Statistical Tests for MIXMAX Pseudorandom Number Generator

Authors

  • Narek H. Martirosyan Yerevan Physics Institute
  • Gevorg A. Karyan Yerevan Physics Institute
  • Norayr Z. Akopov Yerevan Physics Institute

Keywords:

MIXMAX, Statistical tests, MCMC

Abstract

The Pseudo-Random Number Generators (PRNGs) are key tools in Monte Carlo simulations. More recently, the MIXMAX PRNG has been included in ROOT and Class Library for High Energy Physics (CLHEP) software packages and claims to be a state of the art generator due to its long period, high performance and good statistical properties. In this paper the various statistical tests for MIXMAX are performed. The results compared with those obtained from other PRNGs, e.g., Mersenne Twister, Ranlux, LCG reveal better qualities for MIXMAX in generating random numbers. The Mersenne Twister is by far the most widely used PRNG in many software packages including packages in High Energy Physics (HEP), however, the results show that MIXMAX is not inferior to Mersenne Twister

Author Biography

Narek H. Martirosyan, Yerevan Physics Institute

Alikhanian Brothers Street 2, Yerevan, Armenia

References

S. Agostinelli et al., “Geant4: a simulation toolkit”, Nucl. Instrum. Meth. A ,vol. 506, no. 250, 2003.

A.T. Bharucha-Reid, Elements of the Theory of Markov Processes and their Applications, Courier Corporation, 2012.

W.R. Gilks, S. Richardson and D. Spiegelhalter, Markov chain Monte Carlo in practice, CRC press, 1995.

G. O. Roberts and J.S. Rosenthal, “General state space Markov chains and MCMC algorithms”, Probability Surveys, vol. 1, pp. 20-71, 2004.

I. Beichl and F. Sullivan, “The metropolis algorithm”, Computing in Science & Engineering, vol. 2, no. 1, pp. 65-69, 2000. [6]

C. Andrieu, N. de Freitas, A. Doucet and M. Jordan, “An introduction to MCMC for machine learning”, Machine Learning, vol. 50, no.1, pp. 5-43, 2003.

J. Dongarra and F. Sullivan, “Guest editors introduction: The top 10 algorithms”, Computing in Science & Engineering, vol. 2, no. 1, pp. 22-23, 2000.

G. Savvidy and N. Savvidy, “On the Monte Carlo simulation of physical systems”, J.Comput.Phys., vol. 97, no. 2, pp. 566-572, 1991.

N. Akopov, G. Savvidy and N. Savvidy, “Matrix generator of pseudo-random numbers”, J.Comput.Phys., vol. 97, no. 2, pp. 573-579, 1991.

K. Savvidy and G. Savvidy, “Spectrum and entropy of C-systems MIXMAX random number generator”, Chaos, Solitons & Fractals, vol. 91, pp. 33-38, 2016.

A. G¨orlich, M. Kalomenopoulos, K. Savvidy and G. Savvidy, “Distribution of periodic trajectories of Anosov C-system”, arXiv:1608.03496, 2016.

K.G. Savvidy, “The MIXMAX random number generator”, Computer Physics Communications, vol. 196, pp. 161-165, 2015.

G. Marsaglia, “DIEHARD: a battery of tests of randomness”, [Online]. Available: http://stat.fsu.edu/pub/diehard/, 1996.

M. Mascagni and A. Srinivasan, “Algorithm 806: SPRNG: A scalable library for pseudorandom number generation”, ACM Transactions on Mathematical Software (TOMS), vol. 26, no. 3, pp. 436-461, 2000.

A. Rukhin, et al., “A statistical test suite for random and pseudorandom number generators for cryptographic applications”, DTIC Document, 2001.

P. L’ecuyer and R. Simard, “TestU01: A C Library for Empirical Testing of Random Number Generators”, ACM Transactions on Mathematical Software, vol. 33, no. 4, page 22, 2007.

D. Knuth, The Art of Computer Programming, Seminumerical Algorithms, Volume 2, 3rd edition, Massachusetts: Addison Wesley, 1998.

R. J. Serfling, Approximation Theorems of Mathematical Statistics, New York: John Wiley & Sons, 2009.

T. B. Arnold and J. W. Emerson, “Nonparametric goodness-of-fit tests for discrete null distributions”, The R Journal, vol. 3, no. 2, 34-39, 2011.

T. W. Anderson and D. A. Darling, “Asymptotic theory of certain” goodness of fit” criteria based on stochastic processes”, The Annals of Mathematical Statistics, pp. 193-212, 1952.

J. Wang, W. W. Tsang and G. Marsaglia, “Evaluating Kolmogorov’s distribution”, Journal of Statistical Software, vol. 8, no. 18, 2003.

J. A. Peacock, “Two-dimensional goodness-of-fit testing in astronomy”, Monthly Notices Royal Astronomy Society, vol. 202, no. 3, pp. 615-627, 1983.

G. Fasano and A. Franceschini, “A multidimensional version of the KolmogorovSmirnov test”, Monthly Notices Royal Astronomy Society, vol. 225, no. 1, pp. 155-170, 1987.

N.Z. Akopov and N.H. Martirosyan, “The Optimal Approach for Kolmogorov-Smirnov Test Calculation in High Dimensional Space”, Transactions of the IIAP NAS RA, Mathematical Problems of Computer Science, vol. 44, pp. 138-144, 2015.

A. Justel, D. Pena, R. Zamar, “A multivariate Kolmogorov-Smirnov test of goodness of fit”, Statistics & Probability Letters, vol. 35, no. 3, pp. 251-259, 1997.

E. Gosset, “A three-dimensional extended Kolmogorov-Smirnov test as a useful tool in astronomy”, Astronomy and Astrophysics, vol. 188, pp. 258-264, 1987.

D.J. Hudson, Lectures on Elementary Statistics and Probability, CERN: European Organization for Nuclear Research, 1964.

S. Tezuka,Uniform Random Numbers: Theory and Practice, New York:Springer Science & Business Media, 2012.

P. L’ecuyer, “Efficient and portable combined random number generators”, Communications of the ACM, vol. 31, no. 6, pp. 742-751, 1988.

P. L’ecuyer, “Tables of linear congruential generators of different sizes and good lattice structure”, Mathematics of Computation of the American Mathematical Society, vol. 68, no. 225, pp. 249-260, 1999.

U. Dieter, “How to calculate shortest vectors in a lattice”, Mathematics of Computation, vol. 29, no. 131, pp. 827-833, 1975.

G. Fishman, Monte Carlo: Concepts, Algorithms, and Applications, New York: Springer Science & Business Media, 2013.

, P. Hellekalek, “Good random number generators are (not so) easy to find”, Mathematics and Computers in Simulation, vol. 46, no. 5, pp. 485-505,, 1998.

M. Mascagni, “Parallel pseudorandom number generation”, SIAM News, vol. 32, no. 5, pp. 221-251, 1999.

A. Srinivasan, D.M. Ceperley and M. Mascagni, “Random number generators for parallel applications”, Advances in chemical physics, vol. 105, pp. 13-36, 1999.

P. Frederickson, R. Hiromoto, T.L. Jordan, B. Smith and T. Warnock, “Pseudorandom trees in Monte Carlo”, Parallel Computing, vol. 1, no. 2, pp. 175-150, 1984.

D. Istvan, “Uniform random number generators for parallel computers”, Parallel Computing, vol. 15, no. 1-3, pp. 155-164, 1990.

A. Srinivasan, M. Mascagni and D. Ceperley, “Testing parallel random number generators”, Parallel Computing, vol. 29, no. 1, pp. 69-94, 2003.

T. Bradley, J. du Toit, R. Tong, M. Giles and P. Woodhams, “Parallelization techniques for random number generations”, GPU Computing Gems Emerald Edition, vol. 16, pp. 231-246, 2011.

J.K. Salmon, M.A. Moraes, R.O. Dror and D.E. Shaw, “Parallel random numbers: as easy as 1, 2, 3”, International Conference for High Performance Computing, Networking, Storage and Analysis (SC), 2011.

D.R.C. Hill, C. Mazel, J. Passerat-Palmbach and M.K. Traore, “Distribution of random streams for simulation practitioners”, Concurrency and Computation: Practice and Experience, vol. 25, no. 10, pp. 1427-1442, 2013.

N.L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, New York: wiley series in probability and mathematical statistics, 1995.

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Published

2021-12-10

How to Cite

Martirosyan, N. H., Karyan, G. A. ., & Akopov, N. Z. (2021). Statistical Tests for MIXMAX Pseudorandom Number Generator. Mathematical Problems of Computer Science, 47, 37–49. Retrieved from http://mpcs.sci.am/index.php/mpcs/article/view/132