Spherical discrepancy for designs on hyperspheres

The discrepancies have played an important role in quasi-Monte Carlo Methods. There are various discrepancies for measuring the uniformity of a set of points on unit hypercube and simplex. However, there is no discrepancy with explicit expressions on hyperspheres in the literature. In this paper, we propose a new measure of uniformity, called spherical discrepancy (SD), for designs on the unit hypersphere. Different with the pseudo F-discrepancy, the SD is directly defined on the unit hypersphere in terms of spherical coordinates. A computational formula of the new discrepancy is also given by the functional method. The properties of SD and some illustrative examples are also shown.