投影加权对称偏差  

Projection weighted symmetric discrepancy

作　　者：赫兰兰 谢民育[1] 宁建辉[1] Lanlan He;Minyu Xie;Jianhui Ning

机构地区：[1]华中师范大学数学与统计学学院,武汉430079

出　　处：《中国科学：数学》2020年第5期629-644,共16页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11571133和11871237)资助项目。

摘　　要：均匀设计自20世纪80年代被提出以来依靠其稳健和使用方便、灵活的特性而广受欢迎.为获得实验目标区域内散布均匀的设计点集,不同的均匀度量标准被相继提出,其中以从数值积分理论中发展而来的L2-星偏差及相应的改进形式接受度最为广泛.尽管相对于最初的星偏差标准,当前广泛使用的改进偏差有更好的性质和实用性,但这些偏差仍存在一些缺点,例如,中心化偏差(centered discrepancy, CD)对于实验目标区域中心点的特殊依赖性.从各种偏差定义的本源上可以看出,对称偏差(symmetric discrepancy, SD)相较于CD有更好的几何性质, SD给实验目标区域内任何点的权重都是相同的,这在直观上更符合均匀性的要求.但受限于投影均匀性差的缺陷, SD的使用范围十分有限.本文重新研究对称偏差,分析出SD缺陷的本质,并采取加权的办法改进其投影均匀性,提出投影加权对称偏差.本文同时也研究了新偏差的相关优良性质.The uniform design was proposed in the 1980s, and since then lots of researchers have studied it. In order to distribute the points evenly in the experimental domain, many criteria have been forwarded to measure the uniformity of the design array. Among those criteria, the L2-discrepancies, developed from star discrepancy in the number theoretic method, are the most popular measurement, such as the centered L2-discrepancy(CD). There are still some disadvantages with those discrepancies. For example, the central points of the experimental domain plays a much more important role than other points when choosing CD as the criterion. Although, symmetric L2-discrepancy(SD) have better geometric sense than CD, the poor performance at projection uniformity limits the use of SD. In this paper, we propose a projection weighted SD(WSD) to refine the projection properties of SD. Some other properties of SD and WSD are also studied in this work.

关 键 词：均匀设计 偏差 投影均匀性 

分 类 号：O212[理学—概率论与数理统计]