On the Expected ℒ2–Discrepancy of Jittered Sampling

Nathan Kirk, Florian Pausinger
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Abstract

Abstract For m, d ∈ ℕ, a jittered sample of N = md points can be constructed by partitioning [0, 1]d into md axis-aligned equivolume boxes and placing one point independently and uniformly at random inside each box. We utilise a formula for the expected ℒ2−discrepancy of stratified samples stemming from general equivolume partitions of [0, 1]d which recently appeared, to derive a closed form expression for the expected ℒ2−discrepancy of a jittered point set for any m, d ∈ ℕ. As a second main result we derive a similar formula for the expected Hickernell ℒ2−discrepancy of a jittered point set which also takes all projections of the point set to lower dimensional faces of the unit cube into account.
关于抖动采样的期望误差
对于m, d∈_1,将[0,1]d划分为md个与坐标轴对齐的等体积方框,并在每个方框内独立、均匀、随机地放置一个点,可以构造一个N = md个点的抖动样本。我们利用最近出现的来自[0,1]d的一般等体积分区的分层样本的期望∑2−差异的公式,导出了任意m, d∈n的抖动点集的期望∑2−差异的封闭形式表达式。作为第二个主要结果,我们为抖动点集的期望hickerell - 2−差异导出了一个类似的公式,该公式还考虑了点集在单位立方体的低维面上的所有投影。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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