On the connection between maximin distance designs and orthogonal designs

On the connection between maximin distance designs and orthogonal designs SUMMARY Maximin distance designs and orthogonal designs are widely used in computer and physical experiments. We characterize a broad class of maximin distance designs by establishing new bounds on the minimum intersite distance for mirror-symmetric and general U-type designs. We show that maximin distance designs and orthogonal designs are closely related and coincide under some conditions. 1. Introduction Many researchers have proposed methods for constructing orthogonal Latin hypercube designs and orthogonal U-type designs for computer experiments; see Lin & Tang (2015) for a review. The construction of maximin distance designs is more challenging than that of orthogonal designs, and most of the existing approaches for this purpose are based on stochastic algorithms. Despite many attempts, there is no guarantee that the resulting designs truly have the largest minimum distance. Moreover, orthogonal Latin hypercube designs may not be space filling in terms of distance and vice versa (Joseph & Hung, 2008). In this paper, we focus on mirror-symmetric designs, which have often been favoured by researchers. Many constructions of orthogonal designs utilize a mirror-symmetric structure, which can yield higher-order orthogonality (Sun et al., 2009; Tang & Xu, 2014). Morris & Mitchell (1995) observed that many of their maximin distance Latin hypercube designs are mirror-symmetric. An orthogonal mirror-symmetric U-type design of $$2p$$ runs can accommodate at most $$p$$ factors. We show that such designs have maximin distance among a broad class of designs. Loeppky et al. (2009) suggested a rule of thumb that, under effect sparsity, a design with run size around ten times the effective dimension is preferable, given good prior knowledge of the number of active factors, whereas Moon et al. (2012) proposed a two-stage screening procedure and, in agreement with the set-up of the present work, suggested the use of designs with $$2p$$ runs for $$p$$ factors in the first stage. See Woods & Lewis (2017) for some examples. 2. Notation and background Let $$(n,s^p)$$ denote a design with $$n$$ runs and $$p$$ factors each of $$s$$ levels. The $$s$$ levels are labelled symmetrically as $$-(s-1)/2, -(s-3)/2, \ldots, (s-3)/2, (s-1)/2$$. An $$(n,s^p)$$ design is said to be U-type if each of the $$s$$ levels occurs exactly $$n/s$$ times in each column. A U-type design is a Latin hypercube design when $$s=n$$. A design is mirror-symmetric if its mirror image is itself (Tang & Xu, 2014). Up to row permutations, a mirror-symmetric design must have the form $$(X^{{ \mathrm{\scriptscriptstyle T} }},-X^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$$ for even $$n$$ and $$(X^{ \mathrm{\scriptscriptstyle T} }, 0_p^{ \mathrm{\scriptscriptstyle T} }, -X^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$$ for odd $$n$$, where $$ 0_p$$ is a row vector of $$p$$ zeros. A U-type design $${D=(x_{ik})}$$ is column-orthogonal, or orthogonal for short, if $$\sum_{i=1}^{n}x_{ik}x_{ik'}=0$$ for all $$k\neq k'$$. An orthogonal design is $$\ell$$-orthogonal if the sum of elementwise products of any $$\ell$$ columns, of which at least two columns are distinct, is zero. Orthogonal mirror-symmetric designs are $$\ell$$-orthogonal for any odd $$\ell$$ (Tang & Xu, 2014). A $$3$$-orthogonal Latin hypercube design with $$n$$ runs has at most $$\lfloor n/2 \rfloor$$ columns (Sun et al., 2009), where $$\lfloor x \rfloor$$ denotes the largest integer not exceeding $$x$$. This result can be generalized for general U-type designs. Orthogonal mirror-symmetric U-type designs attaining the maximum number of columns are saturated $$3$$-orthogonal designs. For an $$(n,s^p)$$ design $$D$$, the $$L_2$$-distance between two distinct rows $$x_i=(x_{i1},\ldots,x_{ip})$$ and $$x_j=(x_{j1},\ldots,x_{jp})$$ in $$D$$ is defined to be $$d_2( x_i, x_j) = \sum_{k=1}^{p}|x_{ik}-x_{jk}|^2$$, which is the square of the Euclidean distance between $$x_i$$ and $$x_j$$. The $$L_2$$-distance of $$D$$ is defined to be $$d_2(D) = \min \{d_2( x_i, x_j): x_i\neq x_j, x_i, x_j \in D \}$$. A maximin distance design maximizes the $$d_2(D)$$ value. Zhou & Xu (2015) derived the following upper bound for a general U-type design. Lemma 1. For any U-type $$(n,s^p)$$ design $$D$$, $$d_2(D) \leqslant \lfloor n(s^2-1)p/(6n-6) \rfloor$$. For a Latin hypercube design, $$s=n$$, so the bound becomes $$d_2(D) \leqslant \lfloor n(n+1)p/6 \rfloor$$. The upper bound is commonly used for evaluating whether a design has a good minimum distance. However, in many situations it is not attainable. 3. Maximin mirror-symmetric U-type designs First we establish an upper bound on the $$L_2$$-distance of mirror-symmetric U-type designs. Lemma 2. Let $$n>2$$ be even. For any mirror-symmetric U-type $$(n,s^p)$$ design $$D$$, $$d_2(D) \leqslant \lfloor {(s^2-1)p}/{6} \rfloor$$. This bound becomes $$\lfloor {(n^2-1)p}/{6} \rfloor$$ for mirror-symmetric Latin hypercube designs with an even number of runs. Compared with Lemma 1, Lemma 2 gives tighter upper bounds for mirror-symmetric designs. Furthermore, in many cases, the upper bound in Lemma 2 is achievable while that in Lemma 1 is not. Here is an example. Example 1. Consider two Latin hypercube designs with $$n=8$$ runs and $$p=4$$ factors, $$ D_1 = \left( \begin{array}{rrrr} 0.5 & 1.5 & 2.5 & 3.5 \\ 1.5 & -0.5 & -3.5 & 2.5 \\ 2.5 & 3.5 & -0.5 & -1.5 \\ 3.5 & -2.5 & 1.5 & -0.5 \\ -0.5 & -1.5 & -2.5 & -3.5 \\ -1.5 & 0.5 & 3.5 & -2.5 \\ -2.5 & -3.5 & 0.5 & 1.5 \\ -3.5 & 2.5 & -1.5 & 0.5 \end{array} \right), \quad D_2 =\left( \begin{array}{rrrr} 0.5 & 2.5 & 1.5 & 3.5 \\ 3.5 &-0.5 &-2.5 &-1.5 \\ -1.5 &-3.5 & 0.5 & 2.5 \\ 2.5 &-1.5 &-3.5 & 0.5 \\ -0.5 &-2.5 &-1.5 &-3.5 \\ -3.5 & 0.5 & 2.5 & 1.5 \\ 1.5 & 3.5 &-0.5 &-2.5 \\ -2.5 & 1.5 & 3.5 &-0.5 \end{array} \right)\!,$$ where $$D_1$$ is from Sun et al. (2009) and $$D_2$$ is randomly generated. Both designs are mirror-symmetric. The $$L_2$$-distances of $$D_1$$ and $$D_2$$ are $$42$$ and $$7$$, respectively. For Latin hypercube designs with $$n=8$$ and $$p=4$$, the upper bounds in Lemmas 1 and 2 are $$48$$ and $$42$$, respectively. Hence $$D_1$$ attains the upper bound in Lemma 2 and has maximin distance among all mirror-symmetric Latin hypercube designs with $$n=8$$ and $$p=4$$. From the algorithm search result in Table 2(A) of Morris & Mitchell (1995), no Latin hypercube design has distance larger than $$42$$, which suggests that the upper bound in Lemma 1 is not achievable in this case. It is possible that $$D_1$$ has maximin distance among all Latin hypercube designs with $$n=8$$ and $$p=4$$. Consider the designs in Example 1 from the viewpoint of orthogonality. It can be checked that $$D_1$$ is orthogonal but $$D_2$$ is not. In fact $$D_1$$ is a saturated $$3$$-orthogonal Latin hypercube design, so it is optimal in terms of both orthogonality and maximin distance. This does not happen by chance. The following theorem establishes the equivalence between the two criteria for mirror-symmetric U-type designs. Theorem 1. Let $$D$$ be a mirror-symmetric U-type $$(n,s^p)$$ design with $$n=2p$$. (i) If $$D$$ is orthogonal, then $$D$$ has maximin $$L_2$$-distance among all mirror-symmetric U-type $$(n,s^p)$$ designs. (ii) If $$D$$ has maximin $$L_2$$-distance with $$d_2(D)= (s^2-1)p/6$$, then $$D$$ is orthogonal. The condition in Theorem 1(ii), $$d_2(D)=(s^2-1)p/6$$, cannot be relaxed because the value $$(s^2-1)p/6$$ in Lemma 2 may not be an integer for some parameters. If so, even if the distance reaches the upper bound in Lemma 2, the orthogonality of $$D$$ cannot be guaranteed. A counterexample is the following Latin hypercube design with $$n=10$$ and $$p=5$$: $$D_3 = \left( \begin{array}{rrrrrrrrrr} -4.5 & -3.5 & -2.5 & -1.5 & -0.5 & 4.5 & 3.5 & 2.5 & 1.5 & 0.5 \\ 2.5 & -1.5 & -0.5 & -4.5 & 3.5 & -2.5 & 1.5 & 0.5 & 4.5 & -3.5 \\ -1.5 & 0.5 & 4.5 & -3.5 & -2.5 & 1.5 & -0.5 & -4.5 & 3.5 & 2.5 \\ 0.5 & -4.5 & 3.5 & 2.5 & 1.5 & -0.5 & 4.5 & -3.5 & -2.5 & -1.5 \\ -3.5 & 2.5 & 1.5 & 0.5 & 4.5 & 3.5 & -2.5 & -1.5 & -0.5 & -4.5 \end{array} \right)^{ \mathrm{\scriptscriptstyle T} }\!,$$ which is mirror-symmetric with $$L_2$$-distance $$82$$. For $$s=n=10$$ and $$p=5$$, we have $$(s^2-1)p/6=82.5$$. So $$D_3$$ is a maximin distance design among all mirror-symmetric Latin hypercube designs. However, all the off-diagonal entries of $$D_3^{ \mathrm{\scriptscriptstyle T} }D_3$$ are $$\pm 0.5$$, so the design is not orthogonal. Orthogonal mirror-symmetric Latin hypercube designs with $$n=2p$$ runs and $$p$$ factors have been constructed by Sun et al. (2009) for $$p=2^c$$ and any positive integer $$c$$, and by Georgiou & Stylianou (2011) and Sun et al. (2009) for $$p=4$$, $$8$$, $$12$$, $$16$$, $$20$$, $$24$$, and $$32$$. Orthogonal mirror-symmetric U-type designs have been constructed by Bingham et al. (2009), Georgiou et al. (2014a,b) and Stylianou et al. (2015); the Appendix of Georgiou et al. (2014a) provides a collection of designs with run size smaller than $$100$$. All these designs are saturated $$3$$-orthogonal U-type designs. By Theorem 1, all of them have maximin distance among all possible mirror-symmetric designs. Lin et al. (2010) showed that orthogonal Latin hypercube designs with $$n=4m+2$$ do not exist when $$m$$ is an integer. Hence, orthogonal mirror-symmetric Latin hypercube designs with $$n=2p$$ do not exist when $$p$$ is odd. It would be interesting to explore how to construct maximin designs for odd $$p$$. The proof of Theorem 1 indicates that a mirror-symmetric design $$D=(X^{ \mathrm{\scriptscriptstyle T} },-X^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$$ has maximin distance if $$X$$ is row-orthogonal and each row has equal distance from the centre point. Morris & Mitchell (1995) searched maximin Latin hypercube designs using a simulated annealing algorithm and observed that many of the optimal designs obtained share the property that all of the runs have exactly or nearly equal distance from the centre point, especially for those with $$n=p$$ and $$n=2p$$. In addition, nearly all of their $$2p\times p$$ maximin distance designs are mirror-symmetric. Theorem 1 provides a theoretical justification for their findings. From the proof of Theorem 1, the following corollary is obvious, and can be used to obtain larger maximin distance designs from smaller ones. Corollary 1. If $$D$$ is an orthogonal mirror-symmetric U-type $$(n,s^p)$$ design with $$n=2p$$, then $$E=\left(\!\!\begin{array}{rr} D & D\\ D & -D \end{array}\!\!\right)$$ is orthogonal and has maximin distance among all mirror-symmetric U-type $$(2n,s^{2p})$$ designs. Example 2. For the orthogonal Latin hypercube design $$D_1$$ with $$n=8$$ and $$p=4$$ in Example 1, recursively applying Corollary 1 yields a series of maximin distance mirror-symmetric U-type $$(8k,8^{4k})$$ designs for $$k=2,4,8,\ldots$$. In general, starting from an orthogonal mirror-symmetric U-type $$(2p, s^p)$$ design, we obtain a series of maximin distance mirror-symmetric U-type $$(2kp, s^{kp})$$ designs for $$k=2,4,8,\ldots$$. 4. Maximin U-type designs The results in § 3 are obtained under mirror-symmetry. For certain parameters we shall prove an exact equivalence between maximin distance and orthogonality among all possible U-type designs. We first characterize the geometric structure of all maximin distance mirror-symmetric U-type designs with $$n=2p$$ in Theorem 1. A regular cross-polytope in $$\mathbb{R}^{p}$$ is the convex hull of $$p$$ mutually perpendicular line segments of equal length, intersecting at the midpoint of each of them. In other words, a regular cross-polytope has $$2p$$ vertices composing a design $$D = (X^{ \mathrm{\scriptscriptstyle T} },-X^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$$ satisfying $$XX^{ \mathrm{\scriptscriptstyle T} } = X^{ \mathrm{\scriptscriptstyle T} }X = {r^2} I_p$$ for some constant $$r$$. The design $$D$$ is orthogonal and mirror-symmetric, but we emphasize that $$D$$ may not be a U-type or $$s$$-level design. The regular cross-polytope is inscribed in the ball with radius $$r$$, and all its pairwise distances have only two distinct values, $$2r^2$$ and $$4r^2$$. Obviously, the points of any orthogonal mirror-symmetric U-type $$(2p,s^p)$$ design form the vertices of a regular cross-polytope in $$\mathbb{R}^p$$, which indicates that the geometric structure of any maximin $$L_2$$-distance mirror-symmetric U-type $$(2p,s^p)$$ design in Theorem 1 is uniquely determined. We rephrase Theorem 2 of Kuperberg (2007) as a lemma. Lemma 3. If $$n=2p$$ points are placed in the ball of radius $$r$$ in $$\mathbb{R}^p$$, then the maximum attainable minimum pairwise $$L_2$$-distance is $$2r^2$$, and each optimal arrangement must consist of the vertices of a regular cross-polytope inscribed in the ball. Now we can give a theorem showing that for $$s=2,3,4$$, whenever orthogonal mirror-symmetric U-type $$(2p,s^p)$$ designs exist, they are in fact the only maximin distance designs among all possible U-type $$(2p,s^p)$$ designs. Theorem 2. Let $$D$$ be a U-type $$(n,s^p)$$ design with $$n=2p$$ and $$s=2,3$$, or $$4$$. Then $$({\rm i})$$$$d_2(D)\leqslant \lfloor (s^2-1)p/6 \rfloor $$; $$({\rm ii})$$ if $$D$$ is mirror-symmetric and orthogonal, then $$D$$ has maximin $$L_2$$-distance among all possible U-type designs; and $$({\rm iii})$$ if $$D$$ has maximin $$L_2$$-distance with $$d_2(D)= (s^2-1)p/6$$, then $$D$$ is mirror-symmetric and orthogonal. Because any $$p \times p$$ Hadamard matrix $$X$$ is orthogonal, by Theorem 2, $$D=(X^{ \mathrm{\scriptscriptstyle T} },-X^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$$ must have maximin distance among all possible U-type $$(2p,2^p)$$ designs. Butler (2007) showed that when $$n=2p$$, a two-level resolution IV design, regular or nonregular, must be mirror-symmetric. Hence all saturated resolution IV two-level designs have maximin distance among all possible designs. Example 3. Consider the U-type $$(12,3^6)$$ design in Example 3 of Stylianou et al. (2015). It can be checked that the $$L_2$$-distance of the design is $$8$$, which achieves the upper bound $$(s^2-1)p/6$$ in Lemma 2. Since the number of levels is three, from Theorem 2 we can conclude that this design has maximin distance among all possible U-type designs with the same parameters. Theorem 2 also tells us that the design must be mirror-symmetric and orthogonal, that is, all design points form the vertices of a six-dimensional regular cross-polytope. For $$n=2p$$ and $$s\geqslant 5$$, from the constructions in the literature (van Dam et al., 2009; Ba et al., 2015; Xiao & Xu, 2018), no design has been found to have $$L_2$$-distance larger than $${(s^2-1)n}/{6}$$. Thus we conjecture that any orthogonal mirror-symmetric U-type $$(n,s^p)$$ design with $$n=2p$$ and $$s\geqslant 5$$ will have maximin distance among all possible U-type designs. It would be useful to prove this or find a counterexample. Acknowledgement Wang and Yang were supported by the National Natural Science Foundation of China. Yang was also supported by the Tianjin Development Program for Innovation and Entrepreneurship and Tianjin 131 Talents program, and Xu was supported by the U.S. National Science Foundation. Wang is also affiliated with the Center for Statistical Science. The research was carried out when the first two authors were visiting the Department of Statistics at the University of California, Los Angeles. The first two authors contributed equally. The authors thank the editor and reviewers for their helpful comments. Appendix Proof of Lemma 2. Let $$n=2m$$, $$D=(X^{ \mathrm{\scriptscriptstyle T} },-X^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$$ and $$X=( x^{ \mathrm{\scriptscriptstyle T} }_1,\ldots, x^{ \mathrm{\scriptscriptstyle T} }_{m})^{ \mathrm{\scriptscriptstyle T} }$$. To compute the $$L_2$$-distance of the design $$D$$, all distances between any two distinct rows need to be considered. For $$i=1,\ldots,{m}$$, it is easy to see that $$d_2( x_i,- x_i)=4d_2( x_i, 0_p)=4\sum_{{k}=1}^p x_{i{k}}^2$$. Hence, by noting that $$D$$ is a U-type design, the average of all such kinds of distances is \begin{equation} n^{-1}\sum_{i=1}^{m} \{ d_2( x_i,- x_i)+ d_2(- x_i, x_i) \} ={8 n^{-1} \sum_{i=1}^m \sum_{k=1}^{p}}x_{i{k}}^2=(s^2-1)p/{3}\text{.} \end{equation} (A1) Similarly, the average of all other pairwise distances is \begin{align} & \{n(n-2)\}^{-1}\sum_{i=1}^{m}\sum_{j\neq i}2\{d_2( x_i, x_j)+ d_2( x_i,- x_j)\} \nonumber \\ &\quad{} = {4} \{n(n-2)\}^{-1} \sum_{i=1}^{m}\sum_{j\neq i} { \sum_{k=1}^{p} } \left(x_{i{k}}^2+x_{j{k}}^2\right) = (s^2-1)p/{6}\text{.} \end{align} (A2) Because $$d_2(D)$$ is an integer and at most equal to the average distance, the result follows by rounding and comparing (A1) and (A2). □ Proof of Theorem 1. Let $$D=(X^{ \mathrm{\scriptscriptstyle T} }, -X^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$$. (i) Due to the U-type property of $$D$$, the orthogonality of $$D$$ implies that $$X^{ \mathrm{\scriptscriptstyle T} }X = r^2 I_p$$, which is equivalent to $$X X^{ \mathrm{\scriptscriptstyle T} } = r^2 I_p$$, where $$r^2=(s^2-1)p/12$$. This also implies that $$D$$ has only two distinct pairwise distances, $$(s^2-1)p/6$$ and $$(s^2-1)p/3$$. So the $$L_2$$-distance of $$D$$ is $$(s^2-1)p/6$$. By Lemma 2, $$D$$ has maximin distance among all mirror-symmetric U-type $$(n,s^p)$$ designs. (ii) Suppose that the $$L_2$$-distance of $$D$$ is $$(s^2-1)p/6$$. According to the proof of Lemma 2, we have $$d_2( x_i, x_j ) = d_2( x_i, - x_j ) = (s^2-1)p/6$$ for any two distinct rows $$ x_i$$ and $$ x_j$$ in $$X$$. This means that (a) $$d_2( x_i , 0) = (s^2-1)p/12$$ for any row $$ x_i$$ of $$X$$; and (b) $$ x_i x_j^{ \mathrm{\scriptscriptstyle T} }=0$$ for any two distinct rows $$ x_i$$ and $$ x_j$$ of $$X$$. Therefore, we have $$X X^{ \mathrm{\scriptscriptstyle T} } = {r^2} I_p$$, which also implies $$X^{ \mathrm{\scriptscriptstyle T} }X = {r^2} I_p$$, where $$r^2=(s^2-1)p/12$$. This proves that $$D$$ is orthogonal. □ Proof of Theorem 2. We need only show that no U-type design can have distance larger than $$(s^2-1)p/6$$, and the optimal design attaining this bound must comprise the vertices of a regular cross-polytope. For $$s=2$$, the two levels are $$-1/2$$ and $$1/2$$. Therefore, for any run $$ x_i$$ in $$D$$, we have $$d_2( x_i, 0)={r^2}= p/4$$, which means that all two-level designs have points with equal distance from the centre point. By Lemma 3, the maximin distance for two-level designs with $$2p$$ runs and $$p$$ factors is $${2r^2}=p/2$$, which equals $$(s^2-1)p/6$$ for $$s=2$$. Also, if a design attains this bound then it must consist of the vertices of a regular cross-polytope. For $$s=3$$, the levels are $$-1, 0$$ and $$1$$. Given any three-level U-type design, if all runs have equal distance from the centre point, it can be calculated that the radius of the ball that all runs fall on is $${r=(2p/3)^{1/2}}$$. The maximin distance, according to Lemma 3, is $${2r^2}=4p/3$$, which is equal to $$(s^2-1)p/6$$ for $$s=3$$. If not all runs have equal distance from the centre point, then there exists at least one run in $$D$$, without loss of generality say $$ x_1$$, such that $$d_2( x_1, 0)<2p/3$$. This also means that \begin{equation} d_2( x_1, 0)\leqslant {2p}/{3}-1< {2(p-1)}/{3} \end{equation} (A3) since $$n/3 = 2p/3$$ is an integer for a U-type design. Consider the average distance between the run $$ x_1$$ and all other runs $$ x_2,\ldots,x_n$$. Let $$u=(s-1)/2$$. By noting that each of the $$s$$ levels appears exactly $$n/s$$ times in each column of $$D$$, we have $$ (n-1)^{-1}\sum_{i=2}^n \sum_{k=1}^p (x_{ik} - x_{1k})^2 = (n-1)^{-1}\sum_{k =1}^p n s^{-1} \left( \sum_{t=0}^{{u - x_{1k}}} t^2 + \sum_{t=0}^{{u + x_{1k}}} t^2 \right)\!\text{.} $$ Using the formula $$\sum_{t=0}^{q} t^2 = q(q+1)(2q+1)/6$$ and after some algebra, we get \begin{equation} {(n-1)^{-1}\sum_{i=2}^n \sum_{k=1}^p (x_{ik} - x_{1k})^2 = (n-1)^{-1} \{ {np(s^2-1)}/12 + n d_2( x_1, 0) \}\text{.}} \end{equation} (A4) Combining (A3) and (A4) leads to $$ d_2(D)< {(n-1)^{-1} \{ {np(3^2-1)}/12 + 2n(p-1) /3 \} }= {4p}/{3}={(s^2-1)p}/{6} $$ for $$s=3$$, so no U-type design can have $$L_2$$-distance larger than $$(s^2-1)p/6$$ for $$s=3$$. Hence, when the maximin distance is $$(s^2-1)p/6$$, all runs must have equal distance from the centre point and the design points form the vertices of a regular cross-polytope by Lemma 3. For $$s=4$$, the four levels are $$\pm 1/2$$ and $$\pm 3/2$$, which lead to two distinct squared values $$1/4$$ and $$9/4$$. If all runs of $$D$$ have equal distance from the centre point, it can be calculated that the radius of the ball that all runs fall on is $$r=(5p/4)^{1/2}$$. The maximin distance by Lemma 3 is $$2 r^2=5p/2$$, which is equal to $$(s^2-1)p/6$$ for $$s=4$$. If not all runs have equal distance from the centre point, then there exists at least one run, without loss of generality say $$ x_1$$, in $$D$$ such that $$d_2( x_1, 0)<5p/4$$. Because $$|x_{1k}|=1/2$$ or $$3/2$$ for $$k=1,\ldots,p$$, there are at most $$(p/2-1)$$ entries such that $$|x_{1k}|=3/2$$ and at least $$(p/2+1)$$ entries such that $$|x_{1k}|=1/2$$. 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Space-filling properties of good lattice point sets Biometrika 102 , 959 – 66 . Google Scholar CrossRef Search ADS © 2018 Biometrika Trust This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Biometrika Oxford University Press

On the connection between maximin distance designs and orthogonal designs

Biometrika , Volume Advance Article (2) – Feb 28, 2018

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10.1093/biomet/asy005
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Abstract

SUMMARY Maximin distance designs and orthogonal designs are widely used in computer and physical experiments. We characterize a broad class of maximin distance designs by establishing new bounds on the minimum intersite distance for mirror-symmetric and general U-type designs. We show that maximin distance designs and orthogonal designs are closely related and coincide under some conditions. 1. Introduction Many researchers have proposed methods for constructing orthogonal Latin hypercube designs and orthogonal U-type designs for computer experiments; see Lin & Tang (2015) for a review. The construction of maximin distance designs is more challenging than that of orthogonal designs, and most of the existing approaches for this purpose are based on stochastic algorithms. Despite many attempts, there is no guarantee that the resulting designs truly have the largest minimum distance. Moreover, orthogonal Latin hypercube designs may not be space filling in terms of distance and vice versa (Joseph & Hung, 2008). In this paper, we focus on mirror-symmetric designs, which have often been favoured by researchers. Many constructions of orthogonal designs utilize a mirror-symmetric structure, which can yield higher-order orthogonality (Sun et al., 2009; Tang & Xu, 2014). Morris & Mitchell (1995) observed that many of their maximin distance Latin hypercube designs are mirror-symmetric. An orthogonal mirror-symmetric U-type design of $$2p$$ runs can accommodate at most $$p$$ factors. We show that such designs have maximin distance among a broad class of designs. Loeppky et al. (2009) suggested a rule of thumb that, under effect sparsity, a design with run size around ten times the effective dimension is preferable, given good prior knowledge of the number of active factors, whereas Moon et al. (2012) proposed a two-stage screening procedure and, in agreement with the set-up of the present work, suggested the use of designs with $$2p$$ runs for $$p$$ factors in the first stage. See Woods & Lewis (2017) for some examples. 2. Notation and background Let $$(n,s^p)$$ denote a design with $$n$$ runs and $$p$$ factors each of $$s$$ levels. The $$s$$ levels are labelled symmetrically as $$-(s-1)/2, -(s-3)/2, \ldots, (s-3)/2, (s-1)/2$$. An $$(n,s^p)$$ design is said to be U-type if each of the $$s$$ levels occurs exactly $$n/s$$ times in each column. A U-type design is a Latin hypercube design when $$s=n$$. A design is mirror-symmetric if its mirror image is itself (Tang & Xu, 2014). Up to row permutations, a mirror-symmetric design must have the form $$(X^{{ \mathrm{\scriptscriptstyle T} }},-X^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$$ for even $$n$$ and $$(X^{ \mathrm{\scriptscriptstyle T} }, 0_p^{ \mathrm{\scriptscriptstyle T} }, -X^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$$ for odd $$n$$, where $$ 0_p$$ is a row vector of $$p$$ zeros. A U-type design $${D=(x_{ik})}$$ is column-orthogonal, or orthogonal for short, if $$\sum_{i=1}^{n}x_{ik}x_{ik'}=0$$ for all $$k\neq k'$$. An orthogonal design is $$\ell$$-orthogonal if the sum of elementwise products of any $$\ell$$ columns, of which at least two columns are distinct, is zero. Orthogonal mirror-symmetric designs are $$\ell$$-orthogonal for any odd $$\ell$$ (Tang & Xu, 2014). A $$3$$-orthogonal Latin hypercube design with $$n$$ runs has at most $$\lfloor n/2 \rfloor$$ columns (Sun et al., 2009), where $$\lfloor x \rfloor$$ denotes the largest integer not exceeding $$x$$. This result can be generalized for general U-type designs. Orthogonal mirror-symmetric U-type designs attaining the maximum number of columns are saturated $$3$$-orthogonal designs. For an $$(n,s^p)$$ design $$D$$, the $$L_2$$-distance between two distinct rows $$x_i=(x_{i1},\ldots,x_{ip})$$ and $$x_j=(x_{j1},\ldots,x_{jp})$$ in $$D$$ is defined to be $$d_2( x_i, x_j) = \sum_{k=1}^{p}|x_{ik}-x_{jk}|^2$$, which is the square of the Euclidean distance between $$x_i$$ and $$x_j$$. The $$L_2$$-distance of $$D$$ is defined to be $$d_2(D) = \min \{d_2( x_i, x_j): x_i\neq x_j, x_i, x_j \in D \}$$. A maximin distance design maximizes the $$d_2(D)$$ value. Zhou & Xu (2015) derived the following upper bound for a general U-type design. Lemma 1. For any U-type $$(n,s^p)$$ design $$D$$, $$d_2(D) \leqslant \lfloor n(s^2-1)p/(6n-6) \rfloor$$. For a Latin hypercube design, $$s=n$$, so the bound becomes $$d_2(D) \leqslant \lfloor n(n+1)p/6 \rfloor$$. The upper bound is commonly used for evaluating whether a design has a good minimum distance. However, in many situations it is not attainable. 3. Maximin mirror-symmetric U-type designs First we establish an upper bound on the $$L_2$$-distance of mirror-symmetric U-type designs. Lemma 2. Let $$n>2$$ be even. For any mirror-symmetric U-type $$(n,s^p)$$ design $$D$$, $$d_2(D) \leqslant \lfloor {(s^2-1)p}/{6} \rfloor$$. This bound becomes $$\lfloor {(n^2-1)p}/{6} \rfloor$$ for mirror-symmetric Latin hypercube designs with an even number of runs. Compared with Lemma 1, Lemma 2 gives tighter upper bounds for mirror-symmetric designs. Furthermore, in many cases, the upper bound in Lemma 2 is achievable while that in Lemma 1 is not. Here is an example. Example 1. Consider two Latin hypercube designs with $$n=8$$ runs and $$p=4$$ factors, $$ D_1 = \left( \begin{array}{rrrr} 0.5 & 1.5 & 2.5 & 3.5 \\ 1.5 & -0.5 & -3.5 & 2.5 \\ 2.5 & 3.5 & -0.5 & -1.5 \\ 3.5 & -2.5 & 1.5 & -0.5 \\ -0.5 & -1.5 & -2.5 & -3.5 \\ -1.5 & 0.5 & 3.5 & -2.5 \\ -2.5 & -3.5 & 0.5 & 1.5 \\ -3.5 & 2.5 & -1.5 & 0.5 \end{array} \right), \quad D_2 =\left( \begin{array}{rrrr} 0.5 & 2.5 & 1.5 & 3.5 \\ 3.5 &-0.5 &-2.5 &-1.5 \\ -1.5 &-3.5 & 0.5 & 2.5 \\ 2.5 &-1.5 &-3.5 & 0.5 \\ -0.5 &-2.5 &-1.5 &-3.5 \\ -3.5 & 0.5 & 2.5 & 1.5 \\ 1.5 & 3.5 &-0.5 &-2.5 \\ -2.5 & 1.5 & 3.5 &-0.5 \end{array} \right)\!,$$ where $$D_1$$ is from Sun et al. (2009) and $$D_2$$ is randomly generated. Both designs are mirror-symmetric. The $$L_2$$-distances of $$D_1$$ and $$D_2$$ are $$42$$ and $$7$$, respectively. For Latin hypercube designs with $$n=8$$ and $$p=4$$, the upper bounds in Lemmas 1 and 2 are $$48$$ and $$42$$, respectively. Hence $$D_1$$ attains the upper bound in Lemma 2 and has maximin distance among all mirror-symmetric Latin hypercube designs with $$n=8$$ and $$p=4$$. From the algorithm search result in Table 2(A) of Morris & Mitchell (1995), no Latin hypercube design has distance larger than $$42$$, which suggests that the upper bound in Lemma 1 is not achievable in this case. It is possible that $$D_1$$ has maximin distance among all Latin hypercube designs with $$n=8$$ and $$p=4$$. Consider the designs in Example 1 from the viewpoint of orthogonality. It can be checked that $$D_1$$ is orthogonal but $$D_2$$ is not. In fact $$D_1$$ is a saturated $$3$$-orthogonal Latin hypercube design, so it is optimal in terms of both orthogonality and maximin distance. This does not happen by chance. The following theorem establishes the equivalence between the two criteria for mirror-symmetric U-type designs. Theorem 1. Let $$D$$ be a mirror-symmetric U-type $$(n,s^p)$$ design with $$n=2p$$. (i) If $$D$$ is orthogonal, then $$D$$ has maximin $$L_2$$-distance among all mirror-symmetric U-type $$(n,s^p)$$ designs. (ii) If $$D$$ has maximin $$L_2$$-distance with $$d_2(D)= (s^2-1)p/6$$, then $$D$$ is orthogonal. The condition in Theorem 1(ii), $$d_2(D)=(s^2-1)p/6$$, cannot be relaxed because the value $$(s^2-1)p/6$$ in Lemma 2 may not be an integer for some parameters. If so, even if the distance reaches the upper bound in Lemma 2, the orthogonality of $$D$$ cannot be guaranteed. A counterexample is the following Latin hypercube design with $$n=10$$ and $$p=5$$: $$D_3 = \left( \begin{array}{rrrrrrrrrr} -4.5 & -3.5 & -2.5 & -1.5 & -0.5 & 4.5 & 3.5 & 2.5 & 1.5 & 0.5 \\ 2.5 & -1.5 & -0.5 & -4.5 & 3.5 & -2.5 & 1.5 & 0.5 & 4.5 & -3.5 \\ -1.5 & 0.5 & 4.5 & -3.5 & -2.5 & 1.5 & -0.5 & -4.5 & 3.5 & 2.5 \\ 0.5 & -4.5 & 3.5 & 2.5 & 1.5 & -0.5 & 4.5 & -3.5 & -2.5 & -1.5 \\ -3.5 & 2.5 & 1.5 & 0.5 & 4.5 & 3.5 & -2.5 & -1.5 & -0.5 & -4.5 \end{array} \right)^{ \mathrm{\scriptscriptstyle T} }\!,$$ which is mirror-symmetric with $$L_2$$-distance $$82$$. For $$s=n=10$$ and $$p=5$$, we have $$(s^2-1)p/6=82.5$$. So $$D_3$$ is a maximin distance design among all mirror-symmetric Latin hypercube designs. However, all the off-diagonal entries of $$D_3^{ \mathrm{\scriptscriptstyle T} }D_3$$ are $$\pm 0.5$$, so the design is not orthogonal. Orthogonal mirror-symmetric Latin hypercube designs with $$n=2p$$ runs and $$p$$ factors have been constructed by Sun et al. (2009) for $$p=2^c$$ and any positive integer $$c$$, and by Georgiou & Stylianou (2011) and Sun et al. (2009) for $$p=4$$, $$8$$, $$12$$, $$16$$, $$20$$, $$24$$, and $$32$$. Orthogonal mirror-symmetric U-type designs have been constructed by Bingham et al. (2009), Georgiou et al. (2014a,b) and Stylianou et al. (2015); the Appendix of Georgiou et al. (2014a) provides a collection of designs with run size smaller than $$100$$. All these designs are saturated $$3$$-orthogonal U-type designs. By Theorem 1, all of them have maximin distance among all possible mirror-symmetric designs. Lin et al. (2010) showed that orthogonal Latin hypercube designs with $$n=4m+2$$ do not exist when $$m$$ is an integer. Hence, orthogonal mirror-symmetric Latin hypercube designs with $$n=2p$$ do not exist when $$p$$ is odd. It would be interesting to explore how to construct maximin designs for odd $$p$$. The proof of Theorem 1 indicates that a mirror-symmetric design $$D=(X^{ \mathrm{\scriptscriptstyle T} },-X^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$$ has maximin distance if $$X$$ is row-orthogonal and each row has equal distance from the centre point. Morris & Mitchell (1995) searched maximin Latin hypercube designs using a simulated annealing algorithm and observed that many of the optimal designs obtained share the property that all of the runs have exactly or nearly equal distance from the centre point, especially for those with $$n=p$$ and $$n=2p$$. In addition, nearly all of their $$2p\times p$$ maximin distance designs are mirror-symmetric. Theorem 1 provides a theoretical justification for their findings. From the proof of Theorem 1, the following corollary is obvious, and can be used to obtain larger maximin distance designs from smaller ones. Corollary 1. If $$D$$ is an orthogonal mirror-symmetric U-type $$(n,s^p)$$ design with $$n=2p$$, then $$E=\left(\!\!\begin{array}{rr} D & D\\ D & -D \end{array}\!\!\right)$$ is orthogonal and has maximin distance among all mirror-symmetric U-type $$(2n,s^{2p})$$ designs. Example 2. For the orthogonal Latin hypercube design $$D_1$$ with $$n=8$$ and $$p=4$$ in Example 1, recursively applying Corollary 1 yields a series of maximin distance mirror-symmetric U-type $$(8k,8^{4k})$$ designs for $$k=2,4,8,\ldots$$. In general, starting from an orthogonal mirror-symmetric U-type $$(2p, s^p)$$ design, we obtain a series of maximin distance mirror-symmetric U-type $$(2kp, s^{kp})$$ designs for $$k=2,4,8,\ldots$$. 4. Maximin U-type designs The results in § 3 are obtained under mirror-symmetry. For certain parameters we shall prove an exact equivalence between maximin distance and orthogonality among all possible U-type designs. We first characterize the geometric structure of all maximin distance mirror-symmetric U-type designs with $$n=2p$$ in Theorem 1. A regular cross-polytope in $$\mathbb{R}^{p}$$ is the convex hull of $$p$$ mutually perpendicular line segments of equal length, intersecting at the midpoint of each of them. In other words, a regular cross-polytope has $$2p$$ vertices composing a design $$D = (X^{ \mathrm{\scriptscriptstyle T} },-X^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$$ satisfying $$XX^{ \mathrm{\scriptscriptstyle T} } = X^{ \mathrm{\scriptscriptstyle T} }X = {r^2} I_p$$ for some constant $$r$$. The design $$D$$ is orthogonal and mirror-symmetric, but we emphasize that $$D$$ may not be a U-type or $$s$$-level design. The regular cross-polytope is inscribed in the ball with radius $$r$$, and all its pairwise distances have only two distinct values, $$2r^2$$ and $$4r^2$$. Obviously, the points of any orthogonal mirror-symmetric U-type $$(2p,s^p)$$ design form the vertices of a regular cross-polytope in $$\mathbb{R}^p$$, which indicates that the geometric structure of any maximin $$L_2$$-distance mirror-symmetric U-type $$(2p,s^p)$$ design in Theorem 1 is uniquely determined. We rephrase Theorem 2 of Kuperberg (2007) as a lemma. Lemma 3. If $$n=2p$$ points are placed in the ball of radius $$r$$ in $$\mathbb{R}^p$$, then the maximum attainable minimum pairwise $$L_2$$-distance is $$2r^2$$, and each optimal arrangement must consist of the vertices of a regular cross-polytope inscribed in the ball. Now we can give a theorem showing that for $$s=2,3,4$$, whenever orthogonal mirror-symmetric U-type $$(2p,s^p)$$ designs exist, they are in fact the only maximin distance designs among all possible U-type $$(2p,s^p)$$ designs. Theorem 2. Let $$D$$ be a U-type $$(n,s^p)$$ design with $$n=2p$$ and $$s=2,3$$, or $$4$$. Then $$({\rm i})$$$$d_2(D)\leqslant \lfloor (s^2-1)p/6 \rfloor $$; $$({\rm ii})$$ if $$D$$ is mirror-symmetric and orthogonal, then $$D$$ has maximin $$L_2$$-distance among all possible U-type designs; and $$({\rm iii})$$ if $$D$$ has maximin $$L_2$$-distance with $$d_2(D)= (s^2-1)p/6$$, then $$D$$ is mirror-symmetric and orthogonal. Because any $$p \times p$$ Hadamard matrix $$X$$ is orthogonal, by Theorem 2, $$D=(X^{ \mathrm{\scriptscriptstyle T} },-X^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$$ must have maximin distance among all possible U-type $$(2p,2^p)$$ designs. Butler (2007) showed that when $$n=2p$$, a two-level resolution IV design, regular or nonregular, must be mirror-symmetric. Hence all saturated resolution IV two-level designs have maximin distance among all possible designs. Example 3. Consider the U-type $$(12,3^6)$$ design in Example 3 of Stylianou et al. (2015). It can be checked that the $$L_2$$-distance of the design is $$8$$, which achieves the upper bound $$(s^2-1)p/6$$ in Lemma 2. Since the number of levels is three, from Theorem 2 we can conclude that this design has maximin distance among all possible U-type designs with the same parameters. Theorem 2 also tells us that the design must be mirror-symmetric and orthogonal, that is, all design points form the vertices of a six-dimensional regular cross-polytope. For $$n=2p$$ and $$s\geqslant 5$$, from the constructions in the literature (van Dam et al., 2009; Ba et al., 2015; Xiao & Xu, 2018), no design has been found to have $$L_2$$-distance larger than $${(s^2-1)n}/{6}$$. Thus we conjecture that any orthogonal mirror-symmetric U-type $$(n,s^p)$$ design with $$n=2p$$ and $$s\geqslant 5$$ will have maximin distance among all possible U-type designs. It would be useful to prove this or find a counterexample. Acknowledgement Wang and Yang were supported by the National Natural Science Foundation of China. Yang was also supported by the Tianjin Development Program for Innovation and Entrepreneurship and Tianjin 131 Talents program, and Xu was supported by the U.S. National Science Foundation. Wang is also affiliated with the Center for Statistical Science. The research was carried out when the first two authors were visiting the Department of Statistics at the University of California, Los Angeles. The first two authors contributed equally. The authors thank the editor and reviewers for their helpful comments. Appendix Proof of Lemma 2. Let $$n=2m$$, $$D=(X^{ \mathrm{\scriptscriptstyle T} },-X^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$$ and $$X=( x^{ \mathrm{\scriptscriptstyle T} }_1,\ldots, x^{ \mathrm{\scriptscriptstyle T} }_{m})^{ \mathrm{\scriptscriptstyle T} }$$. To compute the $$L_2$$-distance of the design $$D$$, all distances between any two distinct rows need to be considered. For $$i=1,\ldots,{m}$$, it is easy to see that $$d_2( x_i,- x_i)=4d_2( x_i, 0_p)=4\sum_{{k}=1}^p x_{i{k}}^2$$. Hence, by noting that $$D$$ is a U-type design, the average of all such kinds of distances is \begin{equation} n^{-1}\sum_{i=1}^{m} \{ d_2( x_i,- x_i)+ d_2(- x_i, x_i) \} ={8 n^{-1} \sum_{i=1}^m \sum_{k=1}^{p}}x_{i{k}}^2=(s^2-1)p/{3}\text{.} \end{equation} (A1) Similarly, the average of all other pairwise distances is \begin{align} & \{n(n-2)\}^{-1}\sum_{i=1}^{m}\sum_{j\neq i}2\{d_2( x_i, x_j)+ d_2( x_i,- x_j)\} \nonumber \\ &\quad{} = {4} \{n(n-2)\}^{-1} \sum_{i=1}^{m}\sum_{j\neq i} { \sum_{k=1}^{p} } \left(x_{i{k}}^2+x_{j{k}}^2\right) = (s^2-1)p/{6}\text{.} \end{align} (A2) Because $$d_2(D)$$ is an integer and at most equal to the average distance, the result follows by rounding and comparing (A1) and (A2). □ Proof of Theorem 1. Let $$D=(X^{ \mathrm{\scriptscriptstyle T} }, -X^{ \mathrm{\scriptscriptstyle T} })^{ \mathrm{\scriptscriptstyle T} }$$. (i) Due to the U-type property of $$D$$, the orthogonality of $$D$$ implies that $$X^{ \mathrm{\scriptscriptstyle T} }X = r^2 I_p$$, which is equivalent to $$X X^{ \mathrm{\scriptscriptstyle T} } = r^2 I_p$$, where $$r^2=(s^2-1)p/12$$. This also implies that $$D$$ has only two distinct pairwise distances, $$(s^2-1)p/6$$ and $$(s^2-1)p/3$$. So the $$L_2$$-distance of $$D$$ is $$(s^2-1)p/6$$. By Lemma 2, $$D$$ has maximin distance among all mirror-symmetric U-type $$(n,s^p)$$ designs. (ii) Suppose that the $$L_2$$-distance of $$D$$ is $$(s^2-1)p/6$$. According to the proof of Lemma 2, we have $$d_2( x_i, x_j ) = d_2( x_i, - x_j ) = (s^2-1)p/6$$ for any two distinct rows $$ x_i$$ and $$ x_j$$ in $$X$$. This means that (a) $$d_2( x_i , 0) = (s^2-1)p/12$$ for any row $$ x_i$$ of $$X$$; and (b) $$ x_i x_j^{ \mathrm{\scriptscriptstyle T} }=0$$ for any two distinct rows $$ x_i$$ and $$ x_j$$ of $$X$$. Therefore, we have $$X X^{ \mathrm{\scriptscriptstyle T} } = {r^2} I_p$$, which also implies $$X^{ \mathrm{\scriptscriptstyle T} }X = {r^2} I_p$$, where $$r^2=(s^2-1)p/12$$. This proves that $$D$$ is orthogonal. □ Proof of Theorem 2. We need only show that no U-type design can have distance larger than $$(s^2-1)p/6$$, and the optimal design attaining this bound must comprise the vertices of a regular cross-polytope. For $$s=2$$, the two levels are $$-1/2$$ and $$1/2$$. Therefore, for any run $$ x_i$$ in $$D$$, we have $$d_2( x_i, 0)={r^2}= p/4$$, which means that all two-level designs have points with equal distance from the centre point. By Lemma 3, the maximin distance for two-level designs with $$2p$$ runs and $$p$$ factors is $${2r^2}=p/2$$, which equals $$(s^2-1)p/6$$ for $$s=2$$. Also, if a design attains this bound then it must consist of the vertices of a regular cross-polytope. For $$s=3$$, the levels are $$-1, 0$$ and $$1$$. Given any three-level U-type design, if all runs have equal distance from the centre point, it can be calculated that the radius of the ball that all runs fall on is $${r=(2p/3)^{1/2}}$$. The maximin distance, according to Lemma 3, is $${2r^2}=4p/3$$, which is equal to $$(s^2-1)p/6$$ for $$s=3$$. If not all runs have equal distance from the centre point, then there exists at least one run in $$D$$, without loss of generality say $$ x_1$$, such that $$d_2( x_1, 0)<2p/3$$. This also means that \begin{equation} d_2( x_1, 0)\leqslant {2p}/{3}-1< {2(p-1)}/{3} \end{equation} (A3) since $$n/3 = 2p/3$$ is an integer for a U-type design. Consider the average distance between the run $$ x_1$$ and all other runs $$ x_2,\ldots,x_n$$. Let $$u=(s-1)/2$$. By noting that each of the $$s$$ levels appears exactly $$n/s$$ times in each column of $$D$$, we have $$ (n-1)^{-1}\sum_{i=2}^n \sum_{k=1}^p (x_{ik} - x_{1k})^2 = (n-1)^{-1}\sum_{k =1}^p n s^{-1} \left( \sum_{t=0}^{{u - x_{1k}}} t^2 + \sum_{t=0}^{{u + x_{1k}}} t^2 \right)\!\text{.} $$ Using the formula $$\sum_{t=0}^{q} t^2 = q(q+1)(2q+1)/6$$ and after some algebra, we get \begin{equation} {(n-1)^{-1}\sum_{i=2}^n \sum_{k=1}^p (x_{ik} - x_{1k})^2 = (n-1)^{-1} \{ {np(s^2-1)}/12 + n d_2( x_1, 0) \}\text{.}} \end{equation} (A4) Combining (A3) and (A4) leads to $$ d_2(D)< {(n-1)^{-1} \{ {np(3^2-1)}/12 + 2n(p-1) /3 \} }= {4p}/{3}={(s^2-1)p}/{6} $$ for $$s=3$$, so no U-type design can have $$L_2$$-distance larger than $$(s^2-1)p/6$$ for $$s=3$$. Hence, when the maximin distance is $$(s^2-1)p/6$$, all runs must have equal distance from the centre point and the design points form the vertices of a regular cross-polytope by Lemma 3. For $$s=4$$, the four levels are $$\pm 1/2$$ and $$\pm 3/2$$, which lead to two distinct squared values $$1/4$$ and $$9/4$$. If all runs of $$D$$ have equal distance from the centre point, it can be calculated that the radius of the ball that all runs fall on is $$r=(5p/4)^{1/2}$$. The maximin distance by Lemma 3 is $$2 r^2=5p/2$$, which is equal to $$(s^2-1)p/6$$ for $$s=4$$. If not all runs have equal distance from the centre point, then there exists at least one run, without loss of generality say $$ x_1$$, in $$D$$ such that $$d_2( x_1, 0)<5p/4$$. Because $$|x_{1k}|=1/2$$ or $$3/2$$ for $$k=1,\ldots,p$$, there are at most $$(p/2-1)$$ entries such that $$|x_{1k}|=3/2$$ and at least $$(p/2+1)$$ entries such that $$|x_{1k}|=1/2$$. 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BiometrikaOxford University Press

Published: Feb 28, 2018

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