Mixture discrepancy for quasi-random point sets

Abstract

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 Lp-star discrepancy, the center L2-discrepancy (CD) and the wrap-around L2-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.