论文标题
部分可观测时空混沌系统的无模型预测
On the expected L2-discrepancy of jittered sampling
论文作者
论文摘要
对于$ m，d \ in \ mathbb {n} $，可以通过将$ [0,1]^d $划分为$ n = m^d $点的抖动样本构建，以$ [0,1]^d $分隔为$ m^d $ axis平整的equivolume盒，并在每个盒子内随机独立且均匀地将一个透度地放置在每个盒子内。我们将一个公式用于预期的$ \ Mathcal {l} _2- $ $的分层样品的差异，该样品源自$ [0,1]^d $的一般eporivolume分区，该分区最近出现在$ [0,1] \ mathbb {n} $。作为第二个主要结果，我们为预期的Hickernell $ \ Mathcal {l} _2- $ $ $差异的抖动点集的差异得出了类似的公式，该点还将点设置的所有点投影到了单位立方体的较低尺寸面上。
For $m, d \in \mathbb{N}$, a jittered sample of $N=m^d$ points can be constructed by partitioning $[0,1]^d$ into $m^d$ axis-aligned equivolume boxes and placing one point independently and uniformly at random inside each box. We utilise a formula for the expected $\mathcal{L}_2-$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 $\mathcal{L}_2-$discrepancy of a jittered point set for any $m, d \in \mathbb{N}$. As a second main result we derive a similar formula for the expected Hickernell $\mathcal{L}_2-$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.
