球冠偏差的一个计数公式
An enumerative formula for the spherical cap discrepancy
摘要：
球冠差异是一种广泛使用的测量球面上点的样本分布均匀程度的方法。由于计算困难，在设计球面上均匀分布的最优抽样方案时，通常用一些上下估计来代替这种差异度量。在本文中，我们提供了一个完全明确的，易于实现的球冠差异计数公式。毫不奇怪，该公式具有组合性质，因此，其应用仅限于小尺寸和中等样本量的球体。尽管如此，它可以作为一个有用的校准工具，用于测试抽样方案的效率，其明确的字符可能是有用的，也建立必要的最优性条件时，最小化与给定大小的样本的差异。
英文摘要：
The spherical cap discrepancy is a widely used measure for how uniformly a sample of points on the sphere is distributed. Being hard to compute, this discrepancy measure is typically replaced by some lower or upper estimates when designing optimal sampling schemes for the uniform distribution on the sphere. In this paper, we provide a fully explicit, easy to implement enumerative formula for the spherical cap discrepancy. Not surprisingly, this formula is of combinatorial nature and, thus, its application is limited to spheres of small dimension and moderate sample sizes. Nonetheless, it may serve as a useful calibrating tool for testing the efficiency of sampling schemes and its explicit character might be useful also to establish necessary optimality conditions when minimizing the discrepancy with respect to a sample of given size.
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