论文标题
球形帽差异的列举公式
An enumerative formula for the spherical cap discrepancy
论文作者
论文摘要
球形帽差异是一种广泛使用的措施，用于分布球上的点样品。由于很难计算，在设计最佳抽样方案的最佳采样方案以用于球体上的均匀分布时，这种差异度量通常被一些下层或上层估计所取代。在本文中，我们为球形帽差异提供了完全明确，易于实施的枚举公式。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。但是，它可以作为测试采样方案效率的有用校准工具，其明确特征在最小化给定尺寸的样本的差异时，也可以建立必要的最佳条件。
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.
