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The weighted spectral test: diaphony

Authors:

Peter Hellekalek,

Harald NiederreiterAuthors Info & Claims

ACM Transactions on Modeling and Computer Simulation (TOMACS), Volume 8, Issue 1

Pages 43 - 60

https://doi.org/10.1145/272991.273008

Published: 01 January 1998 Publication History

Abstract

In this article, we present a new approach to assessing uniform random number generators, the weighted spectral test, or diaphony. In contrast to the usual spectral test the weighted spectral test is not limited to random number generators with a lattice structure. Its computational complexity is O (s·N²) for any point set of cardinality N in the s-dimensional unit cube. As the main results of this article, we prove an analog of the classical inequality of Erdös-Turán-Koksma, present the necessary tools to transcribe known discrepancy bounds into bounds for diaphony, and provide bounds for the diaphony of multiplicative congruetial pseudorandom numbers. The last section contains numerical results.

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Patel M(2023)The Efficacy of Discrepancy in Computer GraphicsACM SIGGRAPH 2023 Talks10.1145/3587421.3595404(1-2)Online publication date: 6-Aug-2023
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Wyatt CGuimaraes A(2022) 3D MR fingerprinting using Seiffert spirals Magnetic Resonance in Medicine10.1002/mrm.2919788:1(151-163)Online publication date: 24-Mar-2022
https://doi.org/10.1002/mrm.29197
Hornfeck WKuhn P(2015)Diaphony, a measure of uniform distribution, and the Patterson functionActa Crystallographica Section A Foundations and Advances10.1107/S205327331500712371:4(382-391)Online publication date: 29-May-2015
https://doi.org/10.1107/S2053273315007123
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Index Terms

The weighted spectral test: diaphony
1. Computing methodologies
  1. Modeling and simulation
2. Mathematics of computing
  1. Probability and statistics
    1. Probabilistic reasoning algorithms
      1. Random number generation

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Copyright © 1998 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 01 January 1998

Published in TOMACS Volume 8, Issue 1

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Cited By

Patel M(2023)The Efficacy of Discrepancy in Computer GraphicsACM SIGGRAPH 2023 Talks10.1145/3587421.3595404(1-2)Online publication date: 6-Aug-2023
https://dl.acm.org/doi/10.1145/3587421.3595404
Wyatt CGuimaraes A(2022) 3D MR fingerprinting using Seiffert spirals Magnetic Resonance in Medicine10.1002/mrm.2919788:1(151-163)Online publication date: 24-Mar-2022
https://doi.org/10.1002/mrm.29197
Hornfeck WKuhn P(2015)Diaphony, a measure of uniform distribution, and the Patterson functionActa Crystallographica Section A Foundations and Advances10.1107/S205327331500712371:4(382-391)Online publication date: 29-May-2015
https://doi.org/10.1107/S2053273315007123
van Hameren A(2006)Goodness-of-fit tests in many dimensionsNuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment10.1016/j.nima.2005.11.136559:1(167-171)Online publication date: Apr-2006
https://doi.org/10.1016/j.nima.2005.11.136
Choirat CSeri R(2006)The Asymptotic Distribution of Quadratic DiscrepanciesMonte Carlo and Quasi-Monte Carlo Methods 200410.1007/3-540-31186-6_5(61-76)Online publication date: 2006
https://doi.org/10.1007/3-540-31186-6_5
Entacher KSchell TUhl A(2005)Bad Lattice PointsComputing10.1007/s00607-004-0105-z75:4(281-295)Online publication date: 1-Aug-2005
https://dl.acm.org/doi/10.1007/s00607-004-0105-z
Leeb H(2002)Asymptotic properties of the spectral test, diaphony, and related quantitiesMathematics of Computation10.1090/S0025-5718-01-01356-471:237(297-309)Online publication date: 1-Jan-2002
https://dl.acm.org/doi/10.1090/S0025-5718-01-01356-4
Levin M(2001)On the statistical independence of compound pseudorandom numbers over part of the periodACM Transactions on Modeling and Computer Simulation10.1145/502109.50211311:3(294-311)Online publication date: 1-Jul-2001
https://dl.acm.org/doi/10.1145/502109.502113
L'Ecuyer P(1998)Uniform random number generatorsProceedings of the 30th conference on Winter simulation10.5555/293172.293213(97-104)Online publication date: 1-Dec-1998
https://dl.acm.org/doi/10.5555/293172.293213
L'Ecuyer P(1998)Uniform random number generators1998 Winter Simulation Conference. Proceedings (Cat. No.98CH36274)10.1109/WSC.1998.744904(97-104)Online publication date: 1998
https://doi.org/10.1109/WSC.1998.744904
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