Level-augmented uniform designs

Abstract

Most of existing augmented designs are to add some runs in the follow-up stages. While in many cases, the level of factors should be augmented and these augmented designs are called level-augmented designs. According to whether the experimental domain is extended or not, they can be divided into range-extended and range-fixed level-augmented designs. For different types of initial designs, the symmetrical and asymmetrical level-augmented designs are discussed, respectively. Based on the property of robustness, a uniformity criterion is a suitable choice to obtain an optimal level-augmented design when the model is unknown. In this paper, the wrap-around $L_2$-discrepancy (WD) is chosen as the uniformity measure. We give the expressions and the tight lower bounds of WD of level-augmented designs under some special parameters. A method to construct a special case of symmetrical level-augmented designs is given. Some examples and level-augmented uniform designs are also provided.

Appendix

In order to prove Theorems 1 and 2, we give the following lemmas first. The proofs of Lemmas 1 and 2 are straightforward and are omitted.

Lemma 1

For any symmetrical RELAD $D_1\in {{\mathcal {L}}}_e (n + n_1 ;3^{m})$ or symmetrical RFLAD $D_1'\in {{\mathcal {L}}}_f (n + n_1 ;3^{m})$, we have

$$\begin{aligned}&(1)\sum \limits _{i=1}^{n}\sum \limits _{j\ne i=1}^{n}\varphi _{ij0}=\frac{m_1 n(n-3)}{3}, \ (2)\sum \limits _{i=1}^{n}\sum \limits _{j\ne i=1}^{n}\lambda _{ij0}=\frac{m_2 n(n-2)}{2}, \\&(3)\sum \limits _{i=n+1}^{n+n_1}\sum \limits _{j\ne i=n+1}^{n+n_1}\varphi '_{ij0}=\frac{m_1 n_1(n_1-3)}{3},\\&(4)\sum \limits _{i=n+1}^{n+n_1}\sum \limits _{j\ne i=n+1}^{n+n_1}\lambda '_{ij0}=\left[ n_{11}(n_{11}-1)+\frac{(n_{12}-3)n_{12}}{3}\right] m_2, \\&(5)\sum \limits _{i=1}^{n}\sum \limits _{j=n+1}^{n+n_1}\nu _{ij0}=\frac{m_1 n n_1}{3}, \ (6)\sum \limits _{i=1}^{n}\sum \limits _{j=n+1}^{n+n_1}\tau _{ij0}=\frac{m_2 n n_{12}}{3}. \end{aligned}$$

Lemma 2

For any asymmetrical RELAD $D_2\in {{\mathcal {L}}}_e (n + n_1; 2^{m_1}3^{m_2})$ or asymmetrical RFLAD $D_2' \in {{\mathcal {L}}}_f (n + n_1; 2^{m_1}3^{m_2})$, we have

$$\begin{aligned}&(1)\sum \limits _{i=1}^{n}\sum \limits _{j\ne i=1}^{n}\varphi _{ij0}=\frac{m_1 n(n-2)}{2}, \ (2)\sum \limits _{i=1}^{n}\sum \limits _{j\ne i=1}^{n}\lambda _{ij0}=\frac{m_2 n(n-2)}{2}, \\&(3)\sum \limits _{i=n+1}^{n+n_1}\sum \limits _{j\ne i=n+1}^{n+n_1}\varphi '_{ij0}=\frac{m_1 n_1(n_1-2)}{2},\\&(4)\sum \limits _{i=n+1}^{n+n_1}\sum \limits _{j\ne i=n+1}^{n+n_1}\lambda '_{ij0}=\left[ n_{11}(n_{11}-1)+\frac{(n_{12}-3)n_{12}}{3}\right] m_2, \\&(5)\sum \limits _{i=1}^{n}\sum \limits _{j=n+1}^{n+n_1}\nu _{ij0}=\frac{m_1 n n_1}{2}, (6)\sum \limits _{i=1}^{n}\sum \limits _{j=n+1}^{n+n_1}\tau _{ij0}=\frac{m_2 n n_{12}}{3}. \end{aligned}$$

Proof of Theorem 1

Based on Lemma 1, similar to the proof of Theorem 2.1 in Fang et al. (2005), we can obtain the result of Theorem 1 easily. A symmetrical RELAD $D_1 \in {{\mathcal {L}}}_e (n + n_1 ;3^{m})$ or a symmetrical RFLAD $D_1'\in {{\mathcal {L}}}_f (n + n_1 ;3^{m})$ is a uniform design under WD, if all its $F_{ij}^{\alpha }$ distributions, $i \ne j$, are the same. In this case, the WD-value of this design achieves the lower bound. $\square$

Proof of Theorem 2

Based on Lemma 2, similar to the proof of Theorem 1, we can obtain the result of Theorem 2. $\square$

Proof of Theorem 3

(1) Denote $\left( ~\mathbf{1}~~ {{\varvec{d}}}~~ {{\varvec{d}}}~~ {{\varvec{d}}}~\right)$ as ${{\varvec{d}}}_{01}$ and $\left( ~\mathbf {2} ~~ {{\varvec{d}}}~~\phi ^+({{\varvec{d}}}) ~~\phi ^-({{\varvec{d}}}) ~\right)$ as ${{\varvec{d}}}_{02}$. According to Fang et al. (2005), since ${{\varvec{d}}}=({{\varvec{d}}}_{ij})_{1\le i\le n_0,1\le j\le m_0}$ is a three-level uniform design, it is also a Hamming-equidistant design and the coincidence number of any two rows in ${{\varvec{d}}}$ is $\lambda _0=m_0(n_0-3)/[3(n_0-1)]$. The coincidence number between two rows is defined as the number of places where two rows take the same value. The condition $n_0=3^{t-1}, t\ge 2,$ ensures that $\lambda _0$ is an integer and the design ${{\varvec{d}}}$ is available. Then, ${{\varvec{d}}}_{01}$ is also a Hamming-equidistant design with coincidence number of any two rows being $3\lambda _0+1$. According to the definitions of $\phi ^+({{\varvec{d}}})$ and $\phi ^-({{\varvec{d}}})$ in Algorithm 1, they are obtained by the permutation of the levels of ${{\varvec{d}}}$. Hence both the coincidence numbers of any two rows in ${{\varvec{d}}}_{02}$ and in ${{\varvec{d}}}_1$ are also $3\lambda _0+1$. For $i=1,\ldots ,n_0$, the ith rows of ${{\varvec{d}}}$, $\phi ^+({{\varvec{d}}})$ and $\phi ^-({{\varvec{d}}})$ are different from each other and thus the coincidence number of any two among the ith rows in ${{\varvec{d}}}_{01}$, ${{\varvec{d}}}_{02}$ and ${{\varvec{d}}}_1$, is $m_0$, the number of the columns of ${{\varvec{d}}}$. The condition $m_0=(n_0-1)/2$ implies that $m_0=3\lambda _0+1$. For $1\le i \ne j \le n_0$, let $N_{ij}^c=\{k~|~{{\varvec{d}}}_{ik}={{\varvec{d}}}_{jk}, k=1,\ldots ,m_0\}$ and $N_{ij}^h=\{1,\ldots ,m_0\}-N_{ij}^c$. For any $k \in N_{ij}^c$ and $1\le i \le n_0$, ${{\varvec{d}}}_{ik}$, $\phi ^+({{\varvec{d}}}_{ik})$ and $\phi ^-({{\varvec{d}}}_{ik})$ become different from each other. However, for any $l \in N_{ij}^h$ and $1\le i \ne j \le n_0$, if ${{\varvec{d}}}_{jl}$ is one more than or two less than ${{\varvec{d}}}_{il}$, $\phi ^+({{\varvec{d}}}_{jl})\ne {{\varvec{d}}}_{il}$ and $\phi ^-({{\varvec{d}}}_{jl})= {{\varvec{d}}}_{il}$; if ${{\varvec{d}}}_{jl}$ is two more than or one less than ${{\varvec{d}}}_{il}$, $\phi ^+({{\varvec{d}}}_{jl})= {{\varvec{d}}}_{il}$ and $\phi ^-({{\varvec{d}}}_{jl})\ne {{\varvec{d}}}_{il}$. Hence the coincidence number of the ith row in ${{\varvec{d}}}_{01}$ and the jth row in ${{\varvec{d}}}_{02}$ is also $m_0(=3\lambda _0+1)$ for $1\le i \ne j \le n_0$. By similar arguments, we can obtain the same results for ${{\varvec{d}}}_{01}$ and ${{\varvec{d}}}_{1}$, as well as ${{\varvec{d}}}_{02}$ and ${{\varvec{d}}}_{1}$. Therefore, the resulting three-level level-augmented design $D_1$ is a Hamming-equidistant design and hence the lower bound given in Theorem 1 is reachable.

(2) Similar to the proof of (1), since ${{\varvec{d}}}$ is a uniform design in ${{\mathcal {U}}}(n_0; 3^{n_0-1})$, it is also a Hamming-equidistant design and $\lambda _0=(n_0-3)/3$. Since ${{\varvec{d}}}$ is a U-type design, $n_0$ must be a multiple of 3 which ensures that $\lambda _0$ is an integer. The coincidence numbers of any two rows in ${{\varvec{d}}}_{01}$, ${{\varvec{d}}}_{02}$ and ${{\varvec{d}}}_1$ are $n_0-2(=3\lambda _0+1)$. For $1\le i \le n_0$, the coincidence number of any two among the ith rows in ${{\varvec{d}}}_{01}$, ${{\varvec{d}}}_{02}$ and ${{\varvec{d}}}_1$, is $n_0-1$, the number of the columns of ${{\varvec{d}}}$. For $1\le i \ne j \le n_0$, the coincidence number of the ith row in ${{\varvec{d}}}_{01}$ and the jth row in ${{\varvec{d}}}_{02}$ is $n_0-1$, which is also true for ${{\varvec{d}}}_{01}$ and ${{\varvec{d}}}_{1}$, ${{\varvec{d}}}_{02}$ and ${{\varvec{d}}}_{1}$. Hence for $D_1$, the difference of the Hamming distances between its rows is not more than one. For both (1) and (2), the corresponding arguments for $D_1'$ are similar to the case of $D_1$ and we omit it. $\square$