Design Efficiency of the Asymmetric Minimum Projection Uniform Designs

Abstract

Highly efficient designs and uniform designs are widely applied in many fields because of their good properties. The purpose of this paper is to study the issue of design efficiency for asymmetric minimum projection uniform designs. Based on the centered $L_2$ discrepancy, the uniformity of the designs with mixed levels is defined, which is used to measure the projection uniformity of the designs. The analytical relationship between the uniformity pattern and the design efficiency is established for mixed-level orthogonal arrays with a strength of two. Moreover, a tight lower bound of the uniformity pattern is presented. The research is relevant in the field of experimental design by providing a theoretical basis for constructing the minimum number of projection uniform designs with a high design efficiency under a certain condition. These conclusions are verified by some numerical examples, which illustrate the theoretical results obtained in this paper.

1. Introduction

The uniform designs proposed in [1,2] have been widely used in physical and computer experiments. It requires design points that are uniformly scattered over the experimental domain. Generally, the overall uniformity of the design is often considered, and the projection uniformity of the design in low dimensions is ignored. By the effect sparsity principle, the number of relatively important factors is small in an experiment, so it is necessary to study the projection uniformity of the designs. The authors of [3] first defined the projection discrepancy pattern to measure the projection uniformity of designs based on discrete discrepancy. The authors of [4] proposed the minimum projection uniformity criterion under the centered $L_2$ discrepancy to measure the projection uniformity of designs with two levels and established the relationship between the generalized minimum aberration criterion [5] and the orthogonality criterion [6]. Similar conclusions are obtained for multi-level and mixed-level ones [7,8,9,10]. These theoretical results show that the minimum projection uniformity criterion is equivalent to some other design screening criteria, which provides a theoretical basis for the statistical rationality of the projection uniformity of designs.

According to the maximum estimation capacity of the designs, the design efficiency criterion is proposed, which concerns models included the general mean, i.e., all of the main effects and a selection of two-factor interactions (for more information, one can refer to [11]). Design efficiency criterion are closely associated with minimum aberration or generalized minimum aberration criteria [11,12,13]. The authors of [14] studied the design efficiency of minimum projection uniform designs with two levels, which shows that the minimum projection uniformity criterion is equivalent to the design efficiency criterion under a certain condition. The authors of [15] transformed the designs in [14] into q-level designs.

This paper aims at transforming the designs in [15] into mixed-level designs. The relationship between the uniformity pattern, generalized wordlength pattern and design efficiency is established, and the design efficiency of the minimum projection uniform designs are discussed. This paper is organized as follows: Section 2 presents some basic concepts and notations. Section 3 discusses the design efficiency of mixed-level minimum projection uniform designs. Section 4 provides a tight, lower bound uniformity pattern. Some illustrate examples are presented in Section 5. Section 6 presents some concluding remarks.

2. Notations and Preliminaries

Let $\mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$ be a set of n-run, $s(=s_1+s_2)$-factor U-type designs with $q_p$ levels from $\{0,1,\dots ,q_p-1\},p=1,2$. For any design of $d\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, design d is called an orthogonal array with the strength t if all of the possible level combinations of any t columns in a design d occur an equal number of times, denoted as $OA\left(n;q_1^{s_1}\times q_2^{s_2},t\right)$. The U-type designs are an orthogonal array with a strength of one. A typical treatment combination of a design d is defined by $z=(z^{\left(1\right)},z^{\left(2\right)})$, where $z^{\left(p\right)}=(z_1^{\left(p\right)},\dots ,z_{s_p}^{\left(p\right)})$, $0\le z_j^{\left(p\right)}\le q_p-1$, $1\le j\le s_p$. Let $F^{\left(1\right)},F^{\left(2\right)}$ and F, respectively, be the sets of all the $f^{\left(1\right)}=q_1^{s_1},f^{\left(2\right)}=q_2^{s_2}$ and $f=q_1^{s_1}\times q_2^{s_2}$ treatment combinations that are lexicographically ordered.

2.1. Generalized Minimum Aberration Criterion

For any design of $d\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, $v_1=0,\dots ,s_1$, $v_2=0,\dots ,s_2$, the distance distribution of d is defined by

$$\begin{array}{cc}\hfill C_{v_1{v}_2}\left(d\right)=& \frac{1}{n}\left|\right\{(i,k):d_H(i^{\left(1\right)},k^{\left(1\right)})=v_1,d_H(i^{\left(2\right)},k^{\left(2\right)})=v_2,\hfill \\ \hfill & i=(i^{\left(1\right)},i^{\left(2\right)}),k=(k^{\left(1\right)},k^{\left(2\right)})\mathrm{are}\mathrm{two}\mathrm{rows}\mathrm{of}d\left\}\right|,\hfill \end{array}$$

(1)

where $d_H(i,k)$ is the Hamming distance between the i-th and k-th rows (the number of places where they differ), and $|\left\{(i,k)\right\}|$ is the cardinality of the set $\left\{(i,k)\right\}$. ${\delta }_{ik}=s-d_H(i,k)$ is the coincidence number between two rows i and k.

The MacWilliams transforms of the distance distribution are

$$\begin{array}{c}\hfill A_{j_1{j}_2}\left(d\right)=\frac{1}{n}\sum _{v_1=0}^{s_1}\sum _{v_2=0}^{s_2}{P}_{j_1}(v_1;s_1,q_1)P_{j_2}(v_2;s_2,q_2)C_{v_1{v}_2}\left(d\right),\end{array}$$

(2)

for $0\le j_1\le s_1$ and $0\le j_2\le s_2$, where $P_{j_p}(v_p;s_p,q_p)={\sum }_{r=0}^{j_p}{(-1)}^r{(q_p-1)}^{j_p-r}\left(\genfrac{}{}{0pt}{}{v_p}{j_p}\right)\left(\genfrac{}{}{0pt}{}{s_p-v_p}{j_p-r}\right)$ is the Krawtchouk polynomial, $p=1,2$. For $0\le j\le s_1+s_2$, it is defined as

$$\begin{array}{c}\hfill A_j\left(d\right)=\sum _{j_1+j_2=j}{A}_{j_1{j}_2}\left(d\right),\end{array}$$

(3)

the vector $(A_1\left(d\right),\dots ,A_{s_1+s_2}\left(d\right))$ is called the generalized wordlength pattern (GWLP). For two designs, $d_1$ and $d_2$$\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, let r be the smallest integer that makes $A_r\left(d_1\right)<A_r\left(d_2\right)$, $1\le r\le s_1+s_2$, and $A_j\left(d_1\right)=A_j\left(d_2\right)$ for $j=1,\dots ,r-1$; this shows that design $d_1$ has fewer aberrations than design $d_2$ has. In any design at the same scale, no design has fewer aberrations than $d_1$ has; design $d_1$ has a generalized minimum aberration (GMA) (for more information, one can refer to [5]).

2.2. Orthogonality Criterion

For $z\in F$, let $y_d\left(z\right)$ be number of treatment combinations z in $d\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$. For $z^{\left(1\right)}\in F^{\left(1\right)}$, let $y_d\left(z^{\left(1\right)}\right)$ be a $f^{\left(2\right)}\times 1$ vector with elements $y_d(z^{\left(1\right)},z^{\left(2\right)})$ for all of the elements $z^{\left(2\right)}\in F^{\left(2\right)}$ arranged in lexicographic order. Let $y_d$ be a $f\times 1$ vector with elements $y_d\left(z\right)$ arranged in lexicographic order.

We denote $1_v$ as a $v\times 1$ vector with all of the elements in unity and $I_v$ as a $v\times v$ identity matrix, $J_v=1_v{1}_v^{\prime }$, while the s-fold Kronecker products of $1_v$, $I_v$ and $J_v$ are denoted by $1_v^{\left(s\right)}$, $I_v^{\left(s\right)}$ and $J_v^{\left(s\right)}$, respectively. For $p=1,2$, $E^{\left(p\right)}\left(0\right)=q_p^{-1}{J}_{q_p}$, $E^{\left(p\right)}\left(1\right)=I_{q_p}-q_p^{-1}{J}_{q_p}$, $L^{\left(p\right)}\left(0\right)=1_{q_p}^{\prime },L^{\left(p\right)}\left(1\right)=I_{q_p}$, let ${\mathrm{\Omega }}^{\left(p\right)}=\left\{x^{\left(p\right)}=(x_1^{\left(p\right)},\dots ,x_{s_p}^{\left(p\right)}),x_j^{\left(p\right)}\in \left\{0,1\right\},j=1,\dots ,s_p\right\}$, $W^{\left(p\right)}\left(x^{\left(p\right)}\right)=E^{\left(p\right)}\left(x_1^{\left(p\right)}\right)\otimes \cdots \otimes E^{\left(p\right)}\left(x_{s_p}^{\left(p\right)}\right)$,$H^{\left(p\right)}\left(x^{\left(p\right)}\right)=L^{\left(p\right)}\left(x_1^{\left(p\right)}\right)\otimes \cdots \otimes L^{\left(p\right)}\left(x_{s_p}^{\left(p\right)}\right)$, where ⊗ is the Kronecker product. Let $\mathrm{\Omega }=\left\{x=(x^{\left(1\right)},x^{\left(2\right)}):x^{\left(1\right)}\in {\mathrm{\Omega }}^{\left(1\right)},x^{\left(2\right)}\in {\mathrm{\Omega }}^{\left(2\right)}\right\}$ and the members of $\mathrm{\Omega }$ be lexicographically ordered, and the size of $\mathrm{\Omega }$ is $2^{(s_1+s_2)}$. For $j_1=0,\dots ,s_1,j_2=0,\dots ,s_2$, let ${\mathrm{\Omega }}_{j_1{j}_2}$ be the subset of $\mathrm{\Omega }$ consisting of those binary $(s_1+s_2)$-tuples, which has $j_1$ ones in $x^{\left(1\right)}$ and $j_2$ ones in $x^{\left(2\right)}$. We define the $f\times f$ matrix as $W\left(x\right)=W^{\left(1\right)}\left(x^{\left(1\right)}\right)\otimes W^{\left(2\right)}\left(x^{\left(2\right)}\right)$. For $j_1+j_2=j$, we define it as

$$\begin{array}{cc}\hfill & B_{j_1{j}_2}\left(d\right)=\sum _{x\in {\mathrm{\Omega }}_{j_1{j}_2}}{y}_d^{\prime }W\left(x\right)y_d,0\le j_1\le s_1,0\le j_2\le s_2,(j_1,j_2)\ne (0,0),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & B_j\left(d\right)=\sum _{j_1+j_2=j}{B}_{j_1{j}_2}\left(d\right),j=1,\dots ,s.\hfill \end{array}$$

(5)

In [6], the vector $(B_1\left(d\right),\dots ,B_s\left(d\right))$ is called the B vector. The difference between d and the orthogonal array with strength t can be measured by ${\sum }_{j=1}^t{B}_j\left(d\right)$. The orthogonality criterion is to sequentially minimize $(B_1\left(d\right),\dots ,B_s\left(d\right))$.

2.3. Projection-Centered $L_2$ Discrepancy

For $g=1,\dots ,s$, we define it as $J_g=\left\{(g_1,g_2):g_1=0,\dots ,s_1,g_2=0,\dots ,s_2,g_1+g_2=g\right\}$. For any $(g_1,g_2)\in J_g$, $s=s_1+s_2$, let $I_{g_1{g}_2}=\{(u_1,u_2):u_1\in \{1,\dots ,s_1\},u_2\in \{s_1+1,\dots ,s\},$$|u_1|=g_1,|u_2|=g_2\}$, $I_g={\cup }_{(g_1,g_2)\in J_g}{I}_{g_1{g}_2}$. For any design of $d\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, we define it as $u=\left\{(u_1\cup u_2),\left|u\right|=|u_1|+|u_2|=g\right\}$ and let $d_u$ denote the projection designs of d onto u. In [16], the projection-centered $L_2$ discrepancy of design d onto u is denoted by $C{D}_u\left(d\right)$, whose square value can be computed by

$$\begin{array}{c}\hfill {\left[C{D}_u\left(d\right)\right]}^2={\left(\frac{13}{12}\right)}^{g_1+g_2}-\frac{2}{n}\sum _{i=1}^n\prod _{p=1}^2\prod _{j=1}^{g_p}\alpha \left(x_{ij}^{\left(p\right)}\right)+\frac{1}{n^2}\sum _{i=1}^n\sum _{k=1}^n\prod _{p=1}^2\prod _{j=1}^{g_p}\beta \left(x_{ij}^{\left(p\right)},x_{kj}^{\left(p\right)}\right),\end{array}$$

(6)

where $x_{ij}^{\left(p\right)}=\frac{2z_{ij}^{\left(p\right)}+1}{2q_p}$ for any fixed i and $\alpha \left(x_{ij}^{\left(p\right)}\right)=1+\frac{1}{2}\left|x_{ij}^{\left(p\right)}-\frac{1}{2}\right|-\frac{1}{2}{\left|x_{ij}^{\left(p\right)}-\frac{1}{2}\right|}^2$, $\beta (x_{ij}^{\left(p\right)},x_{kj}^{\left(p\right)})=1+\frac{1}{2}\left|x_{ij}^{\left(p\right)}-\frac{1}{2}\right|+\frac{1}{2}\left|x_{kj}^{\left(p\right)}-\frac{1}{2}\right|-\frac{1}{2}\left|x_{ij}^{\left(p\right)}-x_{kj}^{\left(p\right)}\right|$.

2.4. Design Efficiency Criterion

For any design of $d\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, under the effect sparsity principle, the three- or more factor interactions are ignored, and only the main effects and some two-factor interactions are considered. Let $\mathcal{C}\left(h\right)$ be the collection of all of the sets of h two-factor interactions, $1\le h\le H$, $H=\frac{s(s-1)}{2}$. For $c\in \mathcal{C}\left(h\right)$, $M\left(c\right)$ denotes the model composed of only the general mean, all of the main effects and the h two-factor interactions in c; $X\left(c\right)$ is the model matrix under $M\left(c\right)$. The D-criterion aims at maximizing the determinant of the matrix $\left\{X^{\prime }\left(c\right)X\left(c\right)\right\}$ (i.e., $det\left\{X^{\prime }\left(c\right)X\left(c\right)\right\})$. If one wishes to include h two-factor interactions in the model, but they have no prior knowledge on which h should be included, then it makes sense to consider the average of $det\left\{X^{\prime }\left(c\right)X\left(c\right)\right\}$ over all of $c\in \mathcal{C}\left(h\right)$. However, it is difficult to handle the D-criterion algebraically, and the minimization of trace of ${\left\{X^{\prime }\left(c\right)X\left(c\right)\right\}}^2$ (i.e., $tr\left[{\left\{X{\left(c\right)}^{\prime }X\left(c\right)\right\}}^2\right]$) is a good surrogate for the maximization of $det\left\{X^{\prime }\left(c\right)X\left(c\right)\right\}$. In [11], it is defined as

$$\begin{array}{c}\hfill E_h={\left(\genfrac{}{}{0pt}{}{H}{h}\right)}^{-1}\sum _{c\in \mathcal{C}\left(h\right)}tr\left[{\left\{X{\left(c\right)}^{\prime }X\left(c\right)\right\}}^2\right],\end{array}$$

the design efficiency criterion aims at studying designs that keep $E_h$ small for each h, especially for smaller values of h, which are more relevant under the effect sparsity principle.

3. Design Efficiency of Mixed ${\mathbf{q}}_{\mathbf{1}}$- and ${\mathbf{q}}_{\mathbf{2}}$-Level Minimum Projection Uniform Designs

In this section, the design efficiency of mixed $q_1$- and $q_2$-level minimum projection uniform designs is discussed under the centered $L_2$ discrepancy.

For any design of $d\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, when all of the possible permutations for each factor of a design d are considered, we can obtain ${(q_1!)}^{s_1}\times {(q_2!)}^{s_2}$ combinatorially isomorphic designs, and the set of these designs is denoted as $\mathcal{F}\left(d\right)$. Similarly, we denote $\mathcal{F}\left(d_u\right)$ as the set of the projection designs $d_u$ of ${(q_1!)}^{s_1}\times {(q_2!)}^{s_2}$ combinatorially isomorphic designs. The average-centered $L_2$ discrepancy value of all of the designs in $\mathcal{F}\left(d_u\right)$ is denoted by $\overline{{\left[C{D}_u\left(d\right)\right]}^2}$, which is

$$\begin{array}{c}\hfill \overline{{\left[C{D}_u\left(d\right)\right]}^2}=\frac{1}{{(q_1!)}^{g_1}\times {(q_2!)}^{g_2}}\sum _{d_u^{\prime }\in \mathcal{F}\left(d_u\right)}{\left[C{D}_u\left(d\right)\right]}^2.\end{array}$$

(7)

The relationship between $\overline{{\left[C{D}_u\left(d\right)\right]}^2}$ and the distance distribution $C_{j_1{j}_2}\left(d_u\right)$ is presented in the following lemma.

Lemma 1. 

For any design $d\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, $u\in I_g$, $1\le g(=g_1+g_2)\le s$. Then

$\left(i\right)$ when $q_1$ is odd, $q_2$ is even,

$$\begin{array}{cc}\hfill \overline{{\left[C{D}_u\left(d\right)\right]}^2}=& {\left(\frac{13}{12}\right)}^g-2{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{26q_2^2+1}{24q_2^2}\right)}^{g_2}+\frac{1}{n}{\left(\frac{15q_1^2-3}{12q_1^2}\right)}^{g_1}{\left(\frac{5}{4}\right)}^{g_2}\hfill \\ & \times \sum _{j_1=0}^{g_1}\sum _{j_2=0}^{g_2}{\left(\frac{13q_1^2-2q_1-3}{15q_1^2-3}\right)}^{j_1}{\left(\frac{13q_2-2}{15q_2}\right)}^{j_2}{C}_{j_1{j}_2}\left(d_u\right),\hfill \end{array}$$

(8)

$\left(ii\right)$ when $q_1$ and $q_2$ are odd,

$$\begin{array}{cc}\hfill \overline{{\left[C{D}_u\left(d\right)\right]}^2}=& {\left(\frac{13}{12}\right)}^g-2{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2-1}{12q_2^2}\right)}^{g_2}+\frac{1}{n}{\left(\frac{15q_1^2-3}{12q_1^2}\right)}^{g_1}{\left(\frac{15q_2^2-3}{12q_2^2}\right)}^{g_2}\hfill \\ & \times \sum _{j_1=0}^{g_1}\sum _{j_2=0}^{g_2}{\left(\frac{13q_1^2-2q_1-3}{15q_1^2-3}\right)}^{j_1}{\left(\frac{13q_2^2-2q_2-3}{15q_2^2-3}\right)}^{j_2}{C}_{j_1{j}_2}\left(d_u\right),\hfill \end{array}$$

(9)

$\left(iii\right)$ when $q_1$ and $q_2$ are even,

$$\begin{array}{cc}\hfill \overline{{\left[C{D}_u\left(d\right)\right]}^2}=& {\left(\frac{13}{12}\right)}^g-2{\left(\frac{26q_1^2+1}{24q_1^2}\right)}^{g_1}{\left(\frac{26q_2^2+1}{24q_2^2}\right)}^{g_2}+\frac{1}{n}{\left(\frac{5}{4}\right)}^g\hfill \\ & \times \sum _{j_1=0}^{g_1}\sum _{j_2=0}^{g_2}{\left(\frac{13q_1-2}{15q_1}\right)}^{j_1}{\left(\frac{13q_2-2}{15q_2}\right)}^{j_2}{C}_{j_1{j}_2}\left(d_u\right).\hfill \end{array}$$

(10)

Proof of Lemma 1. 

Similar to the proof of Theorem 3.1 in [17], is found $\left(i\right)$ when $q_1$ is odd and $q_2$ is even, and we have

$$\begin{array}{cc}\hfill \sum _{d_u^{\prime }\in \mathcal{F}\left(d_u\right)}{\left[C{D}_u\left(d\right)\right]}^2=& {(q_1!)}^{g_1}\times {(q_2!)}^{g_2}\left[{\left(\frac{13}{12}\right)}^g-2{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{26q_2^2+1}{24q_2^2}\right)}^{g_2}\right]\hfill \\ \hfill & +\frac{{(q_1!)}^{g_1}\times {(q_2!)}^{g_2}}{n^2}{\left(\frac{13q_1^2-2q_1-3}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2-2}{12q_2}\right)}^{g_2}\sum _{i,k=1}^n\hfill \\ \hfill & \times {\left(\frac{15q_1^2-3}{13q_1^2-2q_1-3}\right)}^{{\delta }_{ik}^{u_1}}{\left(\frac{15q_2}{13q_2-2}\right)}^{{\delta }_{ik}^{u_2}}.\hfill \end{array}$$

Combined with the definitions of ${\delta }_{ij}^u$, $C_{v_1{v}_2}\left(d_u\right)$ and (7), (8) proves $\left(ii\right)$ and $\left(iii\right)$, which are are similar to $\left(i\right)$.   □

When design d is an $OA\left(n;q_1^{s_1}\times q_2^{s_2},t\right)$, for $u\in I_g$ and $1\le g(=g_1+g_2)\le t$, all of the possible $q_1^{g_1}\times q_2^{g_2}$-level combinations of the projection design $d_u$ occur equally, often in any of the g columns. With row $i_u=(i_u^{\left(1\right)},i_u^{\left(2\right)})\in d_u$, it is easy to obtain that $\left|\left\{\left(i_u,k_u\right):d_H^{u_1}\left(i,k\right)=v_1,d_H^{u_2}\left(i,k\right)=v_2,k_u\in d\right\}\right|=\left(\genfrac{}{}{0pt}{}{g_1}{v_1}\right)\left(\genfrac{}{}{0pt}{}{g_2}{v_2}\right)\frac{n{(q_1-1)}^{v_1}{(q_2-1)}^{v_2}}{q_1^{g_1}{q}_2^{g_2}}$.

$\left(i\right)$ when $q_1$ is odd, $q_2$ is even, and the third term of (8) can be expressed as

$$\begin{array}{cc}\hfill & \frac{1}{n}{\left(\frac{15q_1^2-3}{12q_1^2}\right)}^{g_1}{\left(\frac{5}{4}\right)}^{g_2}\sum _{j_1=0}^{g_1}\sum _{j_2=0}^{g_2}{\left(\frac{13q_1^2-2q_1-3}{15q_1^2-3}\right)}^{j_1}{\left(\frac{13q_2-2}{15q_2}\right)}^{j_2}{C}_{j_1{j}_2}\left(d_u\right)\hfill \\ \hfill & =\frac{1}{n}{\left(\frac{15q_1^2-3}{12q_1^2}\right)}^{g_1}{\left(\frac{5}{4}\right)}^{g_2}\sum _{j_1=0}^{g_1}\sum _{j_2=0}^{g_2}\left[{\left(\frac{13q_1^2-2q_1-3}{15q_1^2-3}\right)}^{j_1}{\left(\frac{13q_2-2}{15q_2}\right)}^{j_2}\right.\hfill \\ \hfill & \left.\times \left(\genfrac{}{}{0pt}{}{g_1}{j_1}\right)\left(\genfrac{}{}{0pt}{}{g_2}{j_2}\right)\frac{n{\left(q_1-1\right)}^{j_1}{\left(q_2-1\right)}^{j_2}}{q_1^{g_1}{q}_2^{g_2}}\right]\hfill \\ \hfill & ={\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2},\hfill \end{array}$$

so, (8) can be abbreviated as

$$\begin{array}{c}\hfill \overline{{\left[C{D}_u\left(d\right)\right]}^2}={\left(\frac{13}{12}\right)}^g-2{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{26q_2^2+1}{24q_2^2}\right)}^{g_2}+{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}.\end{array}$$

(11)

Similarly for $\left(ii\right)$, when $q_1$ and $q_2$ are odd, (9) can be abbreviated as

$$\begin{array}{c}\hfill \overline{{\left[C{D}_u\left(d\right)\right]}^2}={\left(\frac{13}{12}\right)}^g-2{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2-1}{12q_2^2}\right)}^{g_2}+{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2-1}{12q_2^2}\right)}^{g_2}.\end{array}$$

(12)

$\left(ii\right)$ when $q_1$ and $q_2$ are even, (10) can be abbreviated as

$$\begin{array}{c}\hfill \overline{{\left[C{D}_u\left(d\right)\right]}^2}={\left(\frac{13}{12}\right)}^g-2{\left(\frac{26q_1^2+1}{24q_1^2}\right)}^{g_1}{\left(\frac{26q_2^2+1}{24q_2^2}\right)}^{g_2}+{\left(\frac{13q_1^2+2}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}.\end{array}$$

(13)

The following definition provides the uniformity pattern of design d under the centered $L_2$ discrepancy.

Definition 1. 

For any design $d\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, $u\in I_g$, $1\le g(=g_1+g_2)\le s$,

$\left(i\right)$ when $q_1$ is odd, $q_2$ is even,

$$\begin{array}{cc}\hfill \overline{M{I}_g\left(d\right)}& =\sum _{u\in I_g}\overline{{\left[C{D}_u\left(d\right)\right]}^2}-\sum _{g_1=0}^g\left[{\left(\frac{13}{12}\right)}^g-2{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{26q_2^2+1}{24q_2^2}\right)}^{g_2}\right.\hfill \\ & \left.+{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}\right],\hfill \end{array}$$

$\left(ii\right)$ when $q_1$ and $q_2$ are odd,

$$\begin{array}{cc}\hfill \overline{M{I}_g\left(d\right)}& =\sum _{u\in I_g}\overline{{\left[C{D}_u\left(d\right)\right]}^2}-\sum _{g_1=0}^g\left[{\left(\frac{13}{12}\right)}^g-2{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2-1}{12q_2^2}\right)}^{g_2}\right.\hfill \\ & \left.+{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2-1}{12q_2^2}\right)}^{g_2}\right],\hfill \end{array}$$

$\left(iii\right)$ while when $q_1$ and $q_2$ are even,

$$\begin{array}{cc}\hfill \overline{M{I}_g\left(d\right)}& =\sum _{u\in I_g}\overline{{\left[C{D}_u\left(d\right)\right]}^2}-\sum _{g_1=0}^g\left[{\left(\frac{13}{12}\right)}^g-2{\left(\frac{26q_1^2+1}{24q_1^2}\right)}^{g_1}{\left(\frac{26q_2^2+1}{24q_2^2}\right)}^{g_2}\right.\hfill \\ & \left.+{\left(\frac{13q_1^2+2}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}\right].\hfill \end{array}$$

The vector $(\overline{M{I}_1\left(d\right)},\dots ,\overline{M{I}_s\left(d\right)})$ is called the uniformity pattern of design d.

According to Definition 1 and (11), (12) and (13), the following theorem can be obtained.

Theorem 1. 

For a design $d\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, if and only if $\overline{M{I}_v\left(d\right)}=0$ for $v=1,\dots ,t$, and $\overline{M{I}_{t+1}\left(d\right)}>0$, design d is an $OA\left(n;q_1^{s_1}\times q_2^{s_2},t\right)$.

Theorem 1 indicates that there is a close relationship between $\overline{M{I}_v\left(d\right)}$ of design d and an orthogonal array with a strength of t, which is to say that the closer $\overline{M{I}_t\left(d\right)}$ is to 0, then the closer the projection design is to the orthogonal arrays with a strength of t. It is shown that when the average-centered $L_2$ discrepancy of the projection designs is small, the orthogonality of the projection designs is also good. From the projection uniformity point of view, the uniformity pattern $\overline{M{I}_v\left(d\right)}$ may be used as a measure used to evaluate and compare the designs.

The definition of the minimum projection uniformity criterion is given below.

Definition 2. 

For two designs $d_1$ and $d_2$ in $\mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, let $\mathcal{R}$ be the smallest integer that makes $\overline{M{I}_{\mathcal{R}}\left(d_1\right)}\ne \overline{M{I}_{\mathcal{R}}\left(d_2\right)}$, and $\overline{M{I}_g\left(d_1\right)}=\overline{M{I}_g\left(d_2\right)}$ for $g=1,\dots ,\mathcal{R}-1$, and then $d_1$ has better projection uniformity than $d_2$ dies if $\overline{M{I}_{\mathcal{R}}\left(d_1\right)}<\overline{M{I}_{\mathcal{R}}\left(d_2\right)}$. In any design of the same scale, no other design = has better projection uniformity than design $d_1$ does; design $d_1$ is called the minimum projection uniform design.

The following theorem builds a relationship between $\overline{M{I}_g\left(d\right)}$ and $A_{j_1{j}_2}\left(d\right)$.

Theorem 2. 

For any design $d\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, $u\in I_g$, $1\le g(=g_1+g_2)\le s$. Then

$\left(i\right)$ when $q_1$ is odd, $q_2$ is even,

$$\begin{array}{cc}\hfill \overline{M{I}_g\left(d\right)}=& \sum _{g_1=0}^g{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}\sum _{(j_1,j_2)\in N}{\left(\frac{2q_1+2}{13q_1^2-1}\right)}^{j_1}{\left(\frac{2q_2+2}{13q_2^2+2}\right)}^{j_2}\hfill \\ & \times \left(\genfrac{}{}{0pt}{}{s_1-j_1}{s_1-g_1}\right)\left(\genfrac{}{}{0pt}{}{s_2-j_2}{s_2-g_2}\right)A_{j_1{j}_2}\left(d\right),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill A_g\left(d\right)=& \sum _{g_1=0}^g{\left(-\frac{13q_1^2-1}{2q_1+2}\right)}^{g_1}{\left(-\frac{13q_2^2+2}{2q_2+2}\right)}^{g_2}\sum _{(j_1,j_2)\in N}{\left(-\frac{12q_1^2}{13q_1^2-1}\right)}^{j_1}{\left(-\frac{12q_2^2}{13q_2^2+2}\right)}^{j_2}\hfill \\ & \times \left(\genfrac{}{}{0pt}{}{s_1-j_1}{s_1-g_1}\right)\left(\genfrac{}{}{0pt}{}{s_2-j_2}{s_2-g_2}\right)\overline{M{I}_{j_1{j}_2}\left(d\right)},\hfill \end{array}$$

(15)

$\left(ii\right)$ when $q_1$ and $q_2$ are odd,

$$\begin{array}{cc}\hfill \overline{M{I}_g\left(d\right)}=& \sum _{g_1=0}^g{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2-1}{12q_2^2}\right)}^{g_2}\sum _{(j_1,j_2)\in N}{\left(\frac{2q_1+2}{13q_1^2-1}\right)}^{j_1}{\left(\frac{2q_2+2}{13q_2^2-1}\right)}^{j_2}\hfill \\ & \times \left(\genfrac{}{}{0pt}{}{s_1-j_1}{s_1-g_1}\right)\left(\genfrac{}{}{0pt}{}{s_2-j_2}{s_2-g_2}\right)A_{j_1{j}_2}\left(d\right),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill A_g\left(d\right)=& \sum _{g_1=0}^g{\left(-\frac{13q_1^2-1}{2q_1+2}\right)}^{g_1}{\left(-\frac{13q_2^2-1}{2q_2+2}\right)}^{g_2}\sum _{(j_1,j_2)\in N}{\left(-\frac{12q_1^2}{13q_1^2-1}\right)}^{j_1}{\left(-\frac{12q_2^2}{13q_2^2-1}\right)}^{j_2}\hfill \\ & \times \left(\genfrac{}{}{0pt}{}{s_1-j_1}{s_1-g_1}\right)\left(\genfrac{}{}{0pt}{}{s_2-j_2}{s_2-g_2}\right)\overline{M{I}_{j_1{j}_2}\left(d\right)},\hfill \end{array}$$

(17)

$\left(iii\right)$ while when $q_1$ and $q_2$ are even,

$$\begin{array}{cc}\hfill \overline{M{I}_g\left(d\right)}=& \sum _{g_1=0}^g{\left(\frac{13q_1^2+2}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}\sum _{(j_1,j_2)\in N}{\left(\frac{2q_1+2}{13q_1^2+2}\right)}^{j_1}{\left(\frac{2q_2+2}{13q_2^2+2}\right)}^{j_2}\hfill \\ & \times \left(\genfrac{}{}{0pt}{}{s_1-j_1}{s_1-g_1}\right)\left(\genfrac{}{}{0pt}{}{s_2-j_2}{s_2-g_2}\right)A_{j_1{j}_2}\left(d\right),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill A_g\left(d\right)=& \sum _{g_1=0}^g{\left(-\frac{13q_1^2+2}{2q_1+2}\right)}^{g_1}{\left(-\frac{13q_2^2+2}{2q_2+2}\right)}^{g_2}\sum _{(j_1,j_2)\in N}{\left(-\frac{12q_1^2}{13q_1^2+2}\right)}^{j_1}{\left(-\frac{12q_2^2}{13q_2^2+2}\right)}^{j_2}\hfill \\ & \times \left(\genfrac{}{}{0pt}{}{s_1-j_1}{s_1-g_1}\right)\left(\genfrac{}{}{0pt}{}{s_2-j_2}{s_2-g_2}\right)\overline{M{I}_{j_1{j}_2}\left(d\right)},\hfill \end{array}$$

(19)

where $N=\left\{(j_1,j_2):j_1=0,\dots ,g_1,j_2=0,\dots ,g_2,(j_1,j_2)\ne (0,0)\right\}$, $\overline{M{I}_j\left(d\right)}=\sum _{j_1+j_2=j}\overline{M{I}_{j_1{j}_2}\left(d\right)}$.

Proof of Theorem 2. 

$\left(i\right)$ when $q_1$ is odd, $q_2$ is even,

$$\begin{array}{cc}\hfill \overline{M{I}_g\left(d\right)}& =\sum _{u\in I_g}\overline{{\left[C{D}_u\left(d\right)\right]}^2}-\sum _{g_1=0}^g\sum _{u\in I_{g_1{g}_2}}\left[{\left(\frac{13}{12}\right)}^g-2{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{26q_2^2+1}{24q_2^2}\right)}^{g_2}\right.\hfill \\ & \left.+{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}\right]\hfill \\ \hfill & =\sum _{g_1=0}^g\sum _{u\in I_{g_1{g}_2}}\left[-{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}+\frac{1}{n}{\left(\frac{15q_1^2-3}{12q_1^2}\right)}^{g_1}{\left(\frac{5}{4}\right)}^{g_2}\sum _{v_1=0}^{g_1}\sum _{v_2=0}^{g_2}\right.\hfill \\ & \left.\times {\left(\frac{13q_1^2-2q_1-3}{15q_1^2-3}\right)}^{v_1}{\left(\frac{13q_2-2}{15q_2}\right)}^{v_2}{C}_{v_1{v}_2}\left(d_u\right)\right]\hfill \end{array}$$

if we combine $C_{v_1{v}_2}\left(d_u\right)=n{\left(\frac{1}{q_1}\right)}^{g_1}{\left(\frac{1}{q_2}\right)}^{g_2}\sum _{j_1=0}^{g_1}\sum _{j_2=0}^{g_2}{P}_{v_1}\left(j_1;g_1,q_1\right)P_{v_2}\left(j_2;g_2,q_2\right)A_{j_1{j}_2}\left(d_u\right)$ with ${\displaystyle \sum _{u\in I_{g_1{g}_2}}}{A}_{j_1{j}_2}\left(d_u\right)=\left(\genfrac{}{}{0pt}{}{s_1-j_1}{s_1-g_1}\right)\left(\genfrac{}{}{0pt}{}{s_2-j_2}{s_2-g_2}\right)A_{j_1{j}_2}\left(d\right)$,

$$\begin{array}{cc}\hfill \overline{M{I}_g\left(d\right)}& =\sum _{g_1=0}^g\sum _{u\in I_{g_1{g}_2}}\left[-{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}+{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}\right.\hfill \\ & \left.\times \sum _{j_1=0}^{g_1}\sum _{j_2=0}^{g_2}{\left(\frac{2q_1+2}{13q_1^2-1}\right)}^{j_1}{\left(\frac{2q_2+2}{13q_2^2+2}\right)}^{j_2}{A}_{j_1{j}_2}\left(d_u\right)\right]\hfill \\ \hfill & =\sum _{g_1=0}^g{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}\sum _{\left(j_1,j_2\in N\right)}{\left(\frac{2q_1+2}{13q_1^2-1}\right)}^{j_1}{\left(\frac{2q_2+2}{13q_2^2+2}\right)}^{j_2}\hfill \\ & \times \left(\genfrac{}{}{0pt}{}{s_1-j_1}{s_1-g_1}\right)\left(\genfrac{}{}{0pt}{}{s_2-j_2}{s_2-g_2}\right)A_{j_1{j}_2}\left(d\right),\hfill \end{array}$$

(14) remains the same and (15) can be obtained using simple algebra from (14) and mathematical induction; the proofs of $\left(ii\right)$ and $\left(iii\right)$ are similar to $\left(i\right)$, so Theorem 2 is proved. □

When $q_1=q_2=q$ in Theorem 2, the following corollary is obtained, which is consistent with the conclusion in [15].

Corollary 1. 

For any design $d\in \mathcal{U}\left(n;q^s\right)$, $u\in I_g$, $1\le g\le s$. Then

when q is odd,

$$\begin{array}{c}\hfill \overline{M{I}_g\left(d\right)}={\left(\frac{13q^2-1}{12q^2}\right)}^g\sum _{i=1}^g{\left(\frac{2q+2}{13q^2-1}\right)}^i\left(\genfrac{}{}{0pt}{}{s-i}{s-g}\right)A_i\left(d\right),\end{array}$$

$$\begin{array}{c}\hfill A_g\left(d\right)={\left(-\frac{13q^2-1}{2q+2}\right)}^g\sum _{i=1}^g{\left(-\frac{12q^2}{13q^2-1}\right)}^i\left(\genfrac{}{}{0pt}{}{s-i}{s-g}\right)\overline{M{I}_i\left(d\right)},\end{array}$$

when q is even,

$$\begin{array}{c}\hfill \overline{M{I}_g\left(d\right)}={\left(\frac{13q^2+2}{12q^2}\right)}^g\sum _{i=1}^g{\left(\frac{2q+2}{13q^2+2}\right)}^i\left(\genfrac{}{}{0pt}{}{s-i}{s-g}\right)A_i\left(d\right),\end{array}$$

$$\begin{array}{c}\hfill A_g\left(d\right)={\left(-\frac{13q^2+2}{2q+2}\right)}^g\sum _{i=1}^g{\left(-\frac{12q^2}{13q^2+2}\right)}^i\left(\genfrac{}{}{0pt}{}{s-i}{s-g}\right)\overline{M{I}_i\left(d\right)}.\end{array}$$

In order to discuss the design efficiency of minimum projection uniform designs, the relationship between $\overline{M{I}_g\left(d\right)}$ and $B_{j_1{j}_2}\left(d\right)$ is firstly presented in the following lemma.

Lemma 2. 

For any design $d\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, $u\in I_g$, $1\le g(=g_1+g_2)\le s$. Then

$\left(i\right)$ when $q_1$ is odd, $q_2$ is even,

$$\begin{array}{cc}\hfill \overline{M{I}_g\left(d\right)}=& \frac{q_1^{s_1}\times q_2^{s_2}}{n^2}\sum _{g_1=0}^g{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}\sum _{(j_1,j_2)\in N}{\left(\frac{2q_1+2}{13q_1^2-1}\right)}^{j_1}\hfill \\ & \times {\left(\frac{2q_2+2}{13q_2^2+2}\right)}^{j_2}\left(\genfrac{}{}{0pt}{}{s_1-j_1}{s_1-g_1}\right)\left(\genfrac{}{}{0pt}{}{s_2-j_2}{s_2-g_2}\right)B_{j_1{j}_2}\left(d\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill B_g\left(d\right)=& \frac{n^2}{q_1^{s_1}\times q_2^{s_2}}\sum _{g_1=0}^g{\left(-\frac{13q_1^2-1}{2q_1+2}\right)}^{g_1}{\left(-\frac{13q_2^2+2}{2q_2+2}\right)}^{g_2}\sum _{(j_1,j_2)\in N}{\left(-\frac{12q_1^2}{13q_1^2-1}\right)}^{j_1}\hfill \\ & \times {\left(-\frac{12q_2^2}{13q_2^2+2}\right)}^{j_2}\left(\genfrac{}{}{0pt}{}{s_1-j_1}{s_1-g_1}\right)\left(\genfrac{}{}{0pt}{}{s_2-j_2}{s_2-g_2}\right)\overline{M{I}_{j_1{j}_2}\left(d\right)},\hfill \end{array}$$

(21)

$\left(ii\right)$ when $q_1$ and $q_2$ are odd,

$$\begin{array}{cc}\hfill \overline{M{I}_g\left(d\right)}=& \frac{q_1^{s_1}\times q_2^{s_2}}{n^2}\sum _{g_1=0}^g{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2-1}{12q_2^2}\right)}^{g_2}\sum _{(j_1,j_2)\in N}{\left(\frac{2q_1+2}{13q_1^2-1}\right)}^{j_1}\hfill \\ & \times {\left(\frac{2q_2+2}{13q_2^2-1}\right)}^{g_2}\left(\genfrac{}{}{0pt}{}{s_1-j_1}{s_1-g_1}\right)\left(\genfrac{}{}{0pt}{}{s_2-j_2}{s_2-g_2}\right)B_{j_1{j}_2}\left(d\right),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill B_g\left(d\right)=& \frac{n^2}{q_1^{s_1}\times q_2^{s_2}}\sum _{g_1=0}^g{\left(-\frac{13q_1^2-1}{2q_1+2}\right)}^{g_1}{\left(-\frac{13q_2^2-1}{2q_2+2}\right)}^{g_2}\sum _{(j_1,j_2)\in N}{\left(-\frac{12q_1^2}{13q_1^2-1}\right)}^{j_1}\hfill \\ & \times {\left(-\frac{12q_2^2}{13q_2^2-1}\right)}^{j_2}\left(\genfrac{}{}{0pt}{}{s_1-j_1}{s_1-g_1}\right)\left(\genfrac{}{}{0pt}{}{s_2-j_2}{s_2-g_2}\right)\overline{M{I}_{j_1{j}_2}\left(d\right)},\hfill \end{array}$$

(23)

$\left(iii\right)$ while when $q_1$ and $q_2$ are even,

$$\begin{array}{cc}\hfill \overline{M{I}_g\left(d\right)}=& \frac{q_1^{s_1}\times q_2^{s_2}}{n^2}\sum _{g_1=0}^g{\left(\frac{13q_1^2+2}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}\sum _{(j_1,j_2)\in N}{\left(\frac{2q_1+2}{13q_1^2+2}\right)}^{j_1}\hfill \\ & \times {\left(\frac{2q_2+2}{13q_2^2+2}\right)}^{g_2}\left(\genfrac{}{}{0pt}{}{s_1-j_1}{s_1-g_1}\right)\left(\genfrac{}{}{0pt}{}{s_2-j_2}{s_2-g_2}\right)B_{j_1{j}_2}\left(d\right),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill B_g\left(d\right)=& \frac{n^2}{q_1^{s_1}\times q_2^{s_2}}\sum _{g_1=0}^g{\left(-\frac{13q_1^2+2}{2q_1+2}\right)}^{g_1}{\left(-\frac{13q_2^2+2}{2q_2+2}\right)}^{g_2}\sum _{(j_1,j_2)\in N}{\left(-\frac{12q_1^2}{13q_1^2+2}\right)}^{j_1}\hfill \\ & \times {\left(-\frac{12q_2^2}{13q_2^2+2}\right)}^{j_2}\left(\genfrac{}{}{0pt}{}{s_1-j_1}{s_1-g_1}\right)\left(\genfrac{}{}{0pt}{}{s_2-j_2}{s_2-g_2}\right)\overline{M{I}_{j_1{j}_2}\left(d\right)}.\hfill \end{array}$$

(25)

Proof of Lemma 2. 

In [18], $B_{j_1{j}_2}\left(d\right)=\frac{n^2}{q_1^{s_1}\times q_2^{s_2}}{A}_{j_1{j}_2}\left(d\right)$, and if we combine it with Theorem 2, Lemma 2 is proved. □

When design d is an $OA\left(n;q_1^{s_1}\times q_2^{s_2},2\right)$, for any $h=1,\dots ,H$, $H=\frac{s(s-1)}{2}$, $1\le j<k<l\le s$, we have the relationships between design efficiency $E_h^{\ast }$ and $B_3\left(d\right),B_4\left(d\right)$ in the B vector and the related quantities $B\left(jkl\right)$ in [11].

$$\begin{array}{c}\hfill E_h^{\ast }=c^{\ast }+6f\left[B_3\left(d\right)+\frac{h-1}{H-1}\left(B_4\left(d\right)-2B_3\left(d\right)+\frac{1}{3}\sum _{jkl\in \mathrm{\Delta }\left(3\right)}\left(q_j+q_k+q_l\right)B\left(jkl\right)\right)\right],\end{array}$$

(26)

where $c^{\ast }$ is a constant that may depend on $n,s,q_1,q_2$ and h. $q_j,q_k$ and $q_l$ are the levels of the jth, kth and lth factors in design d, respectively. $f=q_1^{s_1}\times q_2^{s_2}$, $B_3\left(d\right)={\sum }_{jkl\in \mathrm{\Delta }\left(3\right)}B\left(jkl\right)$, $B\left(jkl\right)=y_d^{\prime }W\left(x\left(jkl\right)\right)y_d$, $x\left(jkl\right)$ is the binary s-tuple that has ones in the jth, kth and lth levels and zeros elsewhere, while $\mathrm{\Delta }\left(3\right)$ is the set of all of the ordered triplets $j,k$ and l.

The following theorem builds a relationship between $E_h^{\ast }$, $B\left(jkl\right)$ and $\overline{M{I}_{j_1{j}_2}\left(d\right)}$, where $j_1+j_2=3,4$.

Theorem 3. 

Let design d be an $OA\left(n;q_1^{s_1}\times q_2^{s_2},2\right)$. Then

$\left(i\right)$ when $q_1$ is odd, $q_2$ is even,

$$\begin{array}{c}\hfill E_h^{\ast }=c^{\ast }+6n^2\left[\left(1-2\frac{h-1}{H-1}\right)z_1+\frac{h-1}{H-1}{z}_2\right]+2f\frac{h-1}{H-1}\sum _{jkl\in \mathrm{\Delta }\left(3\right)}\left(q_j+q_k+q_l\right)B\left(jkl\right),\end{array}$$

(27)

$\left(ii\right)$ when $q_1$ and $q_2$ are odd,

$$\begin{array}{c}\hfill E_h^{\ast }=c^{\ast }+6n^2\left[\left(1-2\frac{h-1}{H-1}\right)z_3+\frac{h-1}{H-1}{z}_4\right]+2f\frac{h-1}{H-1}\sum _{jkl\in \mathrm{\Delta }\left(3\right)}\left(q_j+q_k+q_l\right)B\left(jkl\right),\end{array}$$

(28)

$\left(iii\right)$ while when $q_1$ and $q_2$ are even,

$$\begin{array}{c}\hfill E_h^{\ast }=c^{\ast }+6n^2\left[\left(1-2\frac{h-1}{H-1}\right)z_5+\frac{h-1}{H-1}{z}_6\right]+2f\frac{h-1}{H-1}\sum _{jkl\in \mathrm{\Delta }\left(3\right)}\left(q_j+q_k+q_l\right)B\left(jkl\right),\end{array}$$

(29)

where

$$\begin{array}{cc}\hfill z_1=z_3=z_5=& \frac{216q_2^6}{{(q_2+1)}^3}\overline{M{I}_{03}\left(d\right)}+\frac{216q_1^2{q}_2^4}{(q_1+1){(q_2+1)}^2}\overline{M{I}_{12}\left(d\right)}\hfill \\ & +\frac{216q_1^4{q}_2^2}{{(q_1+1)}^2(q_2+1)}\overline{M{I}_{21}\left(d\right)}+\frac{216q_1^6}{{(q_1+1)}^3}\overline{M{I}_{30}\left(d\right)},\hfill \end{array}$$

$$\begin{array}{cc}\hfill z_2=& \frac{1296q_2^8}{{(q_2+1)}^4}\overline{M{I}_{04}\left(d\right)}+\frac{1296q_1^8}{{(q_1+1)}^4}\overline{M{I}_{40}\left(d\right)}+\frac{1296q_1^2{q}_2^6}{(q_1+1){(q_2+1)}^3}\overline{M{I}_{13}\left(d\right)}+\frac{1296q_1^4{q}_2^4}{{(q_1+1)}^2{(q_2+1)}^2}\hfill \\ \hfill & \times \overline{M{I}_{22}\left(d\right)}+\frac{1296q_1^6{q}_2^2}{{(q_1+1)}^3(q_2+1)}\overline{M{I}_{31}\left(d\right)}-\left(\frac{1404q_2^8+216q_2^6}{{(q_2+1)}^4}(s_2-3)+\frac{1404q_1^2{q}_2^6-108q_2^6}{(q_1+1){(q_2+1)}^3}{s}_1\right)\hfill \\ \hfill & \overline{M{I}_{03}\left(d\right)}-\left(\frac{1404q_1^2{q}_2^6+216q_1^2{q}_2^4}{(q_1+1){(q_2+1)}^3}(s_2-2)+\frac{1404q_1^4{q}_2^4-108q_1^2{q}_2^4}{{(q_1+1)}^2{(q_2+1)}^2}(s_1-1)\right)\overline{M{I}_{12}\left(d\right)}\hfill \\ & -\left(\frac{1404q_1^4{q}_2^4+216q_1^4{q}_2^2}{{(q_1+1)}^2{(q_2+1)}^2}(s_2-1)+\frac{1404q_1^6{q}_2^2-108q_1^4{q}_2^2}{{(q_1+1)}^3(q_2+1)}(s_1-2)\right)\overline{M{I}_{21}\left(d\right)}\hfill \\ \hfill & -\left(\frac{1404q_1^6{q}_2^2+216q_1^6}{{(q_1+1)}^3(q_2+1)}{s}_2+\frac{1404q_1^8-108q_1^6}{{(q_1+1)}^4}(s_1-3)\right)\overline{M{I}_{30}\left(d\right)},\hfill \end{array}$$

$$\begin{array}{cc}\hfill z_4=& \frac{1296q_2^8}{{(q_2+1)}^4}\overline{M{I}_{04}\left(d\right)}+\frac{1296q_1^8}{{(q_1+1)}^4}\overline{M{I}_{40}\left(d\right)}+\frac{1296q_1^2{q}_2^6}{(q_1+1){(q_2+1)}^3}\overline{M{I}_{13}\left(d\right)}+\frac{1296q_1^4{q}_2^4}{{(q_1+1)}^2{(q_2+1)}^2}\hfill \\ \hfill & \times \overline{M{I}_{22}\left(d\right)}+\frac{1296q_1^6{q}_2^2}{{(q_1+1)}^3(q_2+1)}\overline{M{I}_{31}\left(d\right)}-\left(\frac{1404q_2^8-108q_2^6}{{(q_2+1)}^4}(s_2-3)+\frac{1404q_1^2{q}_2^6-108q_2^6}{(q_1+1){(q_2+1)}^3}{s}_1\right)\hfill \\ \hfill & \overline{M{I}_{03}\left(d\right)}-\left(\frac{1404q_1^2{q}_2^6-108q_1^2{q}_2^4}{(q_1+1){(q_2+1)}^3}(s_2-2)+\frac{1404q_1^4{q}_2^4-108q_1^2{q}_2^4}{{(q_1+1)}^2{(q_2+1)}^2}(s_1-1)\right)\overline{M{I}_{12}\left(d\right)}\hfill \\ & -\left(\frac{1404q_1^4{q}_2^4-108q_1^4{q}_2^2}{{(q_1+1)}^2{(q_2+1)}^2}(s_2-1)+\frac{1404q_1^6{q}_2^2-108q_1^4{q}_2^2}{{(q_1+1)}^3(q_2+1)}(s_1-2)\right)\overline{M{I}_{21}\left(d\right)}\hfill \\ \hfill & -\left(\frac{1404q_1^6{q}_2^2-108q_1^6}{{(q_1+1)}^3(q_2+1)}{s}_2+\frac{1404q_1^8-108q_1^6}{{(q_1+1)}^4}(s_1-3)\right)\overline{M{I}_{30}\left(d\right)},\hfill \end{array}$$

$$\begin{array}{cc}\hfill z_6=& \frac{1296q_2^8}{{(q_2+1)}^4}\overline{M{I}_{04}\left(d\right)}+\frac{1296q_1^8}{{(q_1+1)}^4}\overline{M{I}_{40}\left(d\right)}+\frac{1296q_1^2{q}_2^6}{(q_1+1){(q_2+1)}^3}\overline{M{I}_{13}\left(d\right)}+\frac{1296q_1^4{q}_2^4}{{(q_1+1)}^2{(q_2+1)}^2}\hfill \\ \hfill & \times \overline{M{I}_{22}\left(d\right)}+\frac{1296q_1^6{q}_2^2}{{(q_1+1)}^3(q_2+1)}\overline{M{I}_{31}\left(d\right)}-\left(\frac{1404q_2^8+216q_2^6}{{(q_2+1)}^4}(s_2-3)+\frac{1404q_1^2{q}_2^6+216q_2^6}{(q_1+1){(q_2+1)}^3}{s}_1\right)\hfill \\ \hfill & \overline{M{I}_{03}\left(d\right)}-\left(\frac{1404q_1^2{q}_2^6+216q_1^2{q}_2^4}{(q_1+1){(q_2+1)}^3}(s_2-2)+\frac{1404q_1^4{q}_2^4+216q_1^2{q}_2^4}{{(q_1+1)}^2{(q_2+1)}^2}(s_1-1)\right)\overline{M{I}_{12}\left(d\right)}\hfill \\ & -\left(\frac{1404q_1^4{q}_2^4+216q_1^4{q}_2^2}{{(q_1+1)}^2{(q_2+1)}^2}(s_2-1)+\frac{1404q_1^6{q}_2^2+216q_1^4{q}_2^2}{{(q_1+1)}^3(q_2+1)}(s_1-2)\right)\overline{M{I}_{21}\left(d\right)}\hfill \\ \hfill & -\left(\frac{1404q_1^6{q}_2^2+216q_1^6}{{(q_1+1)}^3(q_2+1)}{s}_2+\frac{1404q_1^8+216q_1^6}{{(q_1+1)}^4}(s_1-3)\right)\overline{M{I}_{30}\left(d\right)}.\hfill \end{array}$$

Proof of Theorem 3. 

The proof of Theorem 3 is obtained by combining Lemma 2 and (26).  □

Theorem 3 shows that the design efficiency of an orthogonal array with a strength of two depends on $\overline{M{I}_{j_1{j}_2}\left(d\right)}$, where $j_1+j_2=3,4$. In particular, when design d is an $OA\left(n;q_1^{s_1}\times q_2^{s_2},3\right)$,

$\overline{M{I}_{j_1{j}_2}\left(d\right)}=0$, for $j_1+j_2=3$, we can obtain the following corollary.

Corollary 2. 

Let design d be an $OA\left(n;q_1^{s_1}\times q_2^{s_2},3\right)$. Then

$\left(i\right)$ when $q_1$ is odd, $q_2$ is even,

$$\begin{array}{c}\hfill E_h^{\ast }=c^{\ast }+6n^2\left[\frac{h-1}{H-1}{z}_2^{\ast }\right],\end{array}$$

$\left(ii\right)$ when $q_1$ and $q_2$ are odd,

$$\begin{array}{c}\hfill E_h^{\ast }=c^{\ast }+6n^2\left[\frac{h-1}{H-1}{z}_4^{\ast }\right],\end{array}$$

$\left(iii\right)$ while when $q_1$ and $q_2$ are even,

$$\begin{array}{c}\hfill E_h^{\ast }=c^{\ast }+6n^2\left[\frac{h-1}{H-1}{z}_6^{\ast }\right],\end{array}$$

where $z_2^{\ast }$, $z_4^{\ast }$ and $z_6^{\ast }$ are $z_2$, $z_4$ and $z_6$, which satisfy $\overline{M{I}_{j_1{j}_2}\left(d\right)}=0(j_1+j_2=3)$ in Theorem 3.

Corollary 2 indicates that for an orthogonal array with a strength of three, the MPU criterion is completely equivalent to the design efficiency criterion.

4. A Lower Bound of Uniformity Pattern

This section gives a lower bound of the uniformity pattern in Definition 1; the lower bound provides a basis for measuring the uniformity of the projection designs. Two lemmas are given below, which are important to obtain the lower bound of the uniformity pattern.

Lemma 3. 

For any design $d\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, $u\in I_g$, $1\le g(=g_1+g_2)\le s$, we have

$\left(i\right)$ when $q_1$ is odd, $q_2$ is even,

$$\begin{array}{cc}\hfill & \sum _{i=1}^n\sum _{k(\ne i)=1}^n{\delta }_{ik}^{u_1}=\frac{n(n-q_1)g_1}{q_1},\sum _{i=1}^n\sum _{k(\ne i)=1}^n{\delta }_{ik}^{u_2}=\frac{n(n-q_2)g_2}{q_2},\hfill \\ \hfill & \sum _{i=1}^n\sum _{k(\ne i)=1}^n{\varphi }_{ik}^u=ln\left(\frac{15q_1^2-3}{13q_1^2-2q_1-3}\right)\frac{n(n-q_1)g_1}{q_1}+ln\left(\frac{15q_2}{13q_2-2}\right)\frac{n(n-q_2)g_2}{q_2}\triangleq {▽}_i,\hfill \end{array}$$

$\left(ii\right)$ when $q_1$ and $q_2$ are odd,

$$\begin{array}{cc}\hfill & \sum _{i=1}^n\sum _{k(\ne i)=1}^n{\varphi }_{ik}^u=ln\left(\frac{15q_1^2-3}{13q_1^2-2q_1-3}\right)\frac{n(n-q_1)g_1}{q_1}+ln\left(\frac{15q_2^2-3}{13q_2^2-2q_2-3}\right)\frac{n(n-q_2)g_2}{q_2}\triangleq {▽}_{ii},\hfill \end{array}$$

$\left(iii\right)$ while when $q_1$ and $q_2$ are even,

$$\begin{array}{cc}\hfill & \sum _{i=1}^n\sum _{k(\ne i)=1}^n{\varphi }_{ik}^u=ln\left(\frac{15q_1}{13q_1-2}\right)\frac{n(n-q_1)g_1}{q_1}+ln\left(\frac{15q_2}{13q_2-2}\right)\frac{n(n-q_2)g_2}{q_2}\triangleq {▽}_{iii}.\hfill \end{array}$$

Lemma 3 is obvious, so the proof is omitted.

Lemma 4 ([19]). 

Let $a_1,a_2,\dots ,a_n$, $b_1,b_2,\dots ,b_n$ be two sets of non-negative real numbers and satisfy ${\sum }_{i=1}^n{a}_i=r_1$, ${\sum }_{i=1}^n{b}_i=r_2$. For $i=1,2,\dots ,n$, let ${\mathrm{\Gamma }}_i=A{a}_i+B{b}_i$, and $c^{\prime }=A{r}_1+B{r}_2$, where $A>0,B>0$. Denote ${\mathrm{\Gamma }}_{\left(1\right)},\dots ,{\mathrm{\Gamma }}_{\left(i\right)}$ as the ordered arrangements of the distinct possible values of ${\mathrm{\Gamma }}_1,\dots ,{\mathrm{\Gamma }}_n$, where $1\le i\le n$. For any integer r,

$$\begin{array}{c}\hfill \sum _{i=1}^n{\mathrm{\Gamma }}_i^r\ge P{\mathrm{\Gamma }}_{\left(v\right)}^r+Q{\mathrm{\Gamma }}_{(v+1)}^r,\end{array}$$

where v is the largest integer, such that ${\mathrm{\Gamma }}_{\left(v\right)}\le c^{\prime }/n<{\mathrm{\Gamma }}_{(v+1)}$, P and Q are non-negative integers, such that $P+Q=n$ and $P{\mathrm{\Gamma }}_{\left(v\right)}+Q{\mathrm{\Gamma }}_{(v+1)}=c^{\prime }$.

A lower bound of the uniformity pattern $\overline{M{I}_g\left(d\right)}$ of design d is given in the following lemma.

Lemma 5. 

For any design $d\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, $u\in I_g$, $1\le g(=g_1+g_2)\le s$, we have

$$\overline{M{I}_g\left(d\right)}\ge L{B}_1\left(\overline{M{I}_g\left(d\right)}\right),$$

$\left(i\right)$ when $q_1$ is odd, $q_2$ is even,

$$\begin{array}{cc}\hfill L{B}_1\left(\overline{M{I}_g\left(d\right)}\right)=& \frac{1}{n^2}\sum _{g_1=0}^g\sum _{u\in I_{g_1{g}_2}}{\left(\frac{13q_1^2-2q_1-3}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2-2}{12q_2}\right)}^{g_2}\hfill \\ & \times \left[P_u{e}^{{\varphi }_{\left(m_u\right)}^u}+Q_u{e}^{{\varphi }_{\left(m_u+1\right)}^u}\right]+c_i,\hfill \end{array}$$

(30)

where $m_u$ is the largest integer, such that ${\varphi }_{\left(m_u\right)}^u\le \frac{{▽}_i}{n(n-1)}<{\varphi }_{\left(m_u+1\right)}^u$, $P_u$ and $Q_u$ are non-negative real numbers, such that $P_u+Q_u=n(n-1)$ and $P_u{\varphi }_{\left(m_u\right)}^u+Q_u{\varphi }_{\left(m_u+1\right)}^u={▽}_i$, $c_i={\sum }_{g_1=0}^g\left[\frac{1}{n}{\left(\frac{15q_1^2-3}{12q_1^2}\right)}^{g_1}{\left(\frac{5}{4}\right)}^{g_2}-{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}\right]$;

$\left(ii\right)$ when $q_1$ and $q_2$ are odd,

$$\begin{array}{cc}\hfill L{B}_1\left(\overline{M{I}_g\left(d\right)}\right)=& \frac{1}{n^2}\sum _{g_1=0}^g\sum _{u\in I_{g_1{g}_2}}{\left(\frac{13q_1^2-2q_1-3}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2-2q_2-3}{12q_2^2}\right)}^{g_2}\hfill \\ & \times \left[P_u{e}^{{\varphi }_{\left(m_u\right)}^u}+Q_u{e}^{{\varphi }_{\left(m_u+1\right)}^u}\right]+c_{ii},\hfill \end{array}$$

(31)

where $m_u$ is the largest integer, such that ${\varphi }_{\left(m_u\right)}^u\le \frac{{▽}_{ii}}{n(n-1)}<{\varphi }_{\left(m_u+1\right)}^u$, $P_u$ and $Q_u$ are non-negative real numbers, such that $P_u+Q_u=n(n-1)$ and $P_u{\varphi }_{\left(m_u\right)}^u+Q_u{\varphi }_{\left(m_u+1\right)}^u={▽}_{ii}$, $c_{ii}={\sum }_{g_1=0}^g\left[\frac{1}{n}{\left(\frac{15q_1^2-3}{12q_1^2}\right)}^{g_1}{\left(\frac{15q_2^2-3}{12q_2^2}\right)}^{g_2}-{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2-1}{12q_2^2}\right)}^{g_2}\right]$;

$\left(iii\right)$ when $q_1$ and $q_2$ are even,

$$\begin{array}{cc}\hfill L{B}_1\left(\overline{M{I}_g\left(d\right)}\right)=& \frac{1}{n^2}\sum _{g_1=0}^g\sum _{u\in I_{g_1{g}_2}}{\left(\frac{13q_1-2}{12q_1}\right)}^{g_1}{\left(\frac{13q_2-2}{12q_2}\right)}^{g_2}\hfill \\ & \times \left[P_u{e}^{{\varphi }_{\left(m_u\right)}^u}+Q_u{e}^{{\varphi }_{\left(m_u+1\right)}^u}\right]+c_{iii},\hfill \end{array}$$

(32)

where $m_u$ is the largest integer, such that ${\varphi }_{\left(m_u\right)}^u\le \frac{{▽}_{iii}}{n(n-1)}<{\varphi }_{\left(m_u+1\right)}^u$, $P_u$ and $Q_u$ are non-negative real numbers, such that $P_u+Q_u=n(n-1)$ and $P_u{\varphi }_{\left(m_u\right)}^u+Q_u{\varphi }_{\left(m_u+1\right)}^u={▽}_{iii}$, $c_{iii}={\sum }_{g_1=0}^g\left[\frac{1}{n}{\left(\frac{5}{4}\right)}^g-{\left(\frac{13q_1^2+2}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}\right]$.

Proof of Lemma 5. 

$\left(i\right)$ when $q_1$ is odd, $q_2$ is even, and from Definition 1 and Lemma 3,

$$\begin{array}{cc}\hfill \overline{M{I}_g\left(d\right)}& =\sum _{u\in I_g}\overline{{\left[C{D}_u\left(d\right)\right]}^2}-\sum _{g_1=0}^g\sum _{u\in I_{g_1{g}_2}}\left[{\left(\frac{13}{12}\right)}^g-2{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{26q_2^2+1}{24q_2^2}\right)}^{g_2}\right.\hfill \\ & \left.+{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}\right]\hfill \\ \hfill & =c_i+\frac{1}{n^2}\sum _{g_1=0}^g\sum _{u\in I_{g_1{g}_2}}{\left(\frac{13q_1^2-2q_1-3}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2-2}{12q_2}\right)}^{g_2}\sum _{i=1}^n\sum _{k(\ne i)=1}^n{e}^{{\varphi }_{ik}^u}\hfill \\ \hfill & =c_i+\frac{1}{n^2}\sum _{g_1=0}^g\sum _{u\in I_{g_1{g}_2}}{\left(\frac{13q_1^2-2q_1-3}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2-2}{12q_2}\right)}^{g_2}\sum _{i=1}^n\sum _{k(\ne i)=1}^n\left[\sum _{t=0}^{\infty }\frac{{\left({\varphi }_{ik}^u\right)}^t}{t!}\right],\hfill \end{array}$$

and by Lemma 4,

$$\begin{array}{cc}\hfill \overline{M{I}_g\left(d\right)}& \ge c_i+\frac{1}{n^2}\sum _{g_1=0}^g\sum _{u\in I_{g_1{g}_2}}{\left(\frac{13q_1^2-2q_1-3}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2-2}{12q_2}\right)}^{g_2}\left[P_u{e}^{{\varphi }_{\left(m_u\right)}^u}+Q_u{e}^{{\varphi }_{\left(m_u+1\right)}^u}\right].\hfill \end{array}$$

The proofs of $\left(ii\right)$ and $\left(iii\right)$ are similar to $\left(i\right)$, so Lemma 5 is proved. □

Next, another lower bound of the uniformity pattern $\overline{M{I}_g\left(d\right)}$ is obtained.

Lemma 6. 

For any design $d\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, $u\in I_g$, $1\le g(=g_1+g_2)\le s$, we have

$$\overline{M{I}_g\left(d\right)}\ge L{B}_2\left(\overline{M{I}_g\left(d\right)}\right),$$

$\left(i\right)$ when $q_1$ is odd, $q_2$ is even,

$$\begin{array}{cc}\hfill L{B}_2\left(\overline{M{I}_g\left(d\right)}\right)=& \sum _{g_1=0}^g\left[\frac{1}{n^2}{\left(\frac{13q_1^2-2q_1-3}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2-2}{12q_2}\right)}^{g_2}\sum _{j_1=0}^{g_1}\sum _{j_2=0}^{g_2}\left(\genfrac{}{}{0pt}{}{g_1}{j_1}\right)\left(\genfrac{}{}{0pt}{}{g_2}{j_2}\right)\right.\hfill \\ & \times \left.{\left(\frac{2q_1^2+2q_1}{13q_1^2-2q_1-3}\right)}^{j_1}{\left(\frac{2q_2+2}{13q_2-2}\right)}^{j_2}{\theta }_{j_1{j}_2}-{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}\right],\hfill \end{array}$$

(33)

$\left(ii\right)$ when $q_1$ and $q_2$ are odd,

$$\begin{array}{cc}\hfill L{B}_2\left(\overline{M{I}_g\left(d\right)}\right)=& \sum _{g_1=0}^g\left[\frac{1}{n^2}{\left(\frac{13q_1^2-2q_1-3}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2-2q_2-3}{12q_2^2}\right)}^{g_2}\sum _{j_1=0}^{g_1}\sum _{j_2=0}^{g_2}\left(\genfrac{}{}{0pt}{}{g_1}{j_1}\right)\left(\genfrac{}{}{0pt}{}{g_2}{j_2}\right)\right.\hfill \\ & \times \left.{\left(\frac{2q_1^2+2q_1}{13q_1^2-2q_1-3}\right)}^{j_1}{\left(\frac{2q_2^2+2q_2}{13q_2^2-2q_2-3}\right)}^{j_2}{\theta }_{j_1{j}_2}-{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2-1}{12q_2^2}\right)}^{g_2}\right],\hfill \end{array}$$

(34)

$\left(iii\right)$ while when $q_1$ and $q_2$ are even,

$$\begin{array}{cc}\hfill L{B}_2\left(\overline{M{I}_g\left(d\right)}\right)=& \sum _{g_1=0}^g\left[\frac{1}{n^2}{\left(\frac{13q_1-2}{12q_1}\right)}^{g_1}{\left(\frac{13q_2-2}{12q_2}\right)}^{g_2}\sum _{j_1=0}^{g_1}\sum _{j_2=0}^{g_2}\left(\genfrac{}{}{0pt}{}{g_1}{j_1}\right)\left(\genfrac{}{}{0pt}{}{g_2}{j_2}\right)\right.\hfill \\ & \times \left.{\left(\frac{2q_1+2}{13q_1-2}\right)}^{j_1}{\left(\frac{2q_2+2}{13q_2-2}\right)}^{j_2}{\theta }_{j_1{j}_2}-{\left(\frac{13q_1^2+2}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}\right],\hfill \end{array}$$

(35)

where ${\theta }_{j_1{j}_2}=n{\lambda }_{j_1{j}_2}+{\mu }_{j_1{j}_2}\left(1+{\lambda }_{j_1{j}_2}\right),{\mu }_{j_1{j}_2}=n-q_1^{j_1}{q}_2^{j_2}{\lambda }_{j_1{j}_2},{\lambda }_{j_1{j}_2}=⌊n/\left(q_1^{j_1}{q}_2^{j_2}\right)⌋$ is the largest integer that is less than or equal to $n/\left(q_1^{j_1}{q}_2^{j_2}\right)$.

Proof of Lemma 6. 

$\left(i\right)$ when $q_1$ is odd, $q_2$ is even, let

$$D_0^{\left(1\right)}=\left(\begin{array}{cccc}{X}_{11}& X_{12}& \cdots & X_{1q_1}\\ X_{21}& X_{22}& \cdots & X_{2q_1}\\ \cdots & \cdots & \cdots & \cdots \\ X_{q_{1}1}& X_{q_{1}2}& \cdots & X_{q_1{q}_1}\end{array}\right),D_0^{\left(2\right)}=\left(\begin{array}{cccc}{Y}_{11}& Y_{12}& \cdots & Y_{1q_2}\\ Y_{21}& Y_{22}& \cdots & Y_{2q_2}\\ \cdots & \cdots & \cdots & \cdots \\ Y_{q_{2}1}& Y_{q_{2}2}& \cdots & Y_{q_2{q}_2}\end{array}\right),$$

where the diagonal elements in $D_0^{\left(1\right)}$ and $D_0^{\left(2\right)}$ are $\frac{15q_1^2-3}{12q_1^2}$ and $\frac{5}{4}$ respectively, and the rest of them are $\frac{13q_1^2-2q_1-3}{12q_1^2}$ and $\frac{13q_2-2}{12q_2}$ respectively. We denote $D_{g_1}^{\left(1\right)}={⨂}_{j_1=1}^{g_1}{D}_0^{\left(1\right)},D_{g_2}^{\left(2\right)}={⨂}_{j_2=1}^{g_2}{D}_0^{\left(2\right)}$, where

$$\begin{array}{cc}\hfill & D_0^{\left(1\right)}=\frac{13q_1^2-2q_1-3}{12q_1^2}{L}^{\left(1\right)}{\left(0\right)}^{\prime }{L}^{\left(1\right)}\left(0\right)+\frac{q_1+1}{6q_1}{L}^{\left(1\right)}{\left(1\right)}^{\prime }{L}^{\left(1\right)}\left(1\right),\hfill \\ \hfill & D_0^{\left(2\right)}=\frac{13q_2-2}{12q_2}{L}^{\left(2\right)}{\left(0\right)}^{\prime }{L}^{\left(2\right)}\left(0\right)+\frac{q_2+1}{6q_2}{L}^{\left(2\right)}{\left(1\right)}^{\prime }{L}^{\left(2\right)}\left(1\right),\hfill \end{array}$$

Let $D=D_{g_1}^{\left(1\right)}⨂D_{g_2}^{\left(2\right)}$, so

$$\begin{array}{cc}\hfill D=& {\left(\frac{13q_1^2-2q_1-3}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2-2}{12q_2}\right)}^{g_2}\sum _{x^{\left(1\right)}\in {\mathrm{\Omega }}^{\left(1\right)}}\sum _{x^{\left(2\right)}\in {\mathrm{\Omega }}^{\left(2\right)}}{\left(\frac{2q_1^2+2q_1}{13q_1^2-2q_1-3}\right)}^{\sum x_j^{\left(1\right)}}\hfill \\ \hfill & {\left(\frac{2q_2+2}{13q_2-2}\right)}^{\sum x_j^{\left(2\right)}}{H}^{\prime }\left(x\right)H\left(x\right),\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill y_d^{\prime }D{y}_d=& {\left(\frac{13q_1^2-2q_1-3}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2-2}{12q_2}\right)}^{g_2}\sum _{j_1=0}^{g_1}\sum _{j_2=0}^{g_2}{\left(\frac{2q_1^2+2q_1}{13q_1^2-2q_1-3}\right)}^{j_1}\hfill \\ \hfill & {\left(\frac{2q_2+2}{13q_2-2}\right)}^{j_2}\left(\sum _{x\in {\mathrm{\Omega }}_{j_1{j}_2}}{y}_d^{\prime }{H}^{\prime }\left(x\right)H\left(x\right)y_d\right),\hfill \end{array}$$

(37)

for any ${\sum }_{x\in {\mathrm{\Omega }}_{j_1{j}_2}}$, the elements of $\left(q_1^{j_1}{q}_2^{j_2}\right)\times 1$ vector $H\left(x\right)y_d$ are non-negative integers with a sum of n. So, from [18], we have

$$\begin{array}{cc}\hfill y_d^{\prime }{H}^{\prime }\left(X\right)H\left(x\right)y_d& \ge {\lambda }_{j_1{j}_2}^2\left(q_1^{j_1}{q}_2^{j_2}-{\mu }_{j_1{j}_2}\right)+{\left({\lambda }_{j_1{j}_2}+1\right)}^2{\mu }_{j_1{j}_2}\hfill \\ \hfill & =n{\lambda }_{j_1{j}_2}+{\mu }_{j_1{j}_2}\left({\lambda }_{j_1{j}_2}+1\right).\hfill \end{array}$$

(38)

So

$$\begin{array}{cc}\hfill \overline{M{I}_g\left(d\right)}=& \sum _{g_1=0}^g\sum _{u\in I_{g_1{g}_2}}\left[\frac{1}{n^2}{\left(\frac{13q_1^2-2q_1-3}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2-2}{12q_2}\right)}^{g_2}\sum _{i=1}^n\sum _{k=1}^n{\left(\frac{15q_1^2-3}{13q_1^2-2q_1-3}\right)}^{{\delta }_{ik}^{u_1}}\right.\hfill \\ & \times \left.{\left(\frac{15q_2}{13q_2-2}\right)}^{{\delta }_{ik}^{u_2}}-{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}\right]\hfill \\ \hfill =& \sum _{g_1=0}^g\left[\frac{1}{n^2}{y}_d^{\prime }D{y}_d-{\left(\frac{13q_1^2-1}{12q_1^2}\right)}^{g_1}{\left(\frac{13q_2^2+2}{12q_2^2}\right)}^{g_2}\right],\hfill \end{array}$$

if we combine (37) and (38), $\left(i\right)$ is proved, and the proofs of $\left(ii\right)$ and $\left(iii\right)$ are similar to $\left(i\right)$, so Lemma 6 is proved. □

By combining Lemmas 5 and 6, we can give a more general lower bound of the uniformity pattern $\overline{M{I}_g\left(d\right)}$ for any design $d\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, as follows.

Theorem 4. 

For any design $d\in \mathcal{U}\left(n;q_1^{s_1}\times q_2^{s_2}\right)$, $u\in I_g$, $1\le g(=g_1+g_2)\le s$, we have

$$\overline{M{I}_g\left(d\right)}\ge LB\left(\overline{M{I}_g\left(d\right)}\right),$$

where

$$\begin{array}{c}\hfill LB\left(\overline{M{I}_g\left(d\right)}\right)=max\left\{L{B}_1\left(\overline{M{I}_g\left(d\right)}\right),L{B}_2\left(\overline{M{I}_g\left(d\right)}\right)\right\}.\end{array}$$

Theorem 4 gives a lower bound of the uniformity pattern of a design d. The lower bound can be used to measure the uniformity of the projection designs.

5. Numerical Examples

In this section, some numerical examples are provided to illustrate our theoretical results.

Example 1. 

Consider the two designs $d_1$ and $d_2$, which are $OA\left(18;3^5\times 2,2\right)$ as follows. By calculating the degrees of freedom, when the model contains four or more two-factor interactions, the matrix $X{\left(c\right)}^{\prime }X\left(c\right)$ is singular. Hence, we consider $E_h^{\ast }$, $h=1,2,3$. Using (3), (5), (27) and Definition 1, the GWLP, the B vector, the design efficiency and the uniformity pattern of $d_1$ and $d_2$ are obtained, and specific numerical results are listed in

Table 1. Numerical results of Example 1.

		${\mathit{d}}_1$
$A_g\left(d_1\right)$	0	0	8.50000	12	3	2.50000
$B_g\left(d_1\right)$	0	0	5.66700	8	2	1.66700
$\overline{M{I}_g\left(d_1\right)}$	0	0	0.00401	0.01358	0.01531	0.00574
$E_h^{\ast }$	16,524.97000	19,278.83000	22,032.69000
		${\mathit{d}}_{\mathbf{2}}$
$A_g\left(d_2\right)$	0	0	9	10.50000	4.50000	2
$B_g\left(d_2\right)$	0	0	6	7.50000	3	1.33300
$\overline{M{I}_g\left(d_2\right)}$	0	0	0.00478	0.01587	0.01759	0.00650
$E_h^{\ast }$	17,496	20,122.71000	22,749.43000

.

$$d_1={\left[\begin{array}{cccccccccccccccccc}0\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 1\hfill & 2\hfill & 2\hfill & 2\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 1\hfill & 2\hfill & 2\hfill & 2\hfill \\ 0\hfill & 0\hfill & 1\hfill & 0\hfill & 2\hfill & 2\hfill & 1\hfill & 1\hfill & 2\hfill & 1\hfill & 2\hfill & 2\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 2\hfill \\ 0\hfill & 1\hfill & 0\hfill & 2\hfill & 0\hfill & 2\hfill & 1\hfill & 2\hfill & 1\hfill & 2\hfill & 1\hfill & 2\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 2\hfill & 0\hfill \\ 0\hfill & 1\hfill & 1\hfill & 2\hfill & 2\hfill & 1\hfill & 2\hfill & 0\hfill & 0\hfill & 2\hfill & 2\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 2\hfill & 1\hfill & 1\hfill \\ 0\hfill & 1\hfill & 2\hfill & 2\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 2\hfill & 1\hfill & 0\hfill & 2\hfill & 1\hfill & 1\hfill & 2\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill \end{array}\right]}^{\prime },$$

$$d_2={\left[\begin{array}{cccccccccccccccccc}0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill \\ 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 1\hfill & 2\hfill & 0\hfill & 2\hfill & 0\hfill & 1\hfill & 1\hfill & 2\hfill & 0\hfill & 2\hfill & 0\hfill & 1\hfill \\ 0\hfill & 1\hfill & 2\hfill & 1\hfill & 2\hfill & 0\hfill & 0\hfill & 1\hfill & 2\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 1\hfill & 2\hfill & 0\hfill \\ 0\hfill & 1\hfill & 2\hfill & 1\hfill & 2\hfill & 0\hfill & 2\hfill & 0\hfill & 1\hfill & 1\hfill & 2\hfill & 0\hfill & 0\hfill & 1\hfill & 2\hfill & 2\hfill & 0\hfill & 1\hfill \\ 0\hfill & 1\hfill & 2\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 2\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill \end{array}\right]}^{\prime }.$$

From Table 1, it is shown that $d_1$ is better than $d_2$ in terms of the GMA, orthogonality, design efficiency and MPU criteria.

Example 2. 

Considerthe two designs $d_3$ and $d_4$, which are orthogonal arrays $OA\left(81;9\times 3^4,2\right)$ from http://pietereendebak.nl/oapage/ (accessed on 10 December 2022). Using (3), (5), (28) and Definition 1, the GWLP, the B vector, the design efficiency and the uniformity pattern of $d_3$ and $d_4$ are obtained, and specific numerical results are listed in

Table 2. Numerical results of Example 2.

		${\mathit{d}}_3$
$A_g\left(d_3\right)$	0	0	4	2	2
$B_g\left(d_3\right)$	0	0	36	18	18
$\overline{M{I}_g\left(d_3\right)}$	0	0	0.00104	0.00225	0.00122
$E_h^{\ast }$	157,464	201,204	244,944	288,684
		${\mathit{d}}_{\mathbf{4}}$
$A_g\left(d_4\right)$	0	0	8	0	0
$B_g\left(d_4\right)$	0	0	72	0	0
$\overline{M{I}_g\left(d_4\right)}$	0	0	0.00325	0.00701	0.00378
$E_h^{\ast }$	314,928	349,920	384,912	419,904

.

From Table 2, it is shown that $d_3$ is better than $d_4$ in terms of the GMA, orthogonality, design efficiency and MPU criteria.

Example 3. 

Consider the two designs, $d_5$ and $d_6$, which are $OA\left(16;4\times 2^5,2\right)$ as follows. Using (3), (5), (29) and Definition 1, the GWLP, the B vector, the design efficiency and the uniformity pattern of $d_5$ and $d_6$ are obtained, and specific numerical results are listed in

Table 3. Numerical results of Example 3.

		${\mathit{d}}_5$
$A_g\left(d_5\right)$	0	0	3	3	1	0
$B_g\left(d_5\right)$	0	0	6	6	2	0
$\overline{M{I}_g\left(d_5\right)}$	0	0	0.00358	0.01232	0.01416	0.00542
$E_h^{\ast }$	4608	4864
		${\mathit{d}}_{\mathbf{6}}$
$A_g\left(d_6\right)$	0	0	4	3	0	0
$B_g\left(d_6\right)$	0	0	8	6	0	0
$\overline{M{I}_g\left(d_6\right)}$	0	0	0.00553	0.01900	0.02174	0.00828
$E_h^{\ast }$	6144	6180.57100

.

$$d_5={\left[\begin{array}{cccccccccccccccc}0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 2\hfill & 2\hfill & 2\hfill & 2\hfill & 3\hfill & 3\hfill & 3\hfill & 3\hfill \\ 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill \\ 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill \\ 0\hfill & 1\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill \\ 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill \end{array}\right]}^{\prime },$$

$$d_6={\left[\begin{array}{cccccccccccccccc}0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 2\hfill & 2\hfill & 2\hfill & 2\hfill & 3\hfill & 3\hfill & 3\hfill & 3\hfill \\ 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill \\ 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill \\ 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill \end{array}\right]}^{\prime }.$$

From Table 3, it is shown that $d_5$ is better than $d_6$ in terms of the GMA, orthogonality, design efficiency and MPU criteria.

Example 4. 

Considerthe following designs $d_7\in \mathcal{U}\left(18;3^2\times 6\right)$ and $d_8\in \mathcal{U}\left(20;5\times 2^3\right)$.

$$d_7={\left[\begin{array}{cccccccccccccccccc}0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill \\ 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill \\ 0\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 1\hfill & 2\hfill & 2\hfill & 2\hfill & 3\hfill & 3\hfill & 3\hfill & 4\hfill & 4\hfill & 4\hfill & 5\hfill & 5\hfill & 5\hfill \end{array}\right]}^{\prime },$$

$$d_8={\left[\begin{array}{cccccccccccccccccccc}0\hfill & 0\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 1\hfill & 1\hfill & 2\hfill & 2\hfill & 2\hfill & 2\hfill & 3\hfill & 3\hfill & 3\hfill & 3\hfill & 4\hfill & 4\hfill & 4\hfill & 4\hfill \\ 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill \\ 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill \\ 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 0\hfill & 1\hfill \end{array}\right]}^{\prime }.$$

By Definition 1 and Theorem 4, the uniformity patterns and lower bounds of $d_7$ and $d_8$ are listed in

Table 4. Numerical results of Example 4.

g	1	2	3	4
$\overline{M{I}_g\left(d_7\right)}$	0	0	0.00036
$LB\left(\overline{M{I}_g\left(d_7\right)}\right)$	0	0	0.00036
$\overline{M{I}_g\left(d_8\right)}$	0	0	0	0.00016
$LB\left(\overline{M{I}_g\left(d_8\right)}\right)$	0	0	0	0.00016

.

From Table 4, it is obvious that $\overline{M{I}_g\left(d_7\right)}=LB\left(\overline{M{I}_g\left(d_7\right)}\right)$ for $1\le g\le 3$; $\overline{M{I}_g\left(d_8\right)}=LB\left(\overline{M{I}_g\left(d_8\right)}\right)$, while for $1\le g\le 4$. Designs $d_7$ and $d_8$ are the minimum projection uniform designs under the centered $L_2$ discrepancy.

Example 5. 

Consider the following designs $d_9\in \mathcal{U}\left(9;9\times 3^2\right)$ and $d_{10}\in \mathcal{U}\left(27;9\times 3^2\right)$.

$$d_9={\left[\begin{array}{ccccccccc}0\hfill & 1\hfill & 2\hfill & 3\hfill & 4\hfill & 5\hfill & 6\hfill & 7\hfill & 8\hfill \\ 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill \\ 0\hfill & 1\hfill & 2\hfill & 2\hfill & 0\hfill & 1\hfill & 1\hfill & 2\hfill & 0\hfill \end{array}\right]}^{\prime },$$

$$d_{10}={\left[\begin{array}{ccccccccccccccccccccccccccc}0\hfill & 0\hfill & 0\hfill & 1\hfill & 1\hfill & 1\hfill & 2\hfill & 2\hfill & 2\hfill & 3\hfill & 3\hfill & 3\hfill & 4\hfill & 4\hfill & 4\hfill & 5\hfill & 5\hfill & 5\hfill & 6\hfill & 6\hfill & 6\hfill & 7\hfill & 7\hfill & 7\hfill & 8\hfill & 8\hfill & 8\hfill \\ 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill & 0\hfill & 1\hfill & 2\hfill \\ 0\hfill & 1\hfill & 2\hfill & 1\hfill & 2\hfill & 0\hfill & 2\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 2\hfill & 1\hfill & 2\hfill & 0\hfill & 2\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 2\hfill & 1\hfill & 2\hfill & 0\hfill & 2\hfill & 0\hfill & 1\hfill \end{array}\right]}^{\prime }.$$

By Definition 1 and Theorem 4, the uniformity patterns and their lower bounds of $d_9$ and $d_{10}$ are listed in

Table 5. Numerical results of Example 5.

g	1	2	3
$\overline{M{I}_g\left(d_9\right)}$	0	0.00609663	0.00699984
$LB\left(\overline{M{I}_g\left(d_9\right)}\right)$	0	0.00609663	0.00699984
$\overline{M{I}_g\left(d_{10}\right)}$	0	0	0.00022580
$LB\left(\overline{M{I}_g\left(d_{10}\right)}\right)$	0	0	0.00022580

.

From Table 5, it is obvious that $\overline{M{I}_g\left(d_9\right)}=LB\left(\overline{M{I}_g\left(d_9\right)}\right)$, $\overline{M{I}_g\left(d_{10}\right)}=LB\left(\overline{M{I}_g\left(d_{10}\right)}\right)$ for $1\le g\le 3$. Designs $d_9$ and $d_{10}$ are the minimum projection uniform designs under the centered $L_2$ discrepancy.

Example 6. 

Considerthe following designs $d_{11}\in \mathcal{U}\left(8;2^2\times 4\right)$ and $d_{12}\in \mathcal{U}\left(12;2^2\times 6\right)$.

$$d_{11}={\left[\begin{array}{cccccccc}0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill \\ 0\hfill & 1\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill & 1\hfill & 2\hfill & 2\hfill & 3\hfill & 3\hfill \end{array}\right]}^{\prime },$$

$$d_{12}={\left[\begin{array}{cccccccccccc}0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill \\ 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill & 1\hfill & 2\hfill & 2\hfill & 3\hfill & 3\hfill & 4\hfill & 4\hfill & 5\hfill & 5\hfill \end{array}\right]}^{\prime }.$$

By Definition 1 and Theorem 4, the uniformity patterns and their lower bounds of $d_{11}$ and $d_{12}$ are listed in

Table 6. Numerical results of Example 6.

g	1	2	3
$\overline{M{I}_g\left(d_{11}\right)}$	0	0	0.0008138
$LB\left(\overline{M{I}_g\left(d_{11}\right)}\right)$	0	0	0.0008138
$\overline{M{I}_g\left(d_{12}\right)}$	0	0	0.0005064
$LB\left(\overline{M{I}_g\left(d_{12}\right)}\right)$	0	0	0.0005064

.

Table 6 shows that $\overline{M{I}_g\left(d_{11}\right)}=LB\left(\overline{M{I}_g\left(d_{11}\right)}\right)$, $\overline{M{I}_g\left(d_{12}\right)}=LB\left(\overline{M{I}_g\left(d_{12}\right)}\right)$, $1\le g\le 3$. It shows that designs $d_{11}$ and $d_{12}$ are the minimum projection uniform designs under the centered $L_2$ discrepancy.

6. Conclusions

In this paper, the relationship between the projection uniformity and design efficiency of mixed $q_1$- and $q_2$-level designs is explored under the centered $L_2$ discrepancy. The results show that when the design is an orthogonal array with a strength of three, the projection uniformity is equivalent to the design efficiency. Furthermore, a tight lower bound of the uniformity pattern is also obtained, which can serve as a benchmark for measuring the minimum projection uniform designs.