Liu,Cai [19], Liu, Zhang [23], Lu, Hu, Zheng [27], Sun, Lin, Liu [29],Liu,Liu [25]. Saturated OAs are aso a kind of equidistant designs and most of them can be found at the website Based on these existing designs, we may get many more new equidistant (weak equidistant) designs and enarge the ibrary of our known E(f NOD ) optima SSDs. References [1] Ai, M.Y., Fang, K.T., He, S.Y. E(χ 2 )-optima mixed-eve supersaturated designs. J. Statist. Pann. Inference, 137: (2007) [2] Booth, K.H.V., Cox, D.R. Some systematic supersaturated designs. Technometrics, 4: (1962) [3] Chen, J., Liu, M.Q. Optima mixed-eve k-circuant supersaturated designs. J. Statist. Pann. Inference, 138: (2008) [4] Chen, J. and Liu, M.Q. Optima mixed-eve supersaturated design with genera number of runs. Statist. Probab. Lett., 78: (2008) [5] Fang, K.T., Ge, G.N., Liu, M.Q. Construction of E(f NOD )-optima supersaturated designs via Room squares. Cacutta Statist. Assoc. Bu., 52: (2002) [6] Fang, K.T., Ge, G.N., Liu, M.Q. Uniform supersaturated design and its construction. Sci. China Ser. A, 45: (2002) [7] Fang, K.T., Ge, G.N., Liu, M.Q., Qin, H. Construction of minimum generaized aberration designs. Metrika, 57: (2003) [8] Fang, K.T., Ge, G.N., Liu, M.Q. Construction of optima supersaturated designs by the packing method. Sci. China Ser. A, 47: (2004) [9] Fang, K.T., Ge, G.N., Liu, M.Q., Qin, H. Combinatoria constructions for optima supersaturated designs. Discrete Math., 279: (2004) [10] Fang, K.T., Ge, G.N., Liu, M.Q., Qin, H. Construction of uniform designs via super-simpe resovabe t-designs. Uti. Math., 66: (2004) [11] Fang, K.T., Lin, D.K.J., Liu, M.Q. Optima mixed-eve supersaturated design and computer experiment. Technica Report MATH-286, Hong Kong Baptist University, 2000 [12] Fang, K.T., Lin, D.K.J., Liu, M.Q. Optima mixed-eve supersaturated design. Metrika, 58: (2003)

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