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Spherical discrepancy for designs on hyperspheres

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  • Zhang, Mei
  • Zhou, Yong-Dao

Abstract

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.

Suggested Citation

  • Zhang, Mei & Zhou, Yong-Dao, 2016. "Spherical discrepancy for designs on hyperspheres," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 226-234.
    • Handle: RePEc:eee:stapro:v:119:y:2016:i:c:p:226-234
    DOI: 10.1016/j.spl.2016.08.009
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    References listed on IDEAS

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    1. Chuang, S.C. & Hung, Y.C., 2010. "Uniform design over general input domains with applications to target region estimation in computer experiments," Computational Statistics & Data Analysis, Elsevier, vol. 54(1), pages 219-232, January.
    2. David H. Doehlert, 1970. "Uniform Shell Designs," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 19(3), pages 231-239, November.
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