Construction of nearly orthogonal Latin hypercube designs

Abstract

Latin hypercube designs have found wide application in computer experiments. A number of methods have recently been proposed to construct orthogonal Latin hypercube designs. In this paper, we propose an approach for expanding the orthogonal Latin hypercube design in Sun et al. (Biometrika 96:971–974, 2009) to a nearly orthogonal Latin hypercube design of a larger column size. The newly added part has half number of columns of the original part. It can be shown that the upper bound of the maximum correlation between any two distinct columns of the resulting design is very small. Our method also works for expanding any symmetric Latin hypercube designs.

Appendix: Proofs

1.1 1. Proof of Theorem 1

Denote by $l_j=(t_1,\ldots ,t_{2^{c+1}+1})^\mathrm{{\tiny T}}$ the $j$th column of $L_c$. Denote $h=(h_1,\ldots ,h_{2^{c+1}+1})^\mathrm{{\tiny T}}.$ Condition (a) means

$$\begin{aligned} h^\mathrm{{\tiny T}}h=l_j^\mathrm{{\tiny T}}l_j = 2\times \sum _{i=1}^{2^c} i^2=2^c\left(2^c+1\right)\left(2^{c+1}+1\right)/3. \end{aligned}$$

(9)

From the structure of $L_c$ in (3), it is easy to see that, for $i=1,\ldots ,2^c$, $t_i=-t_{i+2^c+1}$. Thus, condition (b) means

$$\begin{aligned} |h^\mathrm{{\tiny T}}l_j| = |\sum _{i=1}^{2^{c+1}}h_it_i| \le \sum _{i=1}^{2^c}|t_i|= \sum _{i=1}^{2^c}i = 2^{c-1}\left(2^c+1\right)\!. \end{aligned}$$

(10)

Combining (9) and (10) together, we have

$$\begin{aligned} |\rho _j|=\frac{|h^\mathrm{{\tiny T}}l_j|}{\sqrt{h^\mathrm{{\tiny T}}h}\sqrt{l_j^\mathrm{{\tiny T}}l_j}}\le \frac{3}{2^{c+2}+2}. \end{aligned}$$

This completes the proof.

1.2 2. Proof of Proposition 1

From Algorithm 1, it is clear that the column $h$ is a permutation of $\{-2^c,-2^c+1,\ldots ,-1,$ $0,1,\ldots ,2^c-1,2^c\}$. For $k=1,\ldots ,2^c$, note that $h_k$ and $h_{2^c+1+k}$ are two different elements of $\mathcal S _{i_k}$. Thus, $|h_i - h_{2^c+1+i}| =1$, where $k=1,\ldots ,2^c$. This completes the proof.

1.3 3. Proof of Theorem 2

The proof is similar to that of Theorem 1 and is omitted here.

1.4 4. Proof of Theorem 3

In order to prove the theorem, we need the following lemma taken from Sun et al. (2009).

Lemma 2

For the $S_c$ and $T_c$ defined in (2), we have $S_c^\mathrm{{\tiny T}}S_c=\lambda _1I_{2^c}$, $T_c^\mathrm{{\tiny T}}T_c=\lambda _2I_{2^c}$ and $S_c^\mathrm{{\tiny T}}T_c+T_c^\mathrm{{\tiny T}}S_c=\lambda _3I_{2^c}$, where $\lambda _1=2^c$, $\lambda _2=2^c(2^c+1)(2^{c+1}+1)/6$ and $\lambda _3=(2^{2c}+2^c)$.

Proof of Theorem 3

For $j=1,\ldots ,2^{c-1}$, note that the $j$th column of $|T_{c-1}|$, $(|t_{1j}|,\ldots ,|t_{2^cj}|)^\mathrm{{\tiny T}}$, is a permutation of $(1,\ldots ,2^{c-1})$. Part (i) follows by the structure of $E_c$ and $F_c$ in Steps 2 and 3 of Algorithm 3. It is clear that $P_c$ in (6) is a Latin hypercube design from Part (i). Hence, for Part (ii), we only need to prove the orthogonality of $P_c$. We do this by showing the orthogonality of $E_c$ and $F_c$, respectively. Consider the $j$th and $j^{\prime }$th columns of $E_c$ with $j<j^{\prime }$. Note that $j$ and $j^{\prime }$ can belong to one of the three cases: (I) $1\le j<j^{\prime }\le 2^{c-1}/2$; (II) $2^{c-1}/2+1\le j<j^{\prime }\le 2^{c-1}$ and (III) $1\le j \le 2^{c-1}/2$ and $2^{c-1}/2+1\le j^{\prime }\le 2^{c-1}$. For Case (I), we have

$$\begin{aligned} \sum _{i=1}^{2^c} e_{ij}e_{ij^{\prime }} = \sum _{i=1}^{2^{c-1}} \left[s_{ij}(2|t_{ij}|-1)s_{ij^{\prime }}(2|t_{ij^{\prime }}|-1)+4s_{ij}|t_{ij}|s_{ij^{\prime }}|t_{ij^{\prime }}|\right]. \end{aligned}$$

(11)

Since $s_{ij}|t_{ij}|=t_{ij}$, (11) becomes

$$\begin{aligned} \sum _{i=1}^{2^{c-1}} \left[8t_{ij}t_{ij^{\prime }}+s_{ij}s_{ij^{\prime }}-2(s_{ij}t_{ij^{\prime }}+s_{ij^{\prime }}t_{ij})\right]\!. \end{aligned}$$

This together with Lemma 2 shows that (11) is 0. Similarly, we have $\sum _{i=1}^{2^c} e_{ij}e_{ij^{\prime }}=0$ for Cases (II) and (III). This shows the orthogonality of $E_c$. Following a similar discussion, we can show the orthogonality of $F_c$. For Part (iii), let $p_1,p_2$ and $p_3$ be any three columns of $P_c$. According to Algorithm 3, it is clear that for any $i = 1, \ldots ,2^{c-1}$, the sum of the $(2i-1)$th and $(2^{c}+2i+1)$th entries of any column of $P_c$ is equal to zero and so are the $2i$th and $(2^{c}+2i)$th entries. Hence we have that the sum of elementwise product of $p_1, p_2$ and $p_3$, no matter whether they are distinct or not, is equal to zero, which implies the second-order orthogonality of $P_c$. This completes the proof.

1.5 5. Proof of Theorem 4

The proof is similar to that of Theorem 3 and is omitted here.