Related papers: Discrepancy of stratified samples from partitions …
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 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…
For $m, d \in {\mathbb N}$, a jittered sampling point set $P$ having $N = m^d$ points in $[0,1)^d$ is constructed by partitioning the unit cube $[0,1)^d$ into $m^d$ axis-aligned cubes of equal size and then placing one point independently…
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 study the expected $\mathcal{L}_2$-discrepancy of stratified samples generated from special equi-volume partitions of the unit square. The partitions are defined via parallel lines that are all orthogonal to the diagonal of the square.…
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…
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…
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 introduce a class of convex equivolume partitions. Expected $L_2-$discrepancy are discussed under these partitions. There are two main results. First, under this kind of partitions, we generate random point sets with smaller expected…
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.
This paper studies the expected $L_p$-discrepancy ($2 \leq p < \infty$) for stratified sampling schemes under importance sampling. We introduce a parametric family of equivolume partitions $\Omega_{\theta,\sim}$ and leverage recent exact…
We study the extreme and the periodic $L_p$ discrepancy of point sets in the $d$-dimensional unit cube. The extreme discrepancy uses arbitrary sub-intervals of the unit cube as test sets, whereas the periodic discrepancy is based on…
The classic problems of testing uniformity of and learning a discrete distribution, given access to independent samples from it, are examined under general $\ell_p$ metrics. The intuitions and results often contrast with the classic…
We consider random discrepancy under weighted importance sampling of a class of stratified input. We give the expected $L_p-$discrepancy($2\leq p<\infty$) upper bound in weighted form under a class of stratified sampling. This result…
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…
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 $1-\exp(-\Theta(s))$…
Stratified sampling is a fast and simple method to generate point sets with uniform distribution in hypercubes. However, for the most common paraxial stratfication it has the prominent drawback that the number of sampled points in n…
In this paper we propose an acceptance-rejection sampler using stratified inputs as diver sequence. We estimate the discrepancy of the points generated by this algorithm. First we show an upper bound on the star discrepancy of order…
Given an $N$-dimensional subspace $X$ of $L_p([0,1])$, we consider the problem of choosing $M$-sampling points which may be used to discretely approximate the $L_p$ norm on the subspace. We are particularly interested in knowing when the…