## U-型设计的对称化L2-偏差的下界

1 新乡学院数学与统计学院, 河南新乡 453003

2 吉首大学数学与统计学院, 湖南吉首 416000

## Lower Bounds for the Symmetric L2-Discrepancy of U-type Designs

1 College of Mathematics and Statistics, Xinxiang University, Henan xinxiang 453003

2 College of Mathematics and Statistics, Jishou University, Hunan Jishou 416000

 基金资助: 国家自然科学基金.  11961027国家自然科学基金.  12161040国家自然科学基金.  11701213湖南省自然科学基金.  2021JJ30550湖南省自然科学基金.  2020JJ4497

 Fund supported: the NSFC.  11961027the NSFC.  12161040the NSFC.  11701213the Natural Science Foundation of Hunan Province.  2021JJ30550the Natural Science Foundation of Hunan Province.  2020JJ4497 Abstract

Uniform design is one of the main methods of fractional factorials, which has been widely used in industrial production, systems engineering, pharmacy and other natural sciences. Various discrepancies are used to measure the uniformity of fractional factorials, the key is to find an accurate lower bound of the discrepancy, because it can be used as a benchmark which measures uniformity of design. In this paper, the lower bounds for the symmetric L2-discrepancy on symmetrical U-type designs with four-level and asymmetrical U-type designs with two and three mixed levels and two and four mixed levels are abtained.

Keywords： U-type design ; Symmetric L2-discrepancy ; Lower bound

Lei Yiju, Ou Zujun. Lower Bounds for the Symmetric L2-Discrepancy of U-type Designs. Acta Mathematica Scientia[J], 2022, 42(6): 1802-1811 doi:

## 3 4水平$U\!$-型设计的对称化$L_2$-偏差的下界

$\sharp A$表示集合$A$中元素的个数, 对$i, j=1, \cdots , n$, 定义

(1)  $\sum\limits_{i=1}^{n}r^{1}_{ii}=\frac{mn}{2}$;

(2)  $\sum\limits_{i=1}^{n}\sum\limits_{j(\neq i)=1}^{n}\alpha^{1}_{ij}=\frac{3mn^2}{8}, \ \alpha^{1}_{ii}=0;$

(3)  $\sum\limits_{i=1}^{n}\sum\limits_{j(\neq i)=1}^{n}\beta^{1}_{ij}=\frac{mn^2}{4}, \ \beta^{1}_{ii}=0;$

(4)  $\sum\limits_{i=1}^{n}\sum\limits_{j(\neq i)=1}^{n}\gamma^{1}_{ij}=\frac{mn^2}{8}, \ \gamma^{1}_{ii}=0.$

$\begin{eqnarray} [SD_2(d)]^2=C_1-\frac{2}{n}\left(\frac{39}{32}\right)^m\sum\limits_{i=1}^{n}\left(\frac{47}{39}\right)^{r^{1}_{ii}} +\frac{2^{m}}{n^{2}}\sum\limits_{i=1}^{n}\sum\limits_{j\neq i}^{n}\left(\frac{3}{4}\right)^{\alpha^{1}_{ij}}\left(\frac{1}{2}\right)^ {\beta^{1}_{ij}}\left(\frac{1}{4}\right)^{\gamma^{1}_{ij}}, \end{eqnarray}$

令$g(r^{1}_{11}, \cdots , r^{1}_{nn})=-\frac{2}{n}\left(\frac{39}{32}\right)^{m}\sum\limits_{i=1}^{n}\left(\frac{47}{39}\right)^{r^{1}_{ii}}$, 由$r^{1}_{ii}$定义可知其为取值$\{0, \cdots , m\}$中的元素且满足$\sum\limits_{i=1}^{n}r^{1}_{ii}=\frac{mn}{2}$.

$L$分别求关于$r^{1}_{ii}, i=1, \cdots , n $$\lambda 的偏导并令其为0, 可得 解上述方程组得 r^{1}_{11}=\cdots=r^{1}_{nn}=\frac{m}{2} , \lambda=\frac{2}{n}\left(\frac{39}{32}\right)^{m}\left(\frac{47}{39}\right)^{\frac{m}{2}}\ln\frac{47}{39} . 所以 g(r^{1}_{11}, \cdots , r^{1}_{nn}) 的最大值点在 (\frac{m}{2}, \cdots , \frac{m}{2}) 处取得, 因为 g(r^{1}_{11}, \cdots , r^{1}_{nn}) 只有一个极值点, 且 g(r^{1}_{11}, \cdots , r^{1}_{nn}) 为凸函数, 所以其最小值在边界点 (\frac{mn}{2}, 0, \cdots , 0) 处取得. 因此 等号成立当且仅当 r^{1}_{11}=\frac{mn}{2}, r^{1}_{22}=\cdots=r^{1}_{nn}=0 . 再由引理3.2易知 所以 [SD_2(d)]^2\geq LB(n, 4^{m}) . 证毕. ### 4 非对称 U\! -型设计的对称化 L_2 -偏差值的下界 下面考虑 2\mbox{、} 3混水平和 2\mbox{、} 4混水平两种非对称U-型设计情况下对称化 L_2 -偏差的下界. d\in{\cal U}(n; q_{1}^{m_{1}}\times q_{2}^{m_{2}}), m_{1}+m_{2}=m , 则其对称化 L_2 -偏差可表示为 ### 4.1 2、3混水平设计 d\in{\cal U}(n; 2^{m_{1}}\times 3^{m_{2}}), m_{1}+m_{2}=m , 即 q_{1}=2, q_{2}=3 , 当 l=1, \cdots , m_1 时, x_{il}\in\{\frac{1}{4}, \frac{3}{4}\} , 因此 1+2x_{il}-2x^2_{il}=\frac{11}{8} l=m_1+1, \cdots , m 时有 x_{il}\in\{\frac{1}{6}, \frac{1}{2}, \frac{5}{6}\} , 因此 对于上述 r^{2}_{ij}, \alpha^{2}_{ij}, \beta^{2}_{ij}, \gamma^{2}_{ij} 有如下结论. 引理4.1 对任意的设计 d\in{\cal U}(n; 2^{m_{1}}\times 3^{m_{2}}) (1) \sum\limits_{i=1}^{n}r^{2}_{ii}=\frac{m_{2}n}{3} ; (2) \sum\limits_{i=1}^{n}\sum\limits_{{j=1 \atop j\neq i}}^{n}\alpha^{2}_{ij}=\frac{4m_2n^2}{9}, \alpha^{2}_{ii}=0, i=1, \cdots , n; (3) \sum\limits_{i=1}^{n}\sum\limits_{{j=1 \atop j\neq i}}^{n}\beta^{2}_{ij}=\frac{2m_2n^2}{9}, \beta^{2}_{ii}=0, i=1, \cdots , n; (4) \sum\limits_{i=1}^{n}\sum\limits_{{j=1 \atop j\neq i}}^{n}\gamma^{2}_{ij}=\frac{m_1n^2}{2}, \gamma^{2}_{ii}=0, i=1, \cdots , n. 对任意的设计 d\in{\cal U}(n; 2^{m_{1}}\times 3^{m_{2}}) , 利用 r^{2}_{ij}, \alpha^{2}_{ij}, \beta^{2}_{ij}, \gamma^{2}_{ij} , 可获得 [SD_2(d)]^2 的如下定理4.1所示的表达式. 定理4.1 对任意的设计 d\in{\cal U}(n; 2^{m_{1}}\times 3^{m_{2}}) \begin{equation} [SD_2(d)]^2=C_1-\frac{2}{n}\Big(\frac{11}{8}\Big)^{m_{1}}\Big(\frac{23}{18}\Big)^{m_{2}}\sum\limits_{i=1}^{n} \left(\frac{27}{23}\right)^{r^{2}_{ii}} +\frac{2^{m}}{n^{2}}\sum\limits_{i=1}^{n}\sum\limits_{j\neq i}^{n}\left(\frac{2}{3}\right)^{\alpha^{2}_{ij}}\left(\frac{1}{3}\right)^ {\beta^{2}_{ij}}\left(\frac{1}{2}\right)^{\gamma^{2}_{ij}}, \end{equation} 其中 C_1 如定理3.1所示. 由定理4.1中 [SD_2(d)]^2 的表达式可得 [SD_2(d)]^2 如下的下界. 定理4.2 对任意的设计 d\in{\cal U}(n; 2^{m_{1}}\times 3^{m_{2}})$$ [SD_2(d)]^2\geq LB(n, 2^{m_{1}}\times 3^{m_{2}})$, 其中

$m_{2}=0 $$m_{1}=0 时, 设计 d 变成对称U-型设计, 其对称化 L_2 -偏差有如下推论. 推论4.1 (1) 对任意的设计 d\in{\cal U}(n; 2^m)$$ [SD_2(d)]^2\geq LB(n, 2^m),$其中

(2) 对任意的设计$d\in{\cal U}(n; 3^{m}) $$[SD_2(d)]^2\geq LB(n, 3^{m}), 其中 其中 C_1 如定理3.1所示. ### 4.2 2、4混水平设计 d\in{\cal U}(n; 2^{m_1}\times 4^{m_2}), m_{1}+m_{2}=m , 即 q_{1}=2, q_{2}=4 , 当 l=1, \cdots , m_1 时有 x_{il}\in\{\frac{1}{4}, \frac{3}{4}\} , 因此 1+2x_{il}-2x^2_{il}=\frac{11}{8} l=m_1+1, \cdots , m 时有 x_{il}\in\{\frac{1}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8}\} , 因此 对于上述定义 r^{3}_{ij}, \alpha^{3}_{ij}, \beta^{3}_{ij}, \gamma^{3}_{ij}, \xi_{ij} 有如下结论. 引理4.2 对任意的设计 d\in{\cal U}(n; 2^{m_1}4^{m_2}) (1) \sum\limits_{i=1}^{n}r^{3}_{ii}=\frac{m_2n}{2} ; (2) \sum\limits_{i=1}^{n}\sum\limits_{{j=1 \atop j\neq i}}^{n}\alpha^{3}_{ij}=\frac{3m_2n^2}{8}, \ \alpha^{1}_{ii}=0, i=1, \cdots , n; (3) \sum\limits_{i=1}^{n}\sum\limits_{{j=1 \atop j\neq i}}^{n}\beta^{3}_{ij}=\frac{m_2n^2}{4}, \ \beta^{1}_{ii}=0, i=1, \cdots , n; (4) \sum\limits_{i=1}^{n}\sum\limits_{{j=1 \atop j\neq i}}^{n}\gamma^{3}_{ij}=\frac{m_2n^2}{8}, \ \gamma^{1}_{ii}=0, i=1, \cdots , n; (5) \sum\limits_{i=1}^{n}\sum\limits_{{j=1 \atop j\neq i}}^{n}\xi_{ij}=\frac{m_{1}n^2}{2}, \xi_{ii}=0, i=1, \cdots , n. 对任意的设计 d\in{\cal U}(n; 2^{m_{1}}\times 4^{m_{2}}) , 利用 r^{3}_{ij}, \alpha^{3}_{ij}, \beta^{3}_{ij}, \gamma^{3}_{ij}, \xi_{ij} , 可获得 [SD_2(d)]^2 如下定理4.3所示的表达式. 定理4.3 对任意的设计 d\in{\cal U}(n; 2^{m_{1}}\times 4^{m_{2}}) \begin{equation} [SD_2(d)]^2=C_1-\frac{2}{n}\Big(\frac{11}{8}\Big)^{m_1}\Big(\frac{39}{32}\Big)^{m_2}\sum\limits_{i=1}^{n}\left(\frac{47}{39}\right)^{r^{3}_{ii}} +\frac{2^m}{n^2}\sum\limits_{i=1}^{n}\sum\limits_{j\neq i}^n\left(\frac{1}{2}\right)^ {\xi_{ij}+\beta^{3}_{ij}+2\gamma^{3}_{ij}}\left(\frac{3}{4}\right)^{\alpha^{3}_{ij}}, \end{equation} 其中 C_1 如定理3.1所示. 基于定理4.3所示 [SD_2(d)]^2 的表达式, 可获得 [SD_2(d)]^2 如下定理4.4所示的下界. 定理4.4 对任意的设计 d\in{\cal U}(n; 2^{m_{1}}\times 4^{m_{2}}) 其中 C_1 如定理3.1所示. m_{2}=0$$ m_{1}=0$时, 设计$d$变成对称U-型设计, 其对称化$L_2$-偏差有如下推论.

(2) 对任意的设计$d\in{\cal U}(n; 4^{m})$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Fang K T, Wang Y. Number-Theoretic Methods in Statistics. London: Chapman and Hall, 1994

Hickernell F J .

A generalized discrepancy and quadrature erron bound

Mathematics of Computation, 1998, 67 (221): 299- 322

Hickernell F J. Lattice Rules: How well do they measure up?//Hellekalek P, Larche G. Random and Quasi-Random Point Sets. New York: Springer, 1998: 109-166

Hickernell F J , Liu M Q .

Uniform designs limit aliasing

Biometrika, 2002, 89, 893- 904

Zhou Y D , Ning J H , Song X B .

Lee discrepancy and its applications in experimental designs

Statistics & Probability Letters, 2008, 78, 1933- 1942

Chatterjee K , Qin H .

Generalized discrete discrepancy and its application in experimental designs

Journal of Statistical Planning and Inference, 2011, 141, 951- 960

Fang K T , Mukerjee R .

A connection between uniformity and aberration in regular fractions of two-level factorials

Biometrika, 2000, 87, 193- 198

Fang K T, Ma C X, Mukerjee R. Uniformity in fractional factorials//Fang K T, Hickernell F J, Niederreiter H. Monte Carlo and Quasi-Monte Carlo Methods 2000. Berlin: Springer-Verlag, 2002: 232-241

Fang K T , Lu X , Winker P .

Lower bounds for centered and wrap-around L2-discrepancy and construction of uniform designs by threshold accepting

Journal of Complexity, 2003, 19, 692- 711

Chatterjee K , Fang K T , Qin H .

Uniformity in factorial designs with mixed levels

Journal of Statistical Planning and Inference, 2005, 128, 593- 607

Chatterjee K , Fang K T , Qin H .

A lower bound for centered L2-discrepancy on asymmetric factorials and its application

Metrika, 2006, 63, 243- 255

Wang Z H , Qin H , Chatterjee K .

Lower bounds for the symmetric L2-discrepancy and their application

Communications in Statistics-Theory and Methods, 2007, 36, 2413- 2423

Qin H , Li D .

Connection between uniformity and orthogonality for symmetrical factorial designs

Journal of Statistical Planning and Inference, 2006, 136, 2770- 2782

Qin H , Fang K T .

Discrete discrepancy in factorial designs

Metrika, 2004, 60, 59- 72

Fang K T , Maringer D , Tang Y , Winker P .

Lower bounds and stochastic optimization algorithms for uniform designs with three or four levels

Mathematics of Computation, 2006, 75, 859- 878

Chatterjee K , Ou Z J , Phoa F K H , Qin H .

Uniform four-level designs from two-level designs: a new look

Statistica Sinica, 2017, 27, 171- 186

Qin H , Ou Z J , Chatterjee K .

Construction of four-level designs for computer experiments

Scientia Sinica Mathematica, 2017, 47 (9): 1089- 1100

Hu L P , Li H Y , Ou Z J .

Constructing optimal four-level designs via gray map code

Metrika, 2019, 82 (5): 573- 587

Qin H , Zhang S L , Fang K T .

Constructing uniform design with two or three-level

Acta Mathematica Scientia, 2006, 26, 451- 459

Zhou Y D , Ning J H .

Lower bounds of wrap-around L2-discrepancy and relationships between MLHD and uniform design with a large size

Journal of Statistical Planning and Inference, 2008, 138, 2330- 2339

Zhang Q H , Wang Z H , Hu J W , Qin H .

A new lower bound for wrap-around L2-discrepancy on two and three mixed level factorials

Statistics & Probability Letters, 2015, 96, 133- 140

Lei Y J , Ou Z J .

Lower bound of symmetric L2-discrepancy on three-level U-type designs

Acta Mathematicae Applicatae Sinca,2018, 41 (1): 138- 144

Zhou Y D , Fang K T , Ning J H .

Mixture discrepancy for quasi-random points sets

Journal of Complexity, 2013, 29, 283- 301

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