Projection Uniformity of Asymmetric Fractional Factorials

Abstract

The objective of this paper is to study the issue of the projection uniformity of asymmetric fractional factorials. On the basis of level permutation and mixture discrepancy, the average projection mixture discrepancy to measure the uniformity for low-dimensional projection designs is defined, the uniformity pattern and minimum projection uniformity criterion are presented for evaluating and comparing any asymmetric factorials. Moreover, lower bounds to uniformity pattern have been obtained, and some illustrative examples are also provided.

1. Introduction

Many criteria were proposed for comparing U-type designs, but none of these criteria can directly distinguish non-isomorphic saturated designs. A special criterion can measure all these subdesigns, and the related values are called its projection pattern. We can use the distribution or the vector of these projection values as a tool to distinguish the underlying designs. Ref. [1] firstly defined the projection discrepancy pattern and proposed the minimum projection uniformity (MPU) criterion, which is equivalent to generalized minimum aberration criterion (GMA [2]). Ref. [3] studied the projection discrepancies of two-level fractional factorials in terms of the centered $L_2$-discrepancy (CD [4]). Subsequently, ref. [5] discussed the relationships among criteria of MPU proposed in [1] and minimum generalized aberration [6]. Following this projection discrepancy, [7] studied the projection properties of two-level factorials in view of geometry and proposed the uniformity pattern and MPU criterion to assess and compare two-level factorials. The relations between MPU and minimum aberration, and GMA and orthogonality are clarified; this close relationship raises the hope of improving the connection between uniform design theory and factorial design theory.

Following the uniform pattern and MPU, projection uniformity of asymmetric design based on CD and wrap-around $L_2$-discrepancy (WD [8]) has been studied, respectively. As a measure of uniformity, CD does not have fewer cursed dimensions and WD is not sensitive to a shift for one or more dimensions, Mixture discrepancy (MD [9]) retains the good properties of CD and WD and overcomes the shortcomings of both. Aided by the level permutation technique in [10,11], ref. [12] obtained the relationship between the mean of mixture discrepancies and the generalized word–length pattern for multi-level designs. Ref. [13] defined the MPU criterion for two- and three-level factorials under MD. Refs. [14,15] generalize the findings in [13] to q-level and mixed two- and three-level factorials, respectively. Moreover, ref. [16] proposed the uniform projection design that have the smallest average CD values of all two-dimensional projections and are shown to have good-filling properties over all sub-spaces in terms of the distance, uniformity, and orthogonality. Based on the findings of [16], many applications and studies on uniform projection designs have emerged [17,18,19,20,21,22].

While the work of [13,14,15] discussed the projection uniformity for two-level, three-level, q-level, and mixed two- and three-level designs under MD, respectively, the present paper aims at obtaining further results. We extend the findings in [13,14,15] to general asymmetrical factorials. First, the uniformity pattern and MPU criterion are proposed for selecting asymmetrical designs. Second, we build some analytic linkages between uniformity pattern, orthogonality, and generalized word–length pattern. Third, we integrate two lower bound methods in [23], which can be served as a benchmark for searching MPU designs. Finally, the results of [13,14,15] can be used as our special cases, and some numerical examples are provided to illustrate our theoretical results.

This paper is organized as follows: Section 2 describes some notations and basic concepts such as distance distribution and generalized word–length pattern, which are useful throughout in this paper. Section 3 defines the average projection mixture discrepancy and related uniformity pattern, presents a statistical justification of MPU criterion, and establishes a connection between MPU and GMA. Section 4 provides a lower bound of the uniformity pattern. Some illustrative examples to verify our theoretical results are presented in Section 5.

2. Notations and Preliminaries

Consider a class of U-type designs, denoted by $\mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$, of mixed $q_1$- and $q_2$-level factorials in n runs and $s(=s_1+s_2)$ factors, where each factor of the first $s_1$ factors takes values from a set of $\{0,1,\dots ,q_1-1\}$ equally often and each factor of the last $s_2$ factors takes values from a set of $\{0,1,\dots ,q_2-1\}$ equally often. For any design $d\in \mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$, a typical treatment combination (or run) of design d is defined by $w=(w^{\left(1\right)},w^{\left(2\right)})$, where, for $i=1,2$, $w^{\left(i\right)}=(w_1^{\left(i\right)},\dots ,w_{s_i}^{\left(i\right)})$, $w_j^{\left(1\right)}\in \{0,1,\dots ,q_1-1\}$ and $w_j^{\left(2\right)}\in \{0,1,\dots ,q_2-1\}$. Denote $d=(d^{\left(1\right)},d^{\left(2\right)})$, where $w^{\left(i\right)}\in d^{\left(i\right)}$, $i=1,2$. If all the possible $q_1^{t_1}\times q_2^{t_2}$ level combinations corresponding to any $t(=t_1+t_2)$ columns of design d appear equally often, $0\le t_1\le s_1$, $0\le t_2\le s_2$, design d is called to be an orthogonal array of strength t and denoted by $OA(n;{q_1}^{s_1}\times {q_2}^{s_2},t)$.

For any design $d\in \mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$, its distance distribution is defined by

$$\begin{array}{c}\hfill E_{j_1{j}_2}\left(d\right)=\frac{1}{n}\left|\{(i,k):H_{i_1{k}_1}^{\left(1\right)}=j_1,H_{i_2{k}_2}^{\left(2\right)}=j_2\}\right|,\end{array}$$

where $\left|u\right|$ is the cardinality of the set $\left|u\right|$, $H_{ik}^t$ is the Hamming distance between two runs i and k of design $d^{\left(t\right)}$, $t=1,2$, $0\le j_1\le s_1$, $0\le j_2\le s_2$.

The MacWilliams transforms of the $\left\{E_{j_1{j}_2}\left(d\right)\right\}$ of any design $d\in \mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$ are defined as

$$\begin{array}{c}\hfill E_{i_1{i}_2}^{{}^{\prime }}\left(d\right)=\frac{1}{n}\sum _{j_1=0}^{s_1}\sum _{j_2=0}^{s_2}{P}_{i_1}(j_1;s_1,q_1)P_{i_2}(j_2;s_2,q_2)E_{j_1{j}_2}\left(d\right),i_1=0,\dots ,s_1,i_2=0,\dots ,s_2,\end{array}$$

where $P_i(j;s,q)={\sum }_{r=0}^i{(-1)}^r{(q-1)}^{i-r}\left(\genfrac{}{}{0pt}{}{j}{r}\right)\left(\genfrac{}{}{0pt}{}{s-j}{i-r}\right)$ is the Krawtchouk polynomial, $\left(\genfrac{}{}{0pt}{}{m}{k}\right)=m(m-1)···(m-k+1)/k!$ and $\left(\genfrac{}{}{0pt}{}{m}{k}\right)=0$ for $m<k$.

Ref. [2] showed that the generalized word–length pattern is the MacWilliams transform of the distance distribution, that is,

$$\begin{array}{c}\hfill A_i\left(d\right)=\sum _{i_1+i_2=i}{E}_{i_1{i}_2}^{{}^{\prime }}\left(d\right),\end{array}$$

(1)

where the vector $(A_1\left(d\right),\dots ,A_s\left(d\right))$ is called the generalized word–length pattern. For any two designs $d_1$ and $d_2$ in $\mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$, $d_1$ is said to have less aberration than $d_2$ if there exists a positive integer $t\le s$, such that $A_t\left(d_1\right)<A_t\left(d_2\right)$ and $A_i\left(d_1\right)=A_i\left(d_2\right)$ for $i=1,\dots ,t-1$. The design $d_1$ has generalized minimum aberration if there is no other design with less aberration than $d_1$.

For any positive integer $g\le s$, defined $C_g=\{(g_1,g_2):g_1=0,\dots ,s_1,$$g_2=0,\dots ,s_2,g_1+g_2=g\}$, and for any $(g_1,g_2)\in C_g$, let $S_{g_1{g}_2}$ be the set of all nonempty subsets of $\{1,\dots ,s\}$ with the first $g_1$ elements from $\{1,2,\dots ,s_1\}$ and the next $g_2$ elements from $\{s_1+1,\dots ,s_1+s_2\}$. For any g, $1\le g\le s$, let $S_g$ be the set of all nonempty subsets of $\{1,2,\dots ,s\}$ with cardinality g, it is to be noted that $S_g={\displaystyle \bigcup _{(g_1,g_2)\in C_g}}{S}_{g_1{g}_2}$.

For any design $d\in \mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$, define the nonempty set $u=u_1\cup u_2=\{u_{11},\dots ,u_{1g_1}\}\cup \{u_{21},\dots ,u_{2g_2}\}\in S_{g_1{g}_2}$ and $g=g_1+g_2$, let $d_u$ be the corresponding projection design of d onto factors with indexes from u. A typical treatment combination of $d_u$ is represented as $w_u=(w_u^{\left(1\right)},w_u^{\left(2\right)})$, where $w_u^{\left(i\right)}=(w_{u_{i1}}^{\left(i\right)},\dots ,w_{u_{i{g}_i}}^{\left(i\right)})$, $w_{u_{i{g}_i}}^{\left(i\right)}\in \{0,1,\dots ,q_i-1\}$, $i=1,2$. Let $H_{ik}^u$ be the Hamming distance between two runs $i^u$ and $k^u$ of the projection design $d_u$, denote ${\delta }_{ik}^u=g-H_{ik}^u$ as the coincide number between two runs $i^u$ and $k^u$, where $i^u=(i_1^u,i_2^u)$ and $k^u=(k_1^u,k_2^u)$.

3. Projection Uniformity of $\mathcal{U}(\mathit{n};{{\mathit{q}}_{\mathbf{1}}}^{{\mathit{s}}_{\mathbf{1}}}\times {{\mathit{q}}_{\mathbf{2}}}^{{\mathit{s}}_{\mathbf{2}}})$

For any design $d\in \mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$, $g(=g_1+g_2)\le s$ and $u\in S_g$, let $M{D}_u\left(d\right)$ be the mixture discrepancy value of the corresponding projection design $d_u$; following [9], we can derive the below formula for $M{D}_u\left(d\right)$,

$$\begin{array}{c}\hfill {\left[M{D}_u\left(d\right)\right]}^2={\left(\frac{7}{12}\right)}^g-\frac{2}{n}\sum _{i=1}^n\prod _{j\in u}{f}_1\left(x_{ij}\right)+\frac{1}{n^2}\sum _{i=1}^n\sum _{k=1}^n\prod _{j\in u}f(x_{ij},x_{kj}),\end{array}$$

(2)

where $f_1\left(x_{ij}\right)=\frac{2}{3}-\frac{1}{4}|x_{ij}-\frac{1}{2}|-\frac{1}{4}{|x_{ij}-\frac{1}{2}|}^2$, $f(x_{ij},x_{kj})=\frac{7}{8}-\frac{1}{4}|x_{ij}-\frac{1}{2}|-\frac{1}{4}|x_{kj}-\frac{1}{2}|-\frac{3}{4}|x_{ij}-x_{kj}|+\frac{1}{2}{|x_{ij}-x_{kj}|}^2$, i, $k=1,\dots ,n$.

When considering all $q_1!\times q_2!$ possible level permutations for every factor of $d\in \mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$, there are ${(q_1!)}^{s_1}\times {(q_2!)}^{s_2}$ combinatorially isomorphic designs of d that can be obtained, and denote the set of these designs as $\mathcal{P}\left(d\right)$. Similarly, for any positive integer $g(=g_1+g_2)\le s$ and $u\in S_g$, we can obtain ${(q_1!)}^{g_1}\times {(q_2!)}^{g_2}$ combinatorially isomorphic designs of $d_u$; the corresponding set of these combinatorially isomorphic designs $d_u^{\prime }$ is denoted by $\mathcal{P}\left(d_u\right)$. The mean of projection mixture discrepancies of all the designs in $\mathcal{P}\left(d_u\right)$ is denoted by $AM{D}_u\left(d\right)$, that is,

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

(3)

The following lemma, which can be proved similarly as [14,15], gives the expression for $AM{D}_u\left(d\right)$.

Lemma 1. 

For any design $d\in \mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$, $u\in S_g$ and $1\le g\le s$,

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

$$\begin{array}{cc}\hfill AM{D}_u\left(d\right)=& {\left(\frac{7}{12}\right)}^g-2{\left(\frac{28{q_1}^2+1}{48{q_1}^2}\right)}^{g_1}{\left(\frac{28{q_2}^2+1}{48{q_2}^2}\right)}^{g_2}\hfill \\ & +\frac{1}{n}{\left(\frac{3}{4}\right)}^{g_1}{\left(\frac{3}{4}\right)}^{g_2}\sum _{i_1=0}^{g_1}\sum _{i_2=0}^{g_2}{\left(\frac{7q_1-2}{9q_1}\right)}^{i_1}{\left(\frac{7q_2-2}{9q_2}\right)}^{i_2}{E}_{i_1{i}_2}\left(d_u\right);\hfill \end{array}$$

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

$$\begin{array}{cc}\hfill AM{D}_u\left(d\right)=& {\left(\frac{7}{12}\right)}^g-2{\left(\frac{7{q_1}^2+1}{12{q_1}^2}\right)}^{g_1}{\left(\frac{7{q_2}^2+1}{12{q_2}^2}\right)}^{g_2}+\frac{1}{n}{\left(\frac{6{q_1}^2+1}{8{q_1}^2}\right)}^{g_1}{\left(\frac{6{q_2}^2+1}{8{q_2}^2}\right)}^{g_2}\hfill \\ & \times \sum _{i_1=0}^{g_1}\sum _{i_2=0}^{g_2}{\left(\frac{14{q_1}^2-4q_1+3}{18{q_1}^2+3}\right)}^{i_1}{\left(\frac{14{q_2}^2-4q_2+3}{18{q_2}^2+3}\right)}^{i_2}{E}_{i_1{i}_2}\left(d_u\right);\hfill \end{array}$$

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

$$\begin{array}{cc}\hfill AM{D}_u\left(d\right)=& {\left(\frac{7}{12}\right)}^g-2{\left(\frac{28{q_1}^2+1}{48{q_1}^2}\right)}^{g_1}{\left(\frac{7{q_2}^2+1}{12{q_2}^2}\right)}^{g_2}+\frac{1}{n}{\left(\frac{3}{4}\right)}^{g_1}{\left(\frac{6{q_2}^2+1}{8{q_2}^2}\right)}^{g_2}\hfill \\ \hfill & \times \sum _{i_1=0}^{g_1}\sum _{i_2=0}^{g_2}{\left(\frac{7q_1-2}{9q_1}\right)}^{i_1}{\left(\frac{14{q_2}^2-4q_2+3}{18{q_2}^2+3}\right)}^{i_2}{E}_{i_1{i}_2}\left(d_u\right).\hfill \end{array}$$

(4)

We can obtain the following lemma when the design d is an orthogonal array $OA(n;{q_1}^{s_1}\times {q_2}^{s_2},t)$.

Lemma 2. 

Suppose design d is an orthogonal array $OA(n;{q_1}^{s_1}\times {q_2}^{s_2},t)$, then

$$\begin{array}{c}\hfill AM{D}_u\left(d\right)={\mathsf{\Phi }}_u,\end{array}$$

where $\left|u\right|=g_1+g_2$, $1\le g_1+g_2\le t$, ${\mathsf{\Phi }}_u$ is a constant only depending on $q_1,q_2,g_1$ and $g_2$. In particular,

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

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_u={\left(\frac{7}{12}\right)}^g-2{\left(\frac{28{q_1}^2+1}{48{q_1}^2}\right)}^{g_1}{\left(\frac{28{q_2}^2+1}{48{q_2}^2}\right)}^{g_2}+{\left(\frac{7{q_1}^2+2}{12{q_1}^2}\right)}^{g_1}{\left(\frac{7{q_2}^2+2}{12{q_2}^2}\right)}^{g_2};\end{array}$$

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

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_u={\left(\frac{7}{12}\right)}^g-2{\left(\frac{7{q_1}^2+1}{12{q_1}^2}\right)}^{g_1}{\left(\frac{7{q_2}^2+1}{12{q_2}^2}\right)}^{g_2}+{\left(\frac{14{q_1}^2+7}{24{q_1}^2}\right)}^{g_1}{\left(\frac{14{q_2}^2+7}{24{q_2}^2}\right)}^{g_2};\end{array}$$

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

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_u={\left(\frac{7}{12}\right)}^g-2{\left(\frac{28{q_1}^2+1}{48{q_1}^2}\right)}^{g_1}{\left(\frac{7{q_2}^2+1}{12{q_2}^2}\right)}^{g_2}+{\left(\frac{7{q_1}^2+2}{12{q_1}^2}\right)}^{g_1}{\left(\frac{14{q_2}^2+7}{24{q_2}^2}\right)}^{g_2}.\end{array}$$

It is well known that strength is an important measure of orthogonality. For comparing the difference between design $d\in \mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$ and orthogonal array $OA(n;{q_1}^{s_1}\times {q_2}^{s_2},t)$ of strength t, the definition of uniformity pattern of design d is given as follows, which provides a measure of the projection uniformity of d onto different dimensions.

Definition 1. 

For any design $d\in \mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$, any positive integer $g(=g_1+g_2)\le s$ and $u\in S_g$, define

$$\begin{array}{c}\hfill M{I}_g\left(d\right)=\sum _{\left|u\right|=g}\left[AM{D}_u\left(d\right)-{\mathsf{\Phi }}_u\right],\end{array}$$

where ${\mathsf{\Phi }}_u$ is shown in Lemma 2. The vector $(M{I}_1\left(d\right),\dots ,M{I}_s\left(d\right))$ is called the uniformity pattern of design d.

We now state the above discussion as the following theorem, which gives a relationship between the uniformity pattern $(M{I}_1\left(d\right),\dots ,M{I}_s\left(d\right))$ of design d and the strength t of orthogonal array $OA(n;{q_1}^{s_1}\times {q_2}^{s_2},t)$.

Theorem 1. 

For any design $d\in \mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$, design d is an orthogonal array $OA(n;{q_1}^{s_1}\times {q_2}^{s_2},t)$ if and only if $M{I}_k\left(d\right)=0$ for $k=1,\dots ,t$ and $M{I}_{t+1}\left(d\right)\ne 0$.

Theorem 1 indicates that there is a close relationship between $M{I}_t\left(d\right)$ and strength t for a design $d\in \mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$, that is, the smaller the value of $M{I}_t\left(d\right)$, the design d will be closer to an orthogonal array of strength t. Based on Theorem 1, $\left\{M{I}_k\left(d\right)\right\}$ may be used as a measure for evaluating designs; it suggests to define some similar criteria, such as MPU.

Definition 2. 

For two designs $d_1,d_2\in \mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$, there is an integer t such that $M{I}_t\left(d_1\right)\ne M{I}_t\left(d_2\right)$ and $M{I}_k\left(d_1\right)=M{I}_k\left(d_2\right)$ for $k=1,\dots ,t-1$; then, $d_1$ is said to have less MPU than $d_2$. If there is no other design in $\mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$ that has less MPU than $d_1$, then $d_1$ is said to have MPU, or $d_1$ is an MPU design.

Here, we mainly establish the connections between projection uniformity and orthogonality, and some relationships between criteria of MPU and GMA will also be included.

Theorem 2. 

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

$$\begin{array}{c}\hfill M{I}_g\left(d\right)=\sum _{\left|u\right|=g}{\alpha }_{g_1{g}_2}\sum _{(r_1,r_2)\in \mathcal{R}}{\beta }_{r_1{r}_2}\left(\genfrac{}{}{0pt}{}{s_1-r_1}{s_1-g_1}\right)\left(\genfrac{}{}{0pt}{}{s_2-r_2}{s_2-g_2}\right)A_{r_1+r_2}\left(d\right),\end{array}$$

where $\mathcal{R}=\{(r_1,r_2):r_1=0,\dots ,g_1,r_2=0,\dots ,g_2,(r_1,r_2)\ne (0,0)\}$, and

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

$$\begin{array}{c}\hfill {\alpha }_{g_1{g}_2}={\left(\frac{7{q_1}^2+2}{12{q_1}^2}\right)}^{g_1}{\left(\frac{7{q_2}^2+2}{12{q_2}^2}\right)}^{g_2},{\beta }_{r_1{r}_2}={\left(\frac{2q_1+2}{7{q_1}^2+2}\right)}^{r_1}{\left(\frac{2q_2+2}{7{q_2}^2+2}\right)}^{r_2};\end{array}$$

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

$$\begin{array}{c}\hfill {\alpha }_{g_1{g}_2}={\left(\frac{14{q_1}^2+7}{24{q_1}^2}\right)}^{g_1}{\left(\frac{14{q_2}^2+7}{24{q_2}^2}\right)}^{g_2},{\beta }_{r_1{r}_2}={\left(\frac{4q_1+4}{14{q_1}^2+7}\right)}^{r_1}{\left(\frac{4q_2+4}{14{q_2}^2+7}\right)}^{r_2};\end{array}$$

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

$$\begin{array}{c}\hfill {\alpha }_{g_1{g}_2}={\left(\frac{7{q_1}^2+2}{12{q_1}^2}\right)}^{g_1}{\left(\frac{14{q_2}^2+7}{24{q_2}^2}\right)}^{g_2},{\beta }_{r_1{r}_2}={\left(\frac{2q_1+2}{7{q_1}^2+2}\right)}^{r_1}{\left(\frac{4q_2+4}{14{q_2}^2+7}\right)}^{r_2}.\end{array}$$

4. A Lower Bound of Uniformity Pattern

This section provides a lower bound of uniformity pattern defined in Definition 1. It is very important that the lower bounds of uniformity pattern can be served as a benchmark not only in searching for uniform designs with minimum projection uniformity but also in helping to validate that some good designs are in fact uniform.

Define ${\mathrm{\Delta }}_u=q_1{\mathrm{e}}^{v_1}+p_1{\mathrm{e}}^{v_3}+q_2({\mathrm{e}}^{v_2}-{\mathrm{e}}^{v_3})$ when $p_1>q_2$, and ${\mathrm{\Delta }}_u=p_2{\mathrm{e}}^{v_1}+q_2{\mathrm{e}}^{v_4}+p_1({\mathrm{e}}^{v_2}-{\mathrm{e}}^{v_4})$ when $p_1\le q_2$.

Theorem 3. 

For any design $d\in \mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$ and positive integer $g(=g_1+g_2)\le s$, we have

$$\begin{array}{c}\hfill M{I}_g\left(d\right)\ge LM{I}_g^{{}^{\prime }}\left(d\right),\end{array}$$

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

$$\begin{array}{c}\hfill LM{I}_g^{{}^{\prime }}\left(d\right)=\sum _{\left|u\right|=g}{\mathrm{\Psi }}_{g_1{g}_2}+\frac{1}{n^2}{\left(\frac{7q_1-2}{12q_1}\right)}^{g_1}{\left(\frac{7q_2-2}{12q_2}\right)}^{g_2}\sum _{u\in S_g}{\mathrm{\Delta }}_u,\end{array}$$

where ${\mathrm{\Psi }}_{g_1{g}_2}=\frac{1}{n}{\left(\frac{3}{4}\right)}^{g_1}{\left(\frac{3}{4}\right)}^{g_2}-{\left(\frac{7{q_1}^2+2}{12{q_1}^2}\right)}^{g_1}{\left(\frac{7{q_2}^2+2}{12{q_2}^2}\right)}^{g_2}$;

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

$$\begin{array}{c}\hfill LM{I}_g^{{}^{\prime }}\left(d\right)=\sum _{\left|u\right|=g}{\mathrm{\Psi }}_{g_1{g}_2}+\frac{1}{n^2}{\left(\frac{14{q_1}^2-4q_1+3}{24{q_1}^2}\right)}^{g_1}{\left(\frac{14{q_2}^2-4q_2+3}{24{q_2}^2}\right)}^{g_2}\sum _{u\in S_g}{\mathrm{\Delta }}_u,\end{array}$$

where ${\mathrm{\Psi }}_{g_1{g}_2}=\frac{1}{n}{\left(\frac{6{q_1}^2+1}{8{q_1}^2}\right)}^{g_1}{\left(\frac{6{q_2}^2+1}{8{q_2}^2}\right)}^{g_2}-{\left(\frac{14{q_1}^2+7}{24{q_1}^2}\right)}^{g_1}{\left(\frac{14{q_2}^2+7}{24{q_2}^2}\right)}^{g_2}$;

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

$$\begin{array}{c}\hfill LM{I}_g^{{}^{\prime }}\left(d\right)=\sum _{\left|u\right|=g}{\mathrm{\Psi }}_{g_1{g}_2}+\frac{1}{n^2}{\left(\frac{7q_1-2}{12q_1}\right)}^{g_1}{\left(\frac{14{q_2}^2-4q_2+3}{24{q_2}^2}\right)}^{g_2}\sum _{u\in S_g}{\mathrm{\Delta }}_u,\end{array}$$

where ${\mathrm{\Psi }}_{g_1{g}_2}=\frac{1}{n}{\left(\frac{3}{4}\right)}^{g_1}{\left(\frac{6{q_2}^2+1}{8{q_2}^2}\right)}^{g_2}-{\left(\frac{7{q_1}^2+2}{12{q_1}^2}\right)}^{g_1}{\left(\frac{14{q_2}^2+7}{24{q_2}^2}\right)}^{g_2}$.

Theorem 4. 

For any design $d\in \mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$ and positive integer $g(=g_1+g_2)\le s$,

$$\begin{array}{c}\hfill M{I}_g\left(d\right)\ge LM{I}_g^{{}^{\prime \prime }}\left(d\right),\end{array}$$

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

$$\begin{array}{cc}\hfill LM{I}_g^{{}^{\prime \prime }}\left(d\right)=& \sum _{\left|u\right|=g}[\frac{1}{n^2}{\left(\frac{7q_1-2}{12q_1}\right)}^{g_1}{\left(\frac{7q_2-2}{12q_2}\right)}^{g_2}\left(\genfrac{}{}{0pt}{}{g_1}{i_1}\right)\left(\genfrac{}{}{0pt}{}{g_2}{i_2}\right)\hfill \\ \hfill & \times {\left(\frac{2q_1+2}{7q_1-2}\right)}^{i_1}{\left(\frac{2q_2+2}{7q_2-2}\right)}^{i_2}{\theta }_{i_1{i}_2}-{\left(\frac{7q_1+2}{12q_1}\right)}^{g_1}{\left(\frac{7q_2+2}{12q_2}\right)}^{g_2}];\hfill \end{array}$$

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

$$\begin{array}{cc}\hfill LM{I}_g^{{}^{\prime \prime }}\left(d\right)=& \sum _{\left|u\right|=g}[\frac{1}{n^2}{\left(\frac{14q_1^2-4q_1+3}{24q_1}\right)}^{g_1}{\left(\frac{14q_2^2-4q_2+3}{24q_2}\right)}^{g_2}\left(\genfrac{}{}{0pt}{}{g_1}{i_1}\right)\left(\genfrac{}{}{0pt}{}{g_2}{i_2}\right)\hfill \\ \hfill & \times {\left(\frac{4q_1^2+4q_1}{14q_1^2-4q_1+3}\right)}^{i_1}{\left(\frac{4q_2^2+4q_2}{14q_2^2-4q_2+3}\right)}^{i_2}{\theta }_{i_1{i}_2}-{\left(\frac{14q_1^2+7}{24q_1^2}\right)}^{g_1}{\left(\frac{14q_2^2+7}{24q_2^2}\right)}^{g_2}];\hfill \end{array}$$

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

$$\begin{array}{cc}\hfill LM{I}_g^{{}^{\prime \prime }}\left(d\right)=& \sum _{\left|u\right|=g}[\frac{1}{n^2}{\left(\frac{7q_1-2}{12q_1}\right)}^{g_1}{\left(\frac{14q_2^2-4q_2+3}{24q_2}\right)}^{g_2}\left(\genfrac{}{}{0pt}{}{g_1}{i_1}\right)\left(\genfrac{}{}{0pt}{}{g_2}{i_2}\right)\hfill \\ \hfill & \times {\left(\frac{2q_1+2}{7q_1-2}\right)}^{i_1}{\left(\frac{4q_2^2+4q_2}{14q_2^2-4q_2+3}\right)}^{i_2}{\theta }_{i_1{i}_2}-{\left(\frac{7q_1+2}{12q_1}\right)}^{g_1}{\left(\frac{14q_2^2+7}{24q_2^2}\right)}^{g_2}],\hfill \end{array}$$

where ${\theta }_{i_1{i}_2}=n{\lambda }_{i_1{i}_2}+{\mu }_{i_1{i}_2}(1+{\lambda }_{i_1{i}_2}),{\mu }_{i_1{i}_2}=n-q_1^{i_1}{q}_2^{i_2}{\lambda }_{i_1{i}_2}$, ${\lambda }_{i_1{i}_2}$ be the largest integer contained in $n/\left(q_1^{i_1}{q}_2^{i_2}\right)$.

Note that Theorem 3 is based on Hamming distances between any two runs of d, but Theorem 4 comes from the quadratic form $y_d^{T}D{y}_d$ in Appendix A Equation (A1). Some numerical examples show that these two lower bounds are not tight simultaneously. Therefore, we give another lower bound of uniformity pattern as the following theorem:

Theorem 5. 

For any design $d\in \mathcal{U}(n;{q_1}^{s_1}\times {q_2}^{s_2})$ and positive integer $g(=g_1+g_2)\le s$, we have

$$\begin{array}{c}\hfill M{I}_g\left(d\right)\ge LM{I}_g^{*}\left(d\right),\end{array}$$

where $LM{I}_g^{*}\left(d\right)=max\{LM{I}_g^{{}^{\prime }}\left(d\right),LM{I}_g^{{}^{\prime \prime }}\left(d\right)\}$.

5. Illustrative Examples

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

Example 1. 

Consider a design $d_1\in \mathcal{U}(4;2^3\times 4^3)$, which are given below:

$$\begin{array}{c}\hfill d_1=\left[\begin{array}{cccccc}0& 0& 0& 0& 3& 2\\ 1& 0& 1& 2& 0& 1\\ 0& 1& 1& 1& 2& 0\\ 1& 1& 0& 3& 1& 3\end{array}\right].\end{array}$$

The number of columns in design $d_1$ is greater than the number of rows, its uniformity pattern in Definition 1, and its lower bound values in Theorems 3–5 are listed in Table 1.

Table 1. Numerical results of designs $d_1$.

g	1	2	3	4	5	6
$M{I}_g\left(d_1\right)$	0	0.0830	0.2193	0.2170	0.0954	0.0157
$LM{I}_g^{{}^{\prime }}\left(d_1\right)$	0	0.0830	0.2193	0.2170	0.0954	0.0157
$LM{I}_g^{{}^{\prime \prime }}\left(d_1\right)$	0	0.0146	0.0397	0.0429	0.0318	0.0157
$LM{I}_g^{*}\left(d_1\right)$	0	0.0830	0.2193	0.2170	0.0954	0.0157

It is clear that $d_1$ is an orthogonal array of strength 1 and attains the lower bounds in Theorem 3.

Example 2. 

Consider design $d_2\in \mathcal{U}(20;2^3\times 5)$ and $d_3\in \mathcal{U}(48;2^5\times 3)$, which are given below,

$$\begin{array}{cc}\hfill d_2=& {\left[\begin{array}{cccccccccccccccccccc}0& 0& 1& 1& 1& 1& 0& 0& 0& 0& 1& 1& 1& 1& 0& 0& 0& 0& 1& 1\\ 0& 1& 0& 1& 1& 0& 1& 0& 0& 1& 0& 1& 1& 0& 1& 0& 0& 1& 0& 1\\ 0& 1& 1& 0& 1& 0& 0& 1& 0& 1& 1& 0& 1& 0& 0& 1& 0& 1& 1& 0\\ 0& 0& 0& 0& 1& 1& 1& 1& 2& 2& 2& 2& 3& 3& 3& 3& 4& 4& 4& 4\end{array}\right]}^T,\hfill \\ \hfill d_3& ={\left[\begin{array}{ccc}1111111100000000& 0000000011111111& 1111111100000000\\ 1111000011110000& 0000111100001111& 1111000011110000\\ 1100110011001100& 0011001100110011& 1100110011001100\\ 1010101010101010& 0101010101010101& 1010101010101010\\ 1001011001101001& 0110100110010110& 1001011001101001\\ 0000000000000000& 1111111111111111& 2222222222222222\end{array}\right]}^T.\hfill \end{array}$$

The number of rows in designs $d_2$ and $d_3$ are greater than the number of columns, and the numerical results of both are shown in Table 2.

Table 2. Numerical results of designs $d_2$ and $d_3$.

g	1	2	3	4	5	6
$M{I}_g\left(d_2\right)$	0	0	$7.8125\times {10}^{-5}$	$1.2148\times {10}^{-4}$
$LM{I}_g^{{}^{\prime }}\left(d_2\right)$	0	−0.0450	−0.0282	−0.0040
$LM{I}_g^{{}^{\prime \prime }}\left(d_2\right)$	0	0	$7.8125\times {10}^{-5}$	$1.2148\times {10}^{-4}$
$LM{I}_g^{*}\left(d_2\right)$	0	0	$7.8125\times {10}^{-5}$	$1.2148\times {10}^{-4}$
$M{I}_g\left(d_3\right)$	0	0	0	0	$3.3908\times {10}^{-6}$	$4.0973\times {10}^{-6}$
$LM{I}_g^{{}^{\prime }}\left(d_3\right)$	0	−0.1837	−0.1742	−0.1300	−0.0365	−0.0044
$LM{I}_g^{{}^{\prime \prime }}\left(d_3\right)$	0	0	0	0	$3.3908\times {10}^{-6}$	$4.0973\times {10}^{-6}$
$LM{I}_g^{*}\left(d_3\right)$	0	0	0	0	$3.3908\times {10}^{-6}$	$4.0973\times {10}^{-6}$

As can be seen from Table 2, designs $d_2$ and $d_3$ are an orthogonal array with strengths of 2 and 4, respectively, and both reach the lower bound in Theorem 4.

It can be seen from Table 1 and Table 2 that the lower bounds of uniformity pattern of designs $d_1$, $d_2$, and $d_3$ are achieved, so $d_1$, $d_2$, and $d_3$ are all MPU designs. We can also see that $LM{I}_g^{{}^{\prime \prime }}\left(d\right)$ is better than $LM{I}_g^{{}^{\prime }}\left(d\right)$ for large n and smaller s. Similar to the findings of Fang et al. (2018) [24], none of the lower bounds in Theorems 3 and 4 are absolutely dominant for all combinations of the number of runs n and of factors s. Therefore, we choose the maximum value of Theorems 3–5.

6. Conclusions

In this paper, the projection uniformity and related properties under mixture discrepancy of asymmetric factorials are explored. The relationship between uniformity pattern and generalized minimum aberration is established. A lower bound of uniformity pattern is also obtained, which can be served as a benchmark for searching minimum projection uniformity designs. These results provide a theoretical basis for searching optimal asymmetric designs with minimum projection uniformity measured by average projection mixture discrepancy. Overall, this paper extends the results of [13,14,15] to the asymmetric case, which makes the corresponding theory more flexible.

The results in this paper can be extended to any asymmetric designs $d\in \mathcal{U}(N;$$q_1^{s_1}\times ···\times q_n^{s_n})$. Taking the first t factors as even and the last $n-t$ factors as odd, and using some simple calculation of tired multiplication, similar definition and results of uniformity pattern and lower bounds can be obtained.