平衡型与BP-最优设计  

Balance Pattern and BP-Optimal Designs

作　　者：陆璇[1] 方开泰[2] 徐勤峰[3] 殷剑新[4] 

机构地区：[1]清华大学数学科学系,北京100084 [2]香港浸会大学数学系 [3]复旦大学统计系,上海200433 [4]苏州大学数学系,江苏苏州215006

出　　处：《数学年刊（A辑）》2007年第2期191-202,共12页Chinese Annals of Mathematics

基　　金：国家自然科学基金(No.10071056);香港RGC基金资助的项目

摘　　要：对Lu和Sun[1],Lu等[2]中提出的E(d2)准则进行了扩展,定义了一个新准则,用以评价和构造因子设计．对一个含有k个因子的因子设计,定义了“平衡型”向量B=(B(1),B(2),…,B(k)),其中B(m)表示与m-平衡(即强度为m的正交性)的接近程度,m=1,…,b,而B(2)=E(d2)．这个新准则(称为“最近平衡准则”),顺序最小化向量B的各个分量．对B(m)得到了下界LB(m),m= 1,…,b．证明了:当试验单元间的Hamming距离都相等时,平衡型向量的各分量同时达到其下界．试验单元间的Hamming距离都相等的等水平设计称为“正则表”．饱和的等水平正交表构成了正则表的一个子类．对2≤q≤7给出了一些q-水平正则表的结果,其中某些是新的．正则表可进一步用于构造具有好性质的设计．A criterion is defined for the evaluation and construction of factorial designs, which is an extension of the E（d^2） criterion proposed by Lu and Sun [1] and Lu, et al [2]. For a factorial design involving k factors, a vector B = [B（1）, B（2）,…, B（k）], called ＂balance pattern＂ is defined, where B（m） measures the closeness to ＂m-balance＂ （the orthogonality of strength m） for m = 1,... ,k, where B（2） = E（d^2）. The criterion ＂nearest balance＂ （briefed as NB）, is to minimize the components of B sequentially. Lower bounds LB（m） are given for B（m）, m = 1, 2,... , k. It is proved that the lower bounds can be achieved simultaneously if and only if all the Hamming distances of the run pairs are equal. Equal-level factorial designs with equal Hamming distances of run pairs, called ＂canonical arrays＂, are equivalent to resolvable balanced incomplete block designs. Saturated equal-level orthogonal arrays form a subclass of canonical arrays. Some q-level canonical arrays for q from 2 to 7 are listed, among which some are new. Applications of canonical arrays in the construction of designs with nice properties are introduced.

关 键 词：平衡型 HAMMING距离 正交表 超饱和设计 

分 类 号：O212.6[理学—概率论与数理统计] O213.1[理学—数学]