Mixture discrepancy for quasi-random point sets

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• 作者：Yong-Dao Zhou^a ; ^{ydzhou@scu.edu.cn} ; Kai-Tai Fang^b ; ^c ; ^{ktfang@uic.edu.hk} ; Jian-Hui Ning^d ; ^{jhning@mail.ccnu.edu.cn}
• 关键词：Centered L2L2-discrepancy ; Curse of dimensionality ; Generalized wordlength pattern ; Mixture discrepancy ; Quasi-Monte Carlo methods ; Uniform design
• 刊名：Journal of Complexity
• 出版年：2013
• 出版时间：June-August, 2013
• 年：2013
• 卷：29
• 期：3-4
• 页码：283-301
• 全文大小：956 K

文摘

There are various discrepancies that are measures of uniformity for a set of points on the unit hypercube. The discrepancies have played an important role in quasi-Monte Carlo methods. Each discrepancy has its own characteristic and some weakness. In this paper we point out some unreasonable phenomena associated with the commonly used discrepancies in the literature such as the -star discrepancy, the center -discrepancy (CD) and the wrap-around -discrepancy (WD). Then, a new discrepancy, called the mixture discrepancy (MD), is proposed. As shown in this paper, the mixture discrepancy is more reasonable than CD and WD for measuring the uniformity from different aspects such as the intuitive view, the uniformity of subdimension projection, the curse of dimensionality and the geometric property of the kernel function. Moreover, the relationships between MD and other design criteria such as the balance pattern and generalized wordlength pattern are also given.