OPTIMAL MAXIMIN L 1 -DISTANCE LATIN HYPERCUBE DESIGNS BASED ON GOOD LATTICE POINT DESIGNS 1

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1 The Annals of Statistics 018, Vol. 46, No. 6B, Institute of Mathematical Statistics, 018 OPTIMAL MAXIMIN L 1 -DISTANCE LATIN HYPERCUBE DESIGNS BASED ON GOOD LATTICE POINT DESIGNS 1 BY LIN WANG,QIAN XIAO AND HONGQUAN XU University of California, Los Angeles and University of Georgia Maximin distance Latin hypercube designs are commonly used for computer experiments, but the construction of such designs is challenging. We construct a series of maximin Latin hypercube designs via Williams transformations of good lattice point designs. Some constructed designs are optimal under the maximin L 1 -distance criterion, while others are asymptotically optimal. Moreover, these designs are also shown to have small pairwise correlations between columns. 1. Introduction. Computer experiments are increasingly being used to investigate complex systems [Sacks, Schiller and Welch 1989, Santner, Williams and Notz 003, Fang, Li and Sudjianto 006, Morris and Moore 015]. A general design approach to planning computer experiments is to seek design points that fill a design region as uniformly as possible [Lin and Tang 015]. Representative designs include Latin hypercube designs LHDs and their modifications, maximin distance designs [Johnson, Moore and Ylvisaker 1990] and uniform designs [Fang and Wang 1994]. LHDs have uniform one-dimensional projections and orthogonal-array based LHDs [Tang 1993, He and Tang 013, 014, He, Cheng and Tang 018] have improved two- or three-dimensional projections. Many researchers have constructed orthogonal or nearly orthogonal LHDs; see, among others, Ye 1998, Steinberg and Lin 006, Cioppa and Lucas 007, Lin, Mukerjee and Tang 009, Sun, Liu and Lin 009, Yang and Liu 01, Georgiou and Efthimiou 014, Lin and Tang 015 and Sun and Tang 017. However, these LHDs are often not space-filling in high dimensions [Joseph and Hung 008, Xiao and Xu 018]. A maximin distance design spreads design points over the design space in such a way that the separation distance, that is, the minimal distance between pairs of points, is maximized. Computer experiments are often modeled as Gaussian processes. When the correlations between observations rapidly decrease as the distances between design points increase, maximin distance designs are asymptotically D-optimal in the sense that they maximize the determinant of the correlation matrix [Johnson, Moore and Ylvisaker 1990]. The choice of distances is Received October 017; revised December Supported by NSF Grant DMS MSC010 subject classifications. Primary 6K99. Key words and phrases. Computer experiment, correlation, space-filling design, Williams transformation. 3741

2 374 L. WANG, Q. XIAO AND H. XU application dependent. Some researchers worked on the L -distance and proposed algorithms such as simulated annealing [Morris and Mitchell 1995, Joseph and Hung 008, Ba, Myers and Brenneman 015] and swarm optimization algorithms [Moon, Dean and Santner 011, Chen et al. 013] to construct maximin distance LHDs. However, such methods are not efficient for constructing large designs due to their computational complexity. Nevertheless, large designs are needed for computer experiments; for example, Morris 1991 considered many simulation models involving hundreds of factors. Zhou and Xu 015 studied both L 1 -andl -distances of good lattice point GLP designs. The GLP method was introduced by Korobov 1959 for numerical evaluation of multivariate integrals and has been widely used in quasi-monte Carlo method, uniform designs and computer experiments [Fang and Wang 1994]. Zhou and Xu 015 showed that permuting levels can increase the separation distances of GLP designs. It is infeasible to conduct all level permutations, so they considered only linear permutations, which limits the ability of generating good designs. Xiao and Xu 017 proposed construction methods via Costas arrays and obtained some LHDs with large minimal L 1 -distance. In this paper, we propose a series of systematic methods to construct maximin L 1 -distance LHDs. The L 1 -distance provides a lower bound for the L -distance by the Cauchy Schwarz inequality so that the constructed designs also perform well regarding the L -distance. The proposed method is based on the Williams transformation and its modification. The Williams transformation was first used by Williams 1949 to construct Latin square designs that are balanced for nearest neighbors. Bailey 198 and Edmondson 1993 used the transformation to construct designs orthogonal to polynomial trends. Butler 001 used the transformation to construct optimal and orthogonal LHDs under a second-order cosine model. Our purpose is different from theirs. We apply the Williams transformation to GLP designs and construct a class of asymptotically optimal maximin LHDs. Applying the leave-one-out method we obtain another class of asymptotically optimal maximin LHDs. By modifying the Williams transformation, we obtain a class of exactly optimal maximin LHDs. Moreover, all resulting designs have small pairwise correlations between columns and the average correlations converge to zero as the design sizes increase. This near orthogonality is desirable for estimating potential linear trend efficiently in a Gaussian process. This paper is organized as follows. Section provides the construction methods. Sections 3 and 4 give theoretical results on separation distances and correlations of some special constructed designs. Section 5 extends the theoretical results to a general situation. Concluding remarks are given in Section 6. Proofs are deferred to the Appendix.. Construction methods. An N n LHD is an N n matrix where each column is a permutation of N equally spaced levels, denoted by 0,...,N 1 or 1,...,N.TheL 1 -distance between two vectors x 1 = x 11,...,x 1n and x =

3 OPTIMAL MAXIMIN L 1 -DISTANCE LATIN HYPERCUBE DESIGNS 3743 x 1,...,x n is dx 1,x = n j=1 x 1j x j. For an N n design matrix D, let x i be the ith row, i = 1,...,N,andd ik D be the L 1 -distance between the ith and kth rows of D, thatis,d ik D = dx i,x k.thel 1 -distance of D, denoted by dd = min{d ik D : i k, i,k = 1,...,N}, is the minimum L 1 -distance between any two distinct rows in D. The maximin distance criterion [Johnson, Moore and Ylvisaker 1990] is to maximize dd among all possible designs. For an N n LHD, the average pairwise L 1 -distance between rows is N + 1n/3[Zhou and Xu 015]. Because the minimum pairwise L 1 -distance cannot exceed the integer part of the average, we have the following result. LEMMA 1. For any N n LHD D, dd d upper = N + 1n/3, where x is the integer part of x. Let h = h 1,...,h n be a set of positive integers smaller than and coprime to N. AnN n GLP design D = x ij is defined by x ij = i h j mod N for i = 1,...,N and j = 1,...,n.ThelastrowofD is a vector of zeros. Each column of D is a permutation of {0,...,N 1}. Thus a GLP design is an LHD. We can construct an N n GLP design for any n φn,whereφn is the Euler function, that is, the number of positive integers smaller than and coprime to N. LetD b = D + bmod N for b = 0,...,N 1, that is, D b is a linearly permuted GLP design. Then D b is still an LHD. Zhou and Xu 015 showed that dd b dd for any b and proposed to search b that maximizes dd b..1. Williams transformation. Given an integer N, forx = 0,...,N 1, the Williams transformation is defined by { x for 0 x<n/;.1 Wx= N x 1 for N/ x<n. The Williams transformation is a permutation of {0,...,N 1}. Hence, for an LHD D = x ij, WD= Wx ij is also an LHD. The following example shows that the Williams transformation can further increase the L 1 -distance of linearly permuted GLP designs. EXAMPLE 1. Consider N = 11 and h = 1,...,10. TheGLPdesignD = x ij is an LHD with x ij = i j mod 11 and dd = 30. For each b = 0,...,10, we obtain two designs via linear permutation and Williams transformation, namely, D b = D + b mod 11 and E b = WD b.table1 shows the L 1 -distances of D b and E b. The linearly permuted designs D b s have distances ranging from 30 to 34, while the distances for E b s vary from 10 to 39. The upper bound from Lemma 1 is 40. The best design from D b s is D 1 or D 9 with dd 1 = dd 9 = 34, while the best design from E b s is E 1 or E 4 with de 1 = de 4 = 39.

4 3744 L. WANG, Q. XIAO AND H. XU TABLE 1 The L 1 -distances of D b and E b in Example 1 b dd b de b Example 1 shows that the Williams transformation can generate designs with larger distances than the linear permutation. Inspired by this, we propose a new construction for maximin LHDs: ALGORITHM 1 Williams transformation of linearly permuted GLP designs. Step 1. Given a pair of integers N and n φn, generate an N n GLP design D. Step. For b = 0,...,N 1, generate D b = D + bmod N and E b = WD b. Step 3. Find the best D b and E b which maximize dd b and de b, respectively. As an illustration, we apply Algorithm 1 for N = 7,...,30 and n = φn. Table compares LHDs generated by the linear permutation, the Williams transformation, R package SLHD provided by Ba, Myers and Brenneman 015 and the Gilbert and Golomb methods proposed by Xiao and Xu 017. The SLHD package adopts the L -distance measure, so we ran the command maximinslhd with option t = 1 and default settings for 100 times, and chose the design with the largest L 1 -distance. The Williams transformation always offers better designs than the linear permutation except for N = 13, and consistently outperforms the Gilbert and Golomb methods, which only work for prime N. Compared to the SLHD package, the Williams transformation performs better for designs with moderate to large sizes. The Williams transformation performs specially well when N is a prime... Leave-one-out method. Since the last row of a GLP design D is 0,..., 0, then the last rows of D b and E b are b,...,b and Wb,...,Wb,respectively. The leave-one-out method is to delete the constant row of a design and rearrange the levels so that the resulting design is still an LHD. Specifically, starting from D b, we delete the last row and reduce the levels b + 1,...,N 1 by one, which gives us an N 1 n LHD, denoted by Db. Similarly, from E b, we obtain another N 1 n LHD, denoted by Eb.Table3 compares the

5 OPTIMAL MAXIMIN L 1 -DISTANCE LATIN HYPERCUBE DESIGNS 3745 TABLE Comparison of L 1 -distances of N n LHDs N n LP WT SLHD Gil Gol N n LP WT SLHD Gil Gol Note: LP, linear permutation; WT, Williams transformation; SLHD, R package SLHD; Gil, Gilbert method; Gol, Golomb method. L 1 -distances of Db and E b for N = 7,...,30, as well as the N 1 n designs generated by R package SLHD and the Gilbert and Golomb methods. From Table 3,the leave-one-out Williams transformation generates designs with larger L 1 -distance than other methods in most cases. It performs specially well when N is a prime. TABLE 3 Comparison of L 1 -distances of N 1 n LHDs N n LP-1 WT-1 SLHD Gil Gol N n LP-1 WT-1 SLHD Gil Gol Note: LP-1, leave-one-out linear permutation; WT-1, leave-one-out Williams transformation.

6 3746 L. WANG, Q. XIAO AND H. XU.3. Modified Williams transformation. To construct other maximin LHDs, we propose a modified Williams transformation. For x = 0,...,N 1, define { x for 0 x<n/;. wx = N x for N/ x <N. The following lemma shows an important connection between the original and modified Williams transformations. LEMMA. Let N be an odd prime, D be an N N 1 GLP design and D b = D + bmod N for b = 0,...,N 1. Then d ik wd b = d ik WD b for i + k N and i, k = 1,...,N 1. The wx is always an even number, so wd b is not an LHD. We can construct LHDs by selecting some submatrices of wd/. Let us see an illustrating example. EXAMPLE. Consider N = 11 and the GLP design D. The design matrices of D and wd/ areshownintable4. If we divide the design matrix of wd/ into four blocks as shown in Table 4, then each block is a LHD. Denote H 1 and H as the top two blocks, and H 3 and H 4 as the bottom two blocks, respectively. It can be verified that H 1 and H are 5 5 LHDs with dh 1 = dh = 10, which attains the upper bound of L 1 -distance in Lemma 1. In fact, H 1 and H are the same design up to column permutations; in addition, H 3 and H 4 can be obtained by adding a row of zeros to H 1 and H, respectively. Generally, suppose that N is an odd prime with N = m + 1andD = x ij is the N N 1 GLP design. Since x ij + x N ij = N and x ij + x in j = N for TABLE 4 The design matrices of D and wd/ in Example D wd/

7 OPTIMAL MAXIMIN L 1 -DISTANCE LATIN HYPERCUBE DESIGNS 3747 TABLE 5 Comparison of L 1 -distances of m m LHDs m MWT SLHD Wel Gil Gol m MWT SLHD Wel Gil Gol Note: MWT, modified Williams transformation; Wel, Welch. any i, j = 1,...,N 1, then A 1 N A wa 1 wa.3 D = N A 3 A 4 and wd = wa 3 wa 4, 0 m 0 m 0 m 0 m where A 1 is the m m leading principal submatrix of D,andA, A 3 and A 4 can be obtained from A 1 by reversing the order of columns, rows and both, respectively. In fact, wa 1,..., wa 4 are the same design up to row and column permutations, each column of which is a permutation of {, 4,...,m}. Let.4 H = wa 1 / be an m m LHD from the modified Williams transformation. Table 5 compares LHDs generated by the modified Williams transformation, the R package SLHD and the Welch, Gilbert and Golomb methods from Xiao and Xu 017. The modified Williams transformation always provides better designs than any other methods. In fact, the L 1 -distance of each design generated by the modified Williams transformation in Table 5 attains the upper bound given in Lemma Theoretical results. The Williams transformation leads to a remarkably simple design structure in terms of the L 1 -distance when N is an odd prime. THEOREM 1. Let N be an odd prime, D be an N N 1 GLP design, D b = D + bmod N and E b = WD b for b = 0,...,N 1. Then for i k, N 1 /3 + fb for i = N or k = N, d ik E b = N 1 /3 fb for i = N k, N 1 /3 otherwise,

8 3748 L. WANG, Q. XIAO AND H. XU FIG. 1. The three possible values of pairwise L 1 -distance of E b for N = 11 or 17. and de b = N 1/3 + min{fb, fb}, where fb= Wb N 1/ N 1/1. The pairwise L 1 -distance between any two distinct rows of E b takes on only three possible values. One attains d upper = N 1/3 given in Lemma 1, and the other two vary around d upper. Figure 1 shows the three values for N = 11 and N = 17 for each b = 0,...,N 1. To maximize de b, we need to maximize min{fb, fb}. Let c 0 = N 1/1, { c0 if c0 c = + c N 1 /4; c otherwise, and 3.1 b = W 1 N 1 ± c. It can be verified that either choice of b defined in 3.1 maximizes min{fb, fb} and leads to the best E b. EXAMPLE 3. Consider N = 11. Then c 0 = 11 1/1 =3. Since c0 + c N 1/4, set c = 3. By 3.1, b = 1 or 4. For either b = 1orb = 4, by Theorem 1, fori k, 39 for i = 11 or k = 11, d ik E b = 4 for i = 11 k, 40 otherwise. Hence, de 1 = de 4 = 39. Based on the upper bound in Lemma 1, we define the distance efficiency as 3. d eff D = dd/d upper = dd/ N + 1n/3.

9 OPTIMAL MAXIMIN L 1 -DISTANCE LATIN HYPERCUBE DESIGNS 3749 When N is a prime, n = φn= N 1andN +1n/3 = N 1/3isaninteger. In this case, d eff D = dd/n + 1n/3. For example, for the designs E 1 and E 4 in Example 3, d eff E 1 = d eff E 4 = 39/40 = Generally, we have the following result. THEOREM. For an odd prime N and b defined in 3.1, de b N 1 N 1 and d eff E b N 1. As N, d eff E b 1; so E b is asymptotically optimal under the maximin distance criterion. For the leave-one-out design Eb defined in Section.,wehave the following result. THEOREM 3. For an odd prime N and b defined in 3.1, d Eb N N 1 N When N 7, d eff E b 1 3 1/ 3/N > 1.43/N. For an odd prime N = m + 1andthem m design H constructed in.4, we have even better results. By Lemma and Theorem 1, d ik wd = N 1/3 fori k, i, k = 1,...,m. By the structure of wd shownin.3, d ik wa 1 = d ik wd/ = N 1/6; so H is an equidistant LHD and dh = N 1/1 = m + 1m/3. THEOREM 4. Let N = m + 1 be an odd prime, D = x ij be an N N 1 GLP design, and A 1 be the m m leading principal submatrix of D, that is, A 1 = x ij with i, j = 1,...,m. Then H = wa 1 / is a maximin distance LHD with dh = m + 1m/3. The modified Williams transformation generates exact maximin LHDs when N is an odd prime. The constructed H is a cyclic Latin square, with each level occurring once in each row and once in each column. We can add a row of zeros to H to obtain an m + 1 m LHD, denoted by H. It is easy to see that dh = dh = m + 1m/3 andd eff H = m + 1/m + 1asm. The proposed methods are also useful in the construction of maximin L - distance designs. An upper bound for the L -distance of an N n LHD is d upper = NN + 1n/6[Zhou and Xu 015]. By the Cauchy Schwarz inequality, we have x x 1 / n for any n-vector x, sod eff > /3 d eff,whered eff is the L -distance efficiency. Therefore, for an asymptotically optimal design under the maximin L 1 -distance criterion, its L -distance efficiency will tend to be

10 3750 L. WANG, Q. XIAO AND H. XU greater than /3 > This is a loose lower bound, and yet it illustrates the good performance of our constructed designs regarding the L -distance. Numerical calculation shows that our proposed methods are able to produce designs with L -distance efficiencies greater than 0.95 for large N. 4. Additional results on correlations. We now consider the pairwise correlation between columns for the constructed designs. For any N n design D = x ij, define j k ρ jk 4.1 ρ ave D = nn 1, where ρ jk is the correlation between columns j and k of D. Theρ ave in 4.1 is a performance measure on the overall pairwise column correlations for design D. A good design should have a low ρ ave value to reduce correlations between factors and reduce the variance of coefficients estimates. Consider the ρ ave values for the designs from the Williams transformation. For each prime N, Table6compares the ρ ave values of designs from the linear permutation, Williams transformation [with b chosen by 3.1], Gilbert and Golomb methods. The Williams transformation always generates designs with the smallest ρ ave values. In fact, we have a general result on the average correlation ρ ave E b for any b = 0,...,N 1, not restricted to the b defined in 3.1. THEOREM 5. Let N be an odd prime and D be an N N 1 GLP design, D b = D + bmod N, and E b = WD b for b = 0,...,N 1. Then ρ ave E b < /N. For a prime N, ρ ave E b 0asN for any b = 0,...,N 1. This property makes it possible to generate large LHDs with tiny pairwise column correla- TABLE 6 Comparison of the ρ ave values for N N 1 LHDs N LP WT Gil Gol N LP WT Gil Gol

11 OPTIMAL MAXIMIN L 1 -DISTANCE LATIN HYPERCUBE DESIGNS 3751 TABLE 7 Comparison of the ρ ave values for N 1 N 1 LHDs N LP-1 WT-1 Gil Gol N LP WT-1 Gil Gol tions without any computer search. For the leave-one-out Williams transformation, we have the following result. THEOREM 6. Let N be an odd prime, D be an N N 1 GLP design, D b = D + bmod N, E b = WD b, and Eb be the leave-one-out design obtained from E b for b = 0,...,N 1. Then ρ ave Eb <5N + 1/N for any b = 0,...,N 1. Table 7 compares designs obtained from the leave-one-out linear permutation, leave-one-out Williams transformation, Gilbert and Golomb methods. The leaveone-out Williams transformation generates designs with the smallest ρ ave values except for N = 13. For the modified Williams transformation, we have the following result. THEOREM 7. Let N = m + 1 be an odd prime, D = x ij be an N N 1 GLP design, A 1 be the m m leading principal submatrix of D, that is, A 1 = x ij with i, j = 1,...,m, and H = wa 1 /. Then ρ ave H < /m 1. Table 8 compares the ρ ave values of designs generated by the modified Williams transformation and some other available methods. The modified Williams transformation always provides designs with the smallest ρ ave values. 5. Extension. We consider extending the results to a general case where N = kp with k and p being prime numbers. Let 5.1 b = N1 + 1/ 3/4, and E b be the N φn design constructed by the Williams transformation. Figure top shows the values of d eff E b for N = p,3p,5p and 7p and p 00.

12 375 L. WANG, Q. XIAO AND H. XU TABLE 8 Comparison of the ρ ave values for m m LHDs m MWT Wel Gil Gol m MWT Wel Gil Gol The d eff E b increases quickly as N increases and reaches 0.9 when N is around 30. When N>100, the d eff E b values are typically greater than 0.95 and converge to 1 for N = p and N = 7p. Thed eff E b values do not converge to 1 for N = 3p and N = 5p, possibly due to the looseness of the upper bound d upper.in addition, Figure bottom shows that ρ ave E b goes to 0 quickly as N increases. We present the asymptotic optimality of E b for N = p based on the theoretical results in Section 3. It is possible to establish similar results for other cases with more elaborate arguments, which we do not pursue here. THEOREM 8. Let p be an odd prime, N = p, D be an N φn GLP design, D b = D +b mod Nand E b = WD b. For b defined in 5.1, d eff E b = 1 O1/N. As N, d eff E b 1. Now we consider an extension of the leave-one-out procedure. We can generate many asymptotically optimal LHDs by applying the following leave-out-one procedure for rows or columns. When we delete any row from an N n LHD D and rearrange the levels as in the leave-one-out method in Section., the distance of the resulting design will reduce at most by n. When we delete any column from an N n LHD D, the distance will reduce at most by N 1. Deleting multiple columns and rows together is equivalent to repeating the leave-one-out procedure for multiple times. The following result can be derived. THEOREM 9. Let D be an N n LHD. Deleting any k r rows and k c columns and rearranging the levels yields an N k r n k c LHD, denoted by D. Then d eff D d eff D 3k r /N k r 3k c /n k c. For N = kp and n = φn, n as N.Ifk r and k c are fixed constants not increasing with N, d eff D 1asN. This multiple leave-one-out pro-

13 OPTIMAL MAXIMIN L 1 -DISTANCE LATIN HYPERCUBE DESIGNS 3753 FIG.. The values of d eff E b top and ρ ave E b bottom with b defined in 5.1. cedure yields many asymptotically optimal LHDs with different sizes. For example, let k = 3andp = 41, we obtain a LHD with d eff = Delete the last rows and rearrange the levels; we obtain a LHD with d eff = Let k = andp = 61, we obtain a 1 60 LHD with d eff = Delete the last 1 rows and rearrange the levels; we obtain a LHD with d eff = Let k = 5andp = 103, we obtain a LHD with d eff = Delete the last 3 rows and the last 8 columns, and rearrange the levels, we obtain a LHD with d eff = A distinctive feature of our method is the excellent performance for moderate and large designs. Many other methods slow down quickly as the design size increases and usually give designs with poor distance efficiencies. In contrast, our method generates moderate and large designs with guaranteed high distance efficiencies without search, as long as the ratios of k r /N and k c /φn are small. When the ratios are relatively large, this simple procedure may not work well and further research is needed. 6. Concluding remarks. We have proposed a series of systematic methods for the construction of maximin LHDs via the Williams transformation and its modification. The Williams transformation and leave-one-out method produce asymptotically optimal LHDs under the maximin distance criterion, and the modified Williams transformation generates equidistant LHDs under the L 1 -distance.

14 3754 L. WANG, Q. XIAO AND H. XU Xu 1999 showed that equidistant LHDs are universally optimal for computer experiments. The average correlations between columns of the constructed designs converge to zero as the design sizes increase. Moreover, the constructed designs often have larger L 1 -distance and smaller average correlation than existing designs even for designs with small sizes. The Williams transformation can be applied to other designs as well. We have explored the Williams transformation on regular fractional factorial designs and found that it can substantially improve design efficiencies for estimating polynomial models. We will report the results in a separate paper. APPENDIX: PROOFS We need to distinguish two addition operations. To clarify, let be the addition operation over the Galois field {0,...,N 1}. LetD = x ij be the N φn GLP design and D b = x ij b. WhenN is a prime, x i = x i1,...,x in 1 and x i b = x i1 b,...,x in 1 b are the ith row of D and D b, respectively, x i is a permutation of {1,...,N 1} for i =,...,N 1; and x 1 = 1,...,N 1. The designs D and D b have some important properties which are crucial for the proofs of all theoretical results. We first summarize these properties in the following lemma. LEMMA 3. Let N be an odd prime: i For i k and i, k = 1,...,N 1, there exists a unique q {,...,N 1} such that k = iq mod N. For any given b, the two matrices xi b x1 b and x k b x q b are the same up to column permutations. In addition, q = N 1 if and only if i + k = N. ii For any b = 0,...,N 1 and i =,...,N, denote a = 1 ib mod N. The two matrices x1 b b x i b b are the same up to column permutations. and x1 0 x i a a PROOF. Part i is obvious from the definition of D and D b. For ii, denote x i = x i, 0 for i = 1,...,N.Then x i b = i x 1 b a. The result follows by noting that x 1 b is a permutation of x 1 and i x 1 a = x i a = x i a,a. PROOF OF LEMMA. We divide the proof in four steps. Step 1. For i + k N, i k, andi, k = 1,...,N 1, by Lemma 3i, there exists a unique q {,...,N } such that d ik WD b = d 1q WD b and

15 OPTIMAL MAXIMIN L 1 -DISTANCE LATIN HYPERCUBE DESIGNS 3755 d ik wd b = d 1q wd b. Therefore, it suffices to show that d 1i WD b = d 1i wd b for any b = 0,...,N 1andi =,...,N. Step. By Lemma 3ii, to prove d 1i WD b = d 1i wd b, we only need to show that dwx 1, Wx i a + Wa= dwx 1, wx i a + wa for any a = 0,...,N 1. Note that Wa= wa if a<n/, and Wa= wa 1if a>n/. It suffices to show that A.1 d Wx 1, Wx i a { d wx1, wx i a if a<n/; = d wx 1, wx i a + 1 if a>n/. Step 3. Recall that x 1 = 1,...,N 1 and x i a = x i1 a,...,x in 1 a. Then dwx 1, Wx i a = N 1 j=1 Wj Wx ij a and dwx 1, wx i a = N 1 j=1 wj wx ij a. It can be shown that Wj Wx ij a wj wx ij a for j I J ; = wj wx ij a 1 for j U\I; wj wx ij a + 1 for j V \J, where I = { j : j<n/,x ij a < N/ }, J = { j : j>n/,x ij a > N/ }, U = { j : j + x ij a < N } and V = { j : j + x ij a N }. Therefore, to prove A.1, we need to show that if a<n/, U\I and V \J contain the same number of elements; and if a>n/, U\I contains one less element than V \J. Step 4. Denote #S as the number of elements in a set S. Since#U\I= #U #I and #V \J= #V #J, we want to show that { #I = #J if a<n/; #U = #V and #I = #J + 1 if a>n/. Since x i+1j a = { j + xij a for j U; j + x ij a N for j V, then N 1 j=1 x i+1j a = N 1 j=1 x ij a + N 1 j=1 j #VN. Because both x i and x i+1 are permutations of {1,...,N 1}, N 1 j=1 x i+1j a = N 1 j=1 x ij a, which leads to #V = N 1 j=1 j/n = N 1/. Because #U + #V = N 1, #U = #V = N 1/. Denote I 1 ={j : j>n/,x ij a < N/}.Ifa<N/, #I + #I 1 = #J + #I 1 = N 1/so#I = #J.Ifa>N/, #I + #I 1 = N + 1/ and #J + #I 1 = N 1/ so#i = #J + 1. This completes the proof. To prove Theorem 1, we need the following lemma.

16 3756 L. WANG, Q. XIAO AND H. XU LEMMA 4. For all i =,...,N and b = 0,...,N 1, dx 1 b,x i b + dn x 1 b,x i b = N + 1/3 N b. and PROOF. We divide the proofs in three steps. Step 1. By Lemma 3ii, dx 1 b,x i b = dx 1,x i a + a d N x 1 b,x i b + N b =dn x 1,x i a + N a, where a = 1 ib mod N.Then dx 1 b,x i b + d N x 1 b,x i b = dx 1,x i a + dn x 1,x i a + N N b. Hence, it suffices to show that dx 1,x i a+ dn x 1,x i a = N + 1/3 N = N 1N 1/3 foranya = 0,...,N 1. Step. Let g i a = dx 1,x i a + dn x 1,x i a. If we can prove g i 0 = g i 1 = =g i N 1, we will have g i a = 1 N N 1 c=0 g i c = 1 N N 1 c=0 dx1,x i c + dn x 1,x i c. Because N 1 c=0 dn x 1,x i c = N 1 c=0 dx 1,x i c, then g i a = N = N N 1 c=0 N 1 j=1 k=0 dx 1,x i c = N N 1 N 1 N 1 c=0 j=1 j x ij c j k =N 1N 1/3. Step 3. Now we prove that g i 0 = g i 1 = =g i N 1. Itsufficestoshow that g i a + 1 = g i a for any a = 0,...,N. Recall that g i a = dx 1,x i a + dn x 1,x i a = N 1 j=1 j x ij a + N j x ij a. Since j x ij a N j x ij a + 1 j x ij a + N j x ij a for j S 1 S ; = j x ij a + N j x ij a + for j S 3 ; j x ij a + N j x ij a for j S 4,

17 OPTIMAL MAXIMIN L 1 -DISTANCE LATIN HYPERCUBE DESIGNS 3757 where S 1 ={j : j x ij a<n j}, S ={j : N j x ij a<j}, S 3 ={j : x ij a j,x ij a N j}, S 4 ={j : x ij a<j,x ij a<n j}, we only need to show that #S 3 = #S 4. Note that x i 1j a = x ij a j and x i+1j a = x ij a + j for j S 1 ; x i 1j a = x ij a j + N and x i+1j a = x ij a + j N for j S ; x i 1j a = x ij a j and x i+1j a = x ij a + j N for j S 3 ; x i 1j a = x ij a j + N and x i+1j a = x ij a + j for j S 4. Then A. N 1 j=1 N 1 xi 1j a + x i+1j a = x ij a N#S 3 #S 4. Because x i a is a permutation of {0,...,a 1,a+ 1,...,N 1} for any i<n, N 1 j=1 x i 1j a = N 1 j=1 x ij a = N 1 j=1 x i+1j a. ByA., N#S 3 #S 4 = 0so#S 3 = #S 4. This completes the proof. PROOF OF THEOREM 1. For the first case, note that Wx i b is a permutation of {0,...,Wb 1,Wb+ 1,...,N 1}, andwx N b is a constant vector with each component equal to Wb,sod in E b = d Ni E b = N 1 j=0 j Wb =N 1/3 + fb. To prove the result for the second case, i = N k, it suffices to prove the result for the third case. This is because the total pairwise L 1 -distance between distinct rows of WD b is t = N 1 N 1 N 1 j 1 =0 j =0 j 1 j =NN 1 N + 1/6. Out of all the pairs of distinct rows, N 1 pairs belong to the first case with a total distance t 1 = N 1[N 1/3 + fb], N 1N 3/ pairs belong to the third case with a total distance t = N 1N 1N 3/6, and N 1/ pairs belong to the second case. By Lemma 3i, d in i E b = d 1N 1 E b for any i. Therefore, d in i E b = t t 1 t /[N 1/]=N 1/3 fb. Now we prove the result for the last case where i N k, i N,andk N.By Lemmas and 3i, it suffices to consider d 1i E b = dwx 1 b,wx i b = j=1

18 3758 L. WANG, Q. XIAO AND H. XU dwx 1 b,wx i b for i =,...,N. Denote B = B 1 B B 3 B 4 wx1 b wx = 1 b N wx 1 b N wx 1 b, wx i b N wx i b wx i b N wx i b then d 1i E b = db 1. By column permutations, B can be rearranged as x1 b x C = 1 b N x 1 b N x 1 b. x i b N x i b x i b N x i b By Lemma 4, db = dc = 4N + 1/3 N b. Note that db 1 = db 4 and db = db 3.ForB, in both wx 1 b and wx i b, 0and wb appear once and all other even numbers smaller than N appear twice. Then db = N 1 j=1 N wx 1j b wx ij b = N +1 N b. Therefore, d 1i E b = db 1 = db db / = N 1/3. PROOF OF THEOREM. If c0 + c N 1/4, then c 0 N 1/1 /9 /3andc0 N 1/1 4/3 N 1/1. Hence, de b = N 1/4 + c0 N 1/3 4/3 N 1/1. Similarly, if c0 + c <N 1/4, c N 1/1 /9 + 1/3, and c N 1/1 + /3 N 1/1. Then de b = N 1/ c N 1/3 4/3 N 1/1. Therefore, de b N N 1 = N 1 N By the definition in 3., d eff E b = de b /N 1/3 1 / 3N 1. PROOF OF THEOREM 3. Let e i = e i1,...,e in 1 and e k = e k1,..., e kn 1 be two distinct rows of E b for i, k = 1,...,N 1, and ei = ei1,..., ein 1 and e k = e k1,...,e kn 1 be the corresponding rows of E b. For j = 1,...,N 1, if e ij >Wb>e kj or e kj >Wb>e ij, eij e kj = e ij e kj 1; otherwise, eij e kj = e ij e kj. Since the number of j s such that e ij >Wb>e kj [or e kj >Wb>e ij ] cannot exceed min{wb,n 1 Wb}, thendeb de b min{wb,n 1 Wb}. Fortheb defined in 3.1, min{wb,n 1 Wb}=N 1/ c. ThendE b de b N 1 + c de b N 1 + N 1/1 1. ByTheorem, deb N 7/3 + N 1/3/3 N 1. WhenN 7, we

19 OPTIMAL MAXIMIN L 1 -DISTANCE LATIN HYPERCUBE DESIGNS 3759 have d eff E b = de b / NN 1/3 de b /NN 1/3 1+1/ 3N 3/N > 1.43/N. PROOF OF THEOREM 5. Let ρ jk be the correlation between the jth and kth columns of E b. Denote the jth column of D b as z j b for j = 1,...,N 1, then z j b = x j b,b T. By Lemma 3i, there exists a unique q {,...,N 1} such that ρ jk = ρ 1q. Thus, A.3 ρ ave E b = where A.4 N 1 ρ 1j = cor W z 1 b,w z j b = j= ρ 1j, N Ni=1 Wx i1 b N 1 Wx ij b N 1 N 3 N/1 For x [0,N], the Fourier cosine expansion of x N/isgivenby A.5 with a u = N N 0 x N cos x N = uπx a u cos, N u=1 uπx By A.5, for any x [0,N], x N 1 Then the numerator of A.4 is A.6 where N { 0 if u is even; dx = 4N/ u π if u is odd. = x N = uπx a u cos N u=1 N Wx i1 b N 1 Wx ij b N 1 i=1 = a u a v su,v = 16N 1 π u=1 v=1 4 odd u odd v u v su,v, N uπwxi1 b su,v = cos cos N i=1.. vπwxij b N.

20 3760 L. WANG, Q. XIAO AND H. XU By.1, for any x = 0,...,N 1, cosuπwx + 0.5/N = cosuπx + 0.5/N. Then N uπxi1 + b vπxij + b su,v = cos cos N N i=1 = 1 N πjv + ui + c1 A.7 cos N i=1 + 1 N πjv ui + c cos, N i=1 where c 1 = b + 0.5u + v and c = b + 0.5v u. For positive odd numbers u and v, leti 1 ={u, v : u = jv or jv,v 0 mod N} and I = {u, v : u = 0andv = 0 mod N}.Foru, v I 1, su,v N/ because only one of the two items in A.7 can be nonzero. For u, v I, su,v N; for u, v / I 1 I, su,v = 0. Then by A.3, A.4 anda.6, A.8 Since ρ ave E b = = N 1 j= N 1 j= N i=1 Wx i1 b N 1 Wx ij b N 1 N N 3 N/1 19N π 4 N 3 NN 19N π 4 N 1N N 1 j= I 1 N 1 j= 1 u I v + 1 u 1 I v 1 1 odd v v 1 odd u u odd v 1 v odd u = N 1 N π 4 1 u 1 odd v 8 π 4 19, k=0 N 1 u v + N 1 u I v I 1 1 u v + I 1 u v 1 v + kn odd k 1 v 4 1 N odd v 1 v odd k 1 k. 1 k N where we used the fact that odd v 1/v = π /8and odd v 1/v 4 = π 4 /96. Then

21 OPTIMAL MAXIMIN L 1 -DISTANCE LATIN HYPERCUBE DESIGNS 3761 by A.8, ρ ave E b 1 19N N 1 π 4 N π 4 N 1 N 8 π 4 19 = 1 3 N N N < 1 N. PROOF OF THEOREM 6. For any b = 0,...,N 1, let E b = e ij. Because Ni=1 e ij N 1/ = NN 1/1 for any j = 1,...,N 1, by Theorem 5, we have A.9 N 1 N j= i=1 e i1 N 1 e ij N 1 < NN 1 6 Let ρjk be the correlation between the jth and kth columns of E b. Similar to A.3, A.10 Note that N 1 ρ ave E j= ρ 1j b =. N A.11 ρ1j = 1C 0 NN 1N with C 0 = e i1 μe ij μ + e i1 1 μe ij μ = e i1 <Wb e ij <Wb + e i1 <Wb e ij >Wb + e i1 >Wb e ij >Wb N e i1 N 1 i=1 where μ = N /, C 1 = 1 + e i1 <Wb N 1 4 e i1 μe ij 1 μ e i1 >Wb e ij <Wb e i1 1 μe ij 1 μ e ij e ij N 1 e i1 >Wb Wb e ij + + C 1 + C, e ij <Wb e i1. e i1 e ij >Wb

22 376 L. WANG, Q. XIAO AND H. XU and C = 1 4 e i1 <Wb e ij <Wb 1 + e i1 >Wb e ij >Wb 1 e i1 >Wb e ij <Wb 1 e i1 <Wb e ij >Wb 1. It is easy to see that C 1 N 1/4 and C N 1/4. Hence, by A.9, A.10 anda.11, ρ ave E 1 b < NN 1N NN 1 + N N < PROOF OF THEOREM 7. for j = 1,...,N 1/, where N N 1 4 5N + 1 N. N wx i1 N i=1 The proof is similar to that of Theorem 5. ByA.5, wx ij N = 16N π 4 odd v N uπwxi1 su,v = cos cos N i=1 Similar to A.8, we can prove that N 1/ N j= i=1 wx i1 N vπwxij N 1 u v su,v,. wx ij N N 3 4. Since N 1 i=1 wx i1 N + 1 N = wx i1 N i=1 wx ij N + 1 wx ij N N 1 + N , 4

23 OPTIMAL MAXIMIN L 1 -DISTANCE LATIN HYPERCUBE DESIGNS 3763 then Hence, N 1/ N 1 j= i=1 wx i1 N + 1 wx ij N + 1 N 3 N 1 N = N 3 6 5N 1N N + 1N 1N 3. 6 ρ ave H = ρ ave wa1 = N 1 N 1/ j= N 1 i=1 wx i1 N+1 wx ij N+1 m 1N + 1N 1N 3/1 m 1. PROOF OF THEOREM 8. To save space, we sketch only the main steps. Step 1. For N = p, φn = p 1andD = x ij with x ij = ij 1 mod N for i = 1,...,p and j = 1,...,p 1. With proper row and column permutations, D is equivalent to C A.1 mod N, C + p where C = y ij is an p p 1 GLP design with y ij = i j mod p for i = 1,...,p and j = 1,...,p 1. Then E b = WD b is equivalent to WC b Ẽ b = W C b + p. Step. Consider WC b. Ifb is even, C b = C + b/ mod p. Then wc b = w p C + b/ mod p where w is the modified Williams transformation defined in. andw p is the modified transformation with N replaced by p. By Lemma and Theorem 1, d ik wc b = [d ik w p C + b/ mod p]=n 1/3 fori k, i p,k p, andi + k p. Following the lines of Lemma will result d ik WC b = d ik wc b. Then A.13 d ik WC b = N 4 /6 for i k,i p,k p, and i + k p. If b is odd, WC b = N 1 WC b + p and A.13 also holds.

24 3764 L. WANG, Q. XIAO AND H. XU Step 3. If b is even, the last row of WC b is b,...,b and each other row is a permutation of {0, 3, 4,...,p 1 1, p 1}\{b}. Based on this structure, we get A.14 d ip WC b = N 6 N Wb + gb, N A.15 d ip i WC b = 6 + N 1 Wb gb, where gb = Wb N Wb N. 3 Similarly, if b is odd, A.14 anda.15 also hold. Step 4. Because WC b = N 1 WC b + p, WC b + p has the same distance structure as WC b. Step 5. By the structure of WC b + p and WC b, by computation, we can get Ẽb d ip+k A.16 N /4 l 1 b for i = k p; N/ 1l 1 b for i = k = p; = N /6 l 1 b + 1/3 for i, k I 1 ; N /6 N /4 + l b/ l 1 b for i, k I ; N /1 + N/ 1l 1 b + N/ l b for i, k I 3, where l 1 b = N Wb 1, l b = Wb+ gb, I 1 ={i, k : i p,k p,i + k p}, I ={i, k : i p,k = p, or i = p,k p} and I 3 ={i, k : i p,k p,i + k = p}. Step 6. For b = N1 + 1/ 3/4, Wb= b = N1 + 1/ 3/ or N1 + 1/ 3/ +1, so N/ 3 gb 0. Then l 1 b = ON and l b = ON. Since for any N N/ 1 LHD, d upper = N + 1N /6, by A.13 A.16, it can be verified that d eff E b = d eff Ẽ b = 1 O1/N. Acknowledgment. The authors thank an Editor, an Associate Editor and two reviewers for their helpful comments. REFERENCES BA, S., MYERS, W. R. and BRENNEMAN, W. A Optimal sliced Latin hypercube designs. Technometrics MR BAILEY, R. A The decomposition of treatment degrees of freedom in quantitative factorial experiments. J. Roy. Statist. Soc. Ser. B MR

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26 3766 L. WANG, Q. XIAO AND H. XU TANG, B Orthogonal array-based Latin hypercubes. J. Amer. Statist. Assoc WILLIAMS, E. J Experimental designs balanced for the estimation of residual effects of treatments. Aust. J. Sci. Res XIAO, Q. and XU, H Construction of maximin distance Latin squares and related Latin hypercube designs. Biometrika XIAO, Q. and XU, H Construction of maximin distance designs via level permutation and expansion. Statist. Sinica XU, H Universally optimal designs for computer experiments. Statist. Sinica YANG, J.Y.andLIU, M. Q. 01. Construction of orthogonal and nearly orthogonal Latin hypercube designs from orthogonal designs. Statist. Sinica YE, K. Q Orthogonal column Latin hypercubes and their application in computer experiments. J. Amer. Statist. Assoc ZHOU, Y. and XU, H Space-filling properties of good lattice point sets. Biometrika L. WANG H. XU DEPARTMENT OF STATISTICS UNIVERSITY OF CALIFORNIA, LOS ANGELES 815 MATH SCIENCES BLDG. LOS ANGELES, CALIFORNIA USA linhappyforever@ucla.edu hqxu@stat.ucla.edu Q. XIAO DEPARTMENT OF STATISTICS UNIVERSITY OF GEORGIA 101 CEDAR STREET ATHENS, GEORGIA 3060 USA qian.xiao@uga.edu