On the Isomorphism of Fractional Factorial Designs 1

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1 journal of complexity 17, 8697 (2001) doi: jcom , available online at on On the Iomorphim of Fractional Factorial Deign 1 Chang-Xing Ma Department of Statitic, Nankai Univerity, Tianjin , China Kai-Tai Fang Department of Mathematic, Hong Kong Baptit Univerity, Kowloon Tong, Hong Kong, China and Denni K. J. Lin Department of Management Science and Information Sytem, The Pennylvania State Univerity, Univerity Park, Pennylvania Received January 4, 2000 Two fractional factorial deign are called iomorphic if one can be obtained from the other by relabeling the factor, reordering the run, and witching the level of factor. To identify the iomorphim of two -factor n-run deign i known to be an NP hard problem, when n and increae. There i no tractable algorithm for the identification of iomorphic deign. In thi paper, we propoe a new algorithm baed on the centered L 2 -dicrepancy, a meaure of uniformity, for detecting the iomorphim of fractional factorial deign. It i hown that the new algorithm i highly reliable and can ignificantly reduce the complexity of the computation. Theoretical jutification for uch an algorithm i alo provided. The efficiency of the new algorithm i demontrated by uing everal example that have previouly been dicued by many other Academic Pre Key Word: factorial deign; Hamming ditance; iomorphim; uniformity. 1 Thi work wa partially upported by a Hong Kong RGC grant and Science Faculty of Hong Kong Baptit Univerity. Denni Lin i partially upported by the National Science Foundation via Grant DMS and the National Science Council of ROC via Contract NSC M The correponding author i Kai-Tai Fang X Copyright 2001 by Academic Pre All right of reproduction in any form reerved. 86

2 ISOMORPHISM OF FACTORIAL DESIGNS INTRODUCTION Fractional factorial experiment have become important in all kind of tudie. Two factorial deign are called iomorphic if one can be obtained from the other by relabeling the factor, reordering the run, or witching the level of factor. Two iomorphic deign are conidered to be equivalent becaue they hare the ame tatitical propertie in a claical ANOVA model. Therefore, it i important to identify deign-iomorphim. Identifying whether two fractional deign are iomorphic ha received a great deal of attention in the literature (ee, for example, Clark and Dean (2000) and reference therein). Denote by d(n, q, ) a factorial deign of n run and factor each having q level. A deign d(n, q, ) i uually expreed a an n_ matrix with element 0, 1,..., q&1. For identifying two d(n, q, ) deign, a complete earch compare n!(q!)! deign from the definition of iomorphim. For example, it require 13!12!2 12 =1.22_10 22 comparion for two factorial d(13, 2, 12) deign to ee if thee deign are iomorphic. The identification of two factorial deign i conidered to be a NP hard problem when the number of run orand the number of factor increae. A fractional factorial deign that i contructed through defining relation among factor i called regular, otherwie nonreqular. For a q &k regular fractional factorial deign D, let A r (D) be the number of factorial effect, involving r factor, which appear in the defining relation. The equence [A 1 (D),..., A (D)] i called the word-length pattern. The mallet r for A r (D)>0 i called the reolution of D. For a given (n, q, ) one earche a deign d(n, q, ) with highet reolution a it ha le confounding. When two d(n, q, ) deign D 1 and D 2 have the ave level of reolution, there exit a r>0 uch that A j (D 1 )=A j (D 2 )=0 for all j<r and A r (D i )>0, i=1, 2. One want to chooe the deign with maller A r ( } ). The minimum aberration criterion i baed on uch an idea. For a detailed dicuion refer to Frie and Hunter (1980). A neceary condition for two regular factorial deign to be iomorphic i that they have identical word-length pattern. Draper and Mitchell (1968) gave two L 512 (2 12 ) orthogonal deign which have identical word-length pattern, but are not iomorphic. Here, L n (q ) denote an orthogonal array with n run and column. Draper and Mitchell (1970) gave a more enitive criterion for iomorphim, called ``letter pattern comparion,'' and tabulated 1024-run deign of reolution 6. Let a ij be the number of word of length j in which letter i appear in a regular deign D and A=(a ij )be the letter pattern matrix of D. They conjectured that two deign D and D$ are iomorphic if and only if A=PA$, where P i a permutation matrix. Obviouly, two deign having identical letter pattern matrice necearily have identical word-length pattern. Chen and Lin (1991) gave two

3 88 MA, FANG, AND LIN noniomorphic deign with identical letter pattern matrice and thu howed that the criterion ``letter pattern matrix'' i not ufficient for deign iomorphim. Note that both the word-length and letter pattern matrix are not eay to calculate and can be applied only to regular factorial deign. Recently, Clark and Dean (2000), denoted by [CD00] for convenience, gave a ufficient and neceary condition for iomorphim of deign. Let H=(d ij ) be the Hamming ditance matrix of a deign D, where d ij i the Hamming ditance of the ith and jth run of D and i defined a the number of level of the factor where they differ. Thi clever method i invariant under the permutation of level, but the complexity here make the calculation intractable. For example, it may require 12!12!12=2.75_10 18 comparion for two non-iomorphic d(13, 2, 12) deign. In thi paper we propoe a neceary criterion for detecting noniomorphic (regular and nonregular) factorial deign baed on uniformity, a criterion that i crucial in pace-filling deign for computer experiment (Bate et al. (1996)) and in uniform deign (Fang and Wang (1994) and Fang et al. (2000)). The centered L 2 -dicrepancy propoed by Hickernell (1998) i employed a the meaure of uniformity in thi tudy and i introduced in Section 2. An algorithm for detecting the iomorphim of two-level d(n, 2,) deign i alo propoed there. Section 3 applie the propoed algorithm to everal example that were dicued by other. Section 4 dicue the extenion to higher level deign, and an example i given for illutration. The concluion and further dicuion are given in Section ISOMORPHISM OF TWO-LEVEL DESIGNS Recall that Clark and Dean (2000) algorithm i mainly baed on the following lemma. Thi will be called a the HD-method for convenience. Alo note that the HD-method require to find the permutation [c 1,..., c p ] and the permutation matrix R. Lemma 1. Let D 1 and D 2 be two d(n, q, ) deign. Then D 1 and D 2 are iomorphic if and only if there exit an n_n permutation R and a permutation [c 1,..., c ]of[1,..., ] uch that for p=1,..., [1,..., p] H D 1 =RH [c 1,..., c p ] R$, D 2 where H [c 1,..., c p ] D i the Hamming ditance matrix of the deign formed by column [c 1,..., c p ] of deign D.

4 ISOMORPHISM OF FACTORIAL DESIGNS 89 A deign d(n, q, ) can be viewed a n point in the unit cube C =[0, 1), after proper coding, e.g., the q level (0, 1,..., q&1) O (0.5q, 1.5q,..., (q&0.5)q). Let P=[u 1,..., u n ] be a et of n point in C. Many criteria have been propoed for the meaure of uniformity. We hall concentrate on the centered L 2 -dicrepancy (CD 2 for hort) in our tudy here. The CD 2 ha ome nice and unique propertie, uch a it i invariant under reordering the run, relabeling coordinate and reflection of the point about any plane paing through the center of the unit cube and parallel to it face. Furthermore, the CD 2 capture uniformity over the unit cube a well a the uniformity over all projected ubdimenion. Hickernell (1998) gave an analytical formula for the CD 2 a (CD 2 (P)) 2 = & n \13 n : k=1 + 2 n : n 2 k=1 n : j=1 ` j=1 \ u kj&0.5 & 1 2 u kj& ` i=1_ u ki& u ji&0.5 & 1 2 u ki&u ji &, where u k =(u k1,..., u k )$. Recently, Fang and Mukerjee (2000) how the relationhip between uniformity and aberration for two-level regular fraction. Ma et al. (1999) found link between uniformity and orthogonality for ome factorial. For a d(n, q, ) deign D with level 0, 1,..., q&1, when we calculate it CD 2 -value we alway aume to map it q level into 12q, 32q,..., (2q&1)2q. Thi undertanding i ueful in link among uniformity, Hamming ditance, ditance ditribution and weight ditribution of a deign D. (1) Theorem 1. For a two-level d(n, 2, ) deign D, we have CD 2 (D)= 2 \ 13 & \ \ 5 n n 4+ \ n+2 : i=1 i&1 : j=1\ 4 d H (u i, u j ) 5+ +, (2) where u i, i=1,..., n are n run of D and d H (u i, u j ) i the Hamming ditance between u i and u j. Proof. Note that the two level are choen a 14 and 34. The econd term on the right hand ide of (1) become 2( ). Conequently, [ u ki& u ji&0.5 & 1 2 u ki&u ji ]= {54 1 if u ki =u ji, otherwie.

5 90 MA, FANG, AND LIN Therefore, we have ` i=1_ u ki& u ji&0.5 & 1 2 u ki&u ji \ 5 &d H (u i, u j ) 4+ &=. The theorem follow from (1). For any d(n, q, ) deign D, let E i (D) be 1 n time the number of pair of two run whoe Hamming ditance to be i, i.e., E i (D)= 1 n *[(c, d) c, d # D, d H(c, d)=i], where d H (c, d) i the Hamming ditance between two run c and d. The equence [E 0 (D),..., E (D)] i referred to a the ditance ditribution of D. From Theorem 1, we can etablih a link between the ditance ditribution and CD 2 -value of a two-level deign. Theorem 2. For a d(n, 2,) deign D, we have CD 2 (D)= 2 \ 13 & \35 + n\ : E k (D) 5+ \4 k=0 k. (3) Comparing Eq. (2) and (3), although both cot O(n 2 ), but the former 4 need to compute n(n&1)2 power of 5 while the latter need only power of 4 5. The latter reduce the complexity of computation. In fact, if D i a regular two-level deign, the complexity can be further reduced into O(n). Let D=[x 1,..., x n ] be a regular factorial deign with two level 0 and 1. The weight ditribution of D i defined by W k (D)= 1 n *[x i } : x ij =k]. j=1 Theorem 3. Let D=[x 1,..., x n ] be a regular factorial deign with two level 0 and 1. We have CD 2 (D)= 2 \ 13 & \35 + n\ : W k (D) \4 k=0 5+ k. (4) Proof. Becaue run of a regular deign D mut form a linear ubpace of the full deign 2 over GF(2), (x i &x j ) (mod 2) i alo a run of D, i.e., [(x i &x j ) (mod 2) j=1,..., n] are all the deign, D, for any i=1,..., n. The proof i completed by Eq. (3).

6 ISOMORPHISM OF FACTORIAL DESIGNS 91 To compute E k (D), one need to calculate n(n&1)2 ditance, but to find W k (D), one need only n ummation for the run. For two iomorphic d(n, 2,) deign D 1 and D 2 they have the ame et of Hamming ditance, the ame equence of E k, and thu an identical CD 2 -value. Furthermore, they have the ame CD 2 -value ditribution for each of their projection deign. For given k (1k<) there are ( k ) ubdeign for a deign D(n, 2,). The ditribution of CD 2 -value of thee ubdeign i called the k-dimenional CD 2 -value ditribution of D and we denote it by F k (D). The following neceary condition i thu obtained. Uniformity Criterion for Iomorphim (UCI) of Two-Level Deign. The neceary condition for two d(n, 2,) deign D 1 and D 2 to be iomorphic are a) they have the ame CD 2 -value; b) they have the ame ditribution F k (D 1 )=F k (D 2 ) for 1k<. Baed on the UCI we propoe the following algorithm, called NIU algorithm, for detecting iomorphic d(n, 2,) deign. Let D 1 and D 2 be two d(n, 2, ) deign. NIU Algorithm. Step 1. Comparing CD 2 (D 1 )andcd 2 (D 2 ), if CD 2 (D 1 ){CD 2 (D 2 )we conclude D 1 and D 2 are not iomorphic and terminate the proce, otherwie go to Step 2. Step 2. Let [x] be the integer part of a poitive number x. For k=1, &1, 2, &2,..., [2], &[2], comparing F k (D 1 ) and F k (D 2 ), if F k (D 1 ){F k (D 2 ) we conclude D 1 and D 2 are not iomorphic and terminate the proce, otherwie thi tep goe to the next k-value. We next apply the NIU algorithm to everal example that have been tudied by other. A will be een, the NIU algorithm efficiently detect the non-iomorphim of deign, typically at Step 1 or Step 2. Note that the NIU algorithm need O(n 2 2 ) operation to compare 2 +1 CD 2 -value in the wort cae. Thi i polynomial in n and exponential in. Thi i a ignificant improvement over the complete earch (which take n!!2 comparion and i uperexponential in both n and ) and the HD-method (which require (!) 2 comparion and each comparion required O(n!) operation in the wort cae). 3. EXAMPLES OF TWO-LEVEL DESIGNS In thi ection we apply the NIU algorithm to everal deign that have been tudied in the literature and how that the new algorithm i very ueful. The calculation wa carried on a PC-computer with double preciion.

7 92 MA, FANG, AND LIN Example 1. Conider two d(7, 2, 6) deign D 1 and D 2. Their treatment matrice are given below. All their treatment are the ame except the ixth treatment combination D 1 =\ , and D 2 =\ The deign D 1 and D 2 are non-iomorphic becaue CD 2 2 (D 1)= while CD 2 2 (D 2)= The proce terminate at the firt tep. Example 2. The three d(12, 2, 5) deign d 1, d 2 and d 3 conidered in Example 3.1 of [CD00] have different quared CD-value , , and So they are non-iomorphic to each other. Example 3. The NIU algorithm i applied to the three d(18, 2, 17) deign, d 9, d 6, 3 and d 3, 3, 3, dicued by Cohn (1994) (alo given in [CD00] a Example 2.1 and 3.1). Firt of all, all three deign produce an identical quared CD 2 -value of We next calculate their k-dimenional CD 2 -value ditribution. For k=1, the F 1 (D) ditribution i identical for all three deign: 9 projection deign have CD 2 2= and 8 deign have CD 2 =0.0224; the F 2 16(D) ditribution i alo identical: CD 2 2= (twice), CD 2 2= (6 time), CD 2 2= (6 time), and CD 2 = (3 time). For k=2, the F 2 2(D) ditribution i identical for all three deign: CD 2 2= (36 time), CD 2 2= (72 time), and CD 2 = (28 time). However, the F 2 15(D) ditribution of d 3, 3, 3 i different from the other two deign. For k=3, we found that the F 3 (D) ditribution are different between deign d 9 and d 6, 3. Finally, we conclude that the three deign are not iomorphic to each other. Example 4. Conider the two L 512 (2 12 ) deign, D 1 and D 2, given in Draper and Mitchell (1968, a deign 3.4 and 3.5 in Table I). They howed that thee two deign are not iomorphic but hare an identical wordlength pattern. When applying NIU to thee two deign, an identical CD-value of wa obtained. Obviouly, for k=1, 2, all the k-projection deign of D 1 and D 2 are iomorphic, becaue both deign are orthogonal deign. However, F 11 (D 1 ){F 11 (D 2 ). We thu conclude that they are not iomorphic. Specifically, F 11 (D 1 ) ha CD 2 2= (12 time); while F 11 (D 2 )hacd 2 = (3 time), 2 CD2=

8 ISOMORPHISM OF FACTORIAL DESIGNS 93 TABLE I The Ditribution F 12 for the Five Deign D 1 D 2 D 3 D 4 D 5 CD 2 2 Freq. CD 2 2 Freq. CD 2 2 Freq. CD 2 2 Freq. CD 2 2 Freq (6 time) and CD 2 2 = (3 time). Note that it i not feaible to apply the HD-method for uch a large deign. Example 5. It i well known that the number of non-iomorphic Hadamard matrice are 1, 1, 1, 5, 3, and 60 repectively for order n =4, 8, 12, 16, 20 and 24. We have uccefully applied the NIU to all ix of thee cae. The detail for n=16 are given here. Hall (1961) found that there exit exactly five non-iomorphic group of Hadamard matrice of order 16. After deleting the column of 1' from each of the matrice after normalizing, the remaining matrice are denoted by D j, j=1,..., 5, repectively. All five deign have the ame quared CD-value of and identical ditribution of F 14 and F 13 when projected into 14 and 13 ub-dimenion. However, the different F 12 -ditribution given in Table I indicate the non-iomorphim of five deign, except deign D 4 and D 5. In the next tep, we found that F 11 (D 4 ){F 11 (D 5 ), a hown in Fig. 1, and thu concluded the non-iomorphim of five deign. Example 6. Chen and Lin (1991, p. 97, Table 1) gave two noniomorphic L (2 31 ) deign with the ame letter pattern matrix. It require a large amount of computation to detect their non-iomorphim FIG. 1. Plot of Ditribution F 11 for D 4 and D 5.

9 94 MA, FANG, AND LIN by any other exiting algorithm, including the HD-method. The noniomorphim, however, can be eaily detected by the NIU algorithm a follow: (1) two deign have the ame CD 2= ; 2 (2) all the 30-dimenional projection deign have the ame CD 2 = ; 2 (3) all the 29-dimenional projection deign have the ame CD 2 = ; 2 (4) the F 28 -ditribution of the two deign are different a follow: Deign (a) Deign (b) CD 2 2 Freq. CD 2 2 Freq The concluion that two deign are not iomorphic follow by implementing only four tep of the algorithm. 4. FRACTIONAL FACTORIAL DESIGNS OF HIGHER LEVELS In thi ection we conider the problem of detecting non-iomorphic deign for high-level factorial deign. Let D be a d(n, q, ) deign and E k (D)' be it ditance ditribution defined in Section 2. Denote B a (D)= : i=1 E i (D) a i a the ditance enumerator of D (Roman, 1992, p. 226). For a two-level deign D we have from (3) (Cd 2 (D)) 2 = n\ 1 5 B (D)&2 32+ \ \13

10 ISOMORPHISM OF FACTORIAL DESIGNS 95 which provide a link between the ditance enumerator and uniformity. In fact, the UCI i equivalent to the meaure B 45 (D) for two-level deign. Thi meaure can naturally be ued for high-level factorial deign. Given k (1k), the ditribution of B a -value over all k-dimenional projection ubdeign i denoted by F Ba, k (D). We now can have an NIU verion for the high-level deign. A the parameter a i a pre-determined value, we omit a from the notation for implicity. NIU Algorithm for High-Level Deign. Step 1. Comparing B(D 1 )andb(d 2 ), if B(D 1 ){B 2 (D 2 ), we conclude D 1 and D 2 are not iomorphic and terminate the proce. Otherwie go to Step 2. Step 2. For k=1, &1, 2, &2,..., [2], &[2], compare F Bk (D 1 ) and F Bk (D 2 ). If F Bk (D 1 ){F Bk (D 2 ), we conclude D 1 and D 2 are not iomorphic and terminate the proce, otherwie thi tep goe to the next k-value. For a imple illutration, conider the four L 18 (3 7 ) in Table II, where Deign (a) i from Mauyama (1957), Deign (c) i from Fang et al. (2000), and Deign (b) i from Taking TABLE II Four L 18 (3 7 ) Deign No. (a) (b) (c) (d)

11 96 MA, FANG, AND LIN a=45, for example, the four deign have the ame ditance enumerator However, the ditribution of ditance enumerator of all 6-dimenional projection deign are different a indicated below. Therefore, we conclude that Deign (a), (c), and (d) are non-iomorphic. Note that an exhautive comparion indicate that Deign (a) and (b) are indeed iomorphic. B (a) Freq. B (b) Freq. B (c) Freq. B (d) Freq DISCUSSION The uniformity criterion propoed in thi paper i ueful for detecting deign iomorphim. The HD-method give a neceary and ufficient link between the iomorphim and the Hamming ditance matrice of two deign. A a matter of fact, CD 2 (D) i a function of the Hamming ditance matrix of D. Equation (3) and (4) can ignificantly reduce the computation effort. Thi make the NIU algorithm a powerful tool, a clearly een from Example 17. For the reearch on projection propertie of a given fractional factorial deign, we need to claify all it projection deign. For example, Lin and Draper (1992) tudied the PlackettBurman L 12 (2 11 ) deign and it projection deign. For the five-dimenional cae they have claified 462 projection deign into two non-iomorphic group. Thi i computationallyintenive work, uing definition of the iomorphim directly. Moreover, many optimality criteria uch a D-optimality, A-optimality, were ued for the claification of projection deign. Unfortunately, there i no analytic link between iomorphim and D-optimality (or A-optimality), and many non-iomorphic deign reult in an identical information matrix, and thu optimalitie baed on the information matrix, uch a D- or A-optimality, are inappropriate in detecting the deign iomorphim. The NIU algorithm propoed here provide a very efficient and meaningful way for claifying projection deign. The UCI i only a neceary condition for deign iomorphim. We conjecture that UCI i alo a ufficient condition, but fail to prove it at thi tage. If the conjecture i true, the computing complexity of comparing two d(n, q, ) i polynomial in n and q and exponential in. Nonethele, it i probably the mot efficient algorithm for detecting non-iomorphim of deign.

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