Discrepancy bounds for a class of negatively dependent random points including Latin hypercube samples

Gnewuch, Michael; Hebbinghaus, Nils

doi:10.1214/20-aap1638

Cited by 18 publications

(27 citation statements)

References 58 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the upper bound, the original proof of Heinrich et al does not easily reveal information on C. Aistleitner [Ais11] gave an alternative, more direct proof that also shows that with positive probability, D * (P ) ≤ 10 √ dN. The currently strongest estimate, lowering the 10 to 2.525, is due to Gnewuch and Hebbinghaus [GH21]. For the lower bound, the elementary proof of [Doe14] clearly can be made more precise and then give a reasonable constant, but this has not been done so far.…”

Section: Estimates For the Star Discrepancymentioning

confidence: 99%

“…The two most prominent such dependent randomized constructions are Latin hypercube samplings [MBC79] and jittered sampling (also called stratified sampling) [Bel81,CPC84]. The first analysis of the discrepancy of such a construction in the modern paradigm of making the influence on d explicit was conducted by Pausinger and Steinerberger [PS16], who showed the following bounds (we note that Latin hypercube samples were later analyzed in [DDG18,GH21]).…”

Section: Jittered Samplingmentioning

confidence: 99%

“…Heinrich et al [HNWW01] used deep results of Talagrand and Haussler from the theory of empirical processes to give the first proof of the O( √ dN) discrepancy bound for N random points in [0, 1) d . With a non-trivial, purely combinatorial decomposition argument called dyadic chaining, Aistleitner [Ais11] and later Gnewuch and Hebbinghaus [GH21] reproved this bound and gave explicit (and small) values for the leading constant…”

Section: Proof Of the Upper Boundmentioning

confidence: 99%

See 2 more Smart Citations

A Sharp Discrepancy Bound for Jittered Sampling

Doerr

2021

Preprint

View full text Add to dashboard Cite

For m, d ∈ 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 axisaligned cubes of equal size and then placing one point independently and uniformly at random in each cube. We show that there are constants c ≥ 0 and C such that for all d and all m ≥ d the expected non-normalized star discrepancy of a jittered sampling point set satisfiesThis discrepancy is thus smaller by a factor of Θthan the one of a uniformly distributed random point set of m d points. This result improves both the upper and the lower bound for the discrepancy of jittered sampling given by Pausinger and Steinerberger (Journal of Complexity ( 2016)). It also removes the asymptotic requirement that m is sufficiently large compared to d.

show abstract

Section: Estimates For the Star Discrepancymentioning

confidence: 99%

Section: Jittered Samplingmentioning

confidence: 99%

Section: Proof Of the Upper Boundmentioning

confidence: 99%

See 1 more Smart Citation

A Sharp Discrepancy Bound for Jittered Sampling

Doerr

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…There one simply substitutes the random variables U (i) j in (2) by constant values 1 2 . The negative dependence properties of LHS were investigated in [4].…”

Section: Latin Hypercube Samplingmentioning

confidence: 99%

“…Recently, there has been some research in this direction. In [4,5] the authors showed that a specific negative dependence property of RQMC point sets guarantees that they satisfy the same pre-asymptotic probabilistic discrepancy bounds (with explicitly revealed dependence on the number of points N as well as on the dimension d) as MC points. Here the negative dependence property relies on the common distribution of all sample points.…”

Section: Introductionmentioning

confidence: 99%

Note on Pairwise Negative Dependence of Randomized Rank-1 Lattices

Wnuk,

Gnewuch

2019

Preprint

Self Cite

View full text Add to dashboard Cite

In her recent paper [Negative dependence, scrambled nets, and variance bounds. Math. Oper. Res. 43 (2018), 228-251] Christiane Lemieux studied a framework to analyze the dependence structure of sampling schemes. The main goal of the framework is to determine conditions under which the negative dependence structure of a sampling scheme yields estimators with reduced variance compared to Monte Carlo estimators. For instance, she was able to show that in dimension d = 2 scrambled (0, m, d)-nets lead to randomized quasi-Monte Carlo estimators with variance no larger than the variance of Monte Carlo estimators for functions monotone in each variable. Her result relies on a pairwise negative dependence property that is, in particular, satisfied by (0, m, 2)-nets. In this note we establish that the same result holds true in arbitrary dimension d for a type of randomized lattice point sets that we call randomly shifted and jittered rank-1 lattices. We show that the details of the randomization are crucial and that already small modifications may destroy the pairwise negative dependence property.

show abstract

An efficient surrogate model for system reliability with multiple failure modes

Xu,

Gao,

Wang

et al. 2024

Quality & Reliability Eng

View full text Add to dashboard Cite

The failure modes of complex systems are often highly dependent. Aiming at the reliability analysis of engineering systems with multiple failure modes, an efficient surrogate model for system reliability with multiple failure modes is proposed, namely, the Copula Adaptive Kriging surrogate model (CAKSM). First, the initial Kriging model is established based on the initial samples. In order to select more effective added samples to update the Kriging model, a new learning function, namely, Copula Unites with Expected Improvement Function (CUEIF), is established. The function can make a reasonable sample selection by considering the failure correlation. Then, based on the optimal Copula function, the reliability data of each failure mode are coupled and analyzed. In order to adapt to various computing scales, regularization is introduced to improve the prediction speed of the surrogate model, and the probability density function of each performance function is obtained by introducing nonparametric kernel density estimation. Finally, the effectiveness of the CAKSM method and the learning function CUEIF is verified through four examples.

show abstract

Discrepancy bounds for a class of negatively dependent random points including Latin hypercube samples

Cited by 18 publications

References 58 publications

A Sharp Discrepancy Bound for Jittered Sampling

A Sharp Discrepancy Bound for Jittered Sampling

Note on Pairwise Negative Dependence of Randomized Rank-1 Lattices

An efficient surrogate model for system reliability with multiple failure modes

Contact Info

Product

Resources

About