相关论文: A Sharp Discrepancy Bound for Jittered Sampling
We study the discrepancy of jittered sampling sets: such a set $\mathcal{P} \subset [0,1]^d$ is generated for fixed $m \in \mathbb{N}$ by partitioning $[0,1]^d$ into $m^d$ axis aligned cubes of equal measure and placing a random point…
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…
We compare expected star discrepancy under jittered sampling with simple random sampling, and the strong partition principle for the star discrepancy is proved.
We extend the notion of jittered sampling to arbitrary partitions and study the discrepancy of the related point sets. Let $\mathbf{\Omega}=(\Omega_1,\ldots,\Omega_N)$ be a partition of $[0,1]^d$ and let the $i$th point in $\mathcal{P}$ be…
Classical jittered sampling partitions $[0,1]^d$ into $m^d$ cubes for a positive integer $m$ and randomly places a point inside each of them, providing a point set of size $N=m^d$ with small discrepancy. The aim of this note is to provide a…
Jittered Sampling is a refinement of the classical Monte Carlo sampling method. Instead of picking $n$ points randomly from $[0,1]^2$, one partitions the unit square into $n$ regions of equal measure and then chooses a point randomly from…
We prove that classical jittered sampling of the $d$-dimensional unit cube does not yield the smallest expected $\mathcal{L}_2$-discrepancy among all stratified samples with $N=m^d$ points. Our counterexample can be given explicitly and…
We present two main contributions to the expected star discrepancy theory. First, we derive a sharper expected upper bound for jittered sampling, improving the leading constants and logarithmic terms compared to the state-of-the-art [Doerr,…
We investigate the expected star discrepancy under a newly designed class of convex equivolume partition models. The main contributions are two-fold. First, we establish a strong partition principle for the star discrepancy, showing that…
We study the expected star discrepancy under a newly designed class of non-equal volume partitions. The main contributions are twofold. First, we establish a strong partition principle for the star discrepancy, showing that our newly…
We show that there is a constant $K > 0$ such that for all $N, s \in \N$, $s \le N$, the point set consisting of $N$ points chosen uniformly at random in the $s$-dimensional unit cube $[0,1]^s$ with probability at least $1-\exp(-\Theta(s))$…
We introduce a class of convex equivolume partitions. Expected star discrepancy results are compared for stratified samples under these partitions, including simple random samples. There are four main parts of our results. First, among…
The star-discrepancy is a quantitative measure for the irregularity of distribution of a point set in the unit cube that is intimately linked to the integration error of quasi-Monte Carlo algorithms. These popular integration rules are…
By a result of Heinrich, Novak, Wasilkowski and Wo\'zniakowski the inverse of the star discrepancy $n(d,\varepsilon)$ satisfies $n(d,\varepsilon)\leq c_{\abs}d\varepsilon^{-2}$. Equivalently for any $N$ and $d$ there exists a set of $N$…
We study the probabilistic existence of point configurations satisfying the $(0, m, d)$-net property in base $b$ within a randomly generated point set of size $N$ in the $d$-dimensional unit cube. We first derive an upper bound on the…
We study the expected $ L_2-$discrepancy under two classes of partitions, explicit and exact formulas are derived respectively. These results attain better expected $L_2-$discrepancy formulas than jittered sampling.
The inverse of the star-discrepancy problem asks for point sets $P_{N,s}$ of size $N$ in the $s$-dimensional unit cube $[0,1]^s$ whose star-discrepancy $D^\ast(P_{N,s})$ satisfies $$D^\ast(P_{N,s}) \le C \sqrt{s/N},$$ where $C> 0$ is a…
It is well understood that if one is given a set $X \subset [0,1]$ of $n$ independent uniformly distributed random variables, then $$ \sup_{0 \leq x \leq 1} \left| \frac{\# X \cap [0,x]}{\# X} - x \right| \lesssim \frac{\sqrt{\log{n}}}{…
We study the dispersion of a point set, a notion closely related to the discrepancy. Given a real $r\in (0,1)$ and an integer $d\geq 2$, let $N(r,d)$ denote the minimum number of points inside the $d$-dimensional unit cube $[0,1]^d$ such…
In this paper, we consider the upper bound of the probabilistic star discrepancy based on Hilbert space filling curve sampling. This problem originates from the multivariate integral approximation, but the main result removes the strict…