论文标题
分层样品与单位立方体分区的差异
Discrepancy of stratified samples from partitions of the unit cube
论文作者
论文摘要
我们将抖动抽样的概念扩展到任意分区，并研究相关点集的差异。 Let $\mathbfΩ=(Ω_1,\ldots,Ω_N)$ be a partition of $[0,1]^d$ and let the $i$th point in $\mathcal{P}$ be chosen uniformly in the $i$th set of the partition (and stochastically independent of the other points), $i=1,\ldots,N$.对于此类集合的研究，我们介绍了均匀分布的三角形阵列的概念，并将此概念与文献中的相关概念进行了比较。我们证明了预期的$ {\ Mathcal {l} _p} $ - 差异，$ \ Mathbb {e} {\ Mathcal {\ Mathcal {l} _p}（\ Mathcal {p} _ {p} _ {\MathbfΩ}^P $， Equivolume分区$ \MATHBFΩ$总是严格小于预期的$ {\ Mathcal {l} _p} $ - $ p> 1 $的一组$ n $统一随机样本的差异。对于固定的$ n $，我们考虑基于单位立方体的等效分区的分层样本类别，或在其触及范围内均匀的正限制的凸组集合中。结果表明，这些类包含至少一个最小化的$ {\ Mathcal {l} _p} $ - 差异。我们使用针对小$ n $的明确结构来说明结果。此外，我们提出了一个分区家庭，似乎每$ n $都会提高蒙特卡洛采样的预期差异2。
We extend the notion of jittered sampling to arbitrary partitions and study the discrepancy of the related point sets. Let $\mathbfΩ=(Ω_1,\ldots,Ω_N)$ be a partition of $[0,1]^d$ and let the $i$th point in $\mathcal{P}$ be chosen uniformly in the $i$th set of the partition (and stochastically independent of the other points), $i=1,\ldots,N$. For the study of such sets we introduce the concept of a uniformly distributed triangular array and compare this notion to related notions in the literature. We prove that the expected ${\mathcal{L}_p}$-discrepancy, $\mathbb{E} {\mathcal{L}_p}(\mathcal{P}_{\mathbfΩ})^p$, of a point set $\mathcal{P}_\mathbfΩ$ generated from any equivolume partition $\mathbfΩ$ is always strictly smaller than the expected ${\mathcal{L}_p}$-discrepancy of a set of $N$ uniform random samples for $p>1$. For fixed $N$ we consider classes of stratified samples based on equivolume partitions of the unit cube into convex sets or into sets with a uniform positive lower bound on their reach. It is shown that these classes contain at least one minimizer of the expected ${\mathcal{L}_p}$-discrepancy. We illustrate our results with explicit constructions for small $N$. In addition, we present a family of partitions that seems to improve the expected discrepancy of Monte Carlo sampling by a factor of 2 for every $N$.
