Notes for Intel® oneAPI Math Kernel Library Vector Statistics

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Document Table of Contents x
Notes for Intel® oneAPI Math Kernel Library Vector Statistics
Notes for Intel® oneAPI Math Kernel Library Vector Statistics x
Introduction Randomness and Scientific Experiment Random Numbers Figures of Merit for Random Number Generators Vector Statistics Structure Testing of Basic Random Number Generators Testing of Distribution Random Number Generators Bibliography Notices and Disclaimers
Introduction x
What's New About Vector Statistics About This Document Conventions
Figures of Merit for Random Number Generators x
Uniform Probability Distribution and Basic Pseudo- and Quasi-Random Number Generators Figures of Merit for General (Non-Uniform) Distribution Generators
Vector Statistics Structure x
Why Vector Type Generators? Basic Random Number Generators Random Streams and RNGs in Parallel Computation Generating Methods for Random Numbers of Non-Uniform Distribution Accurate and Fast Modes of Random Number Generation Example of VS Use
Random Streams and RNGs in Parallel Computation x
Initializing a BRNG Creating and Initializing Random Streams Creating Random Stream Copy and Copying Stream State Saving and Restoring Random Streams Independent Streams. Block-Splitting and Leapfrogging Abstract Basic Random Number Generators. Abstract Streams
Generating Methods for Random Numbers of Non-Uniform Distribution x
Inverse Transformation Acceptance/Rejection Mixture of Distributions Special Properties
Testing of Basic Random Number Generators x
BRNG Implementations and Categories Interpreting Test Results BRNG Test Descriptions BRNG Properties and Testing Results
Interpreting Test Results x
One-Level (Threshold) Testing Two-Level Testing
BRNG Test Descriptions x
3D Spheres Test Birthday Spacing Test Bitstream Test Rank of 31x31 Binary Matrices Test Rank of 32x32 Binary Matrices Test Rank of 6x8 Binary Matrices Test Count-the-1's Test (Stream of Bits) Count-the-1's Test (Stream of Specific Bytes) Craps Test Parking Lot Test Self-Avoiding Random Walk Test Template Test
BRNG Properties and Testing Results x
MCG31m1 R250 MRG32k3a MCG59 WH MT19937 SFMT19937 MT2203 SOBOL NIEDERREITER Philox4x32-10 ARS5
Testing of Distribution Random Number Generators x
Interpreting Test Results Description of Distribution Generator Tests Continuous Distribution Random Number Generators Discrete Distribution Random Number Generators
Description of Distribution Generator Tests x
Confidence Test Distribution Moments Test Chi-Squared Goodness-of-Fit Test Performance
Continuous Distribution Random Number Generators x
Uniform (VSL_RNG_METHOD_UNIFORM_STD/VSL_RNG_METHOD_UNIFORM_STD_ACCURATE) Gaussian (VSL_RNG_METHOD_GAUSSIAN_BOXMULLER) Gaussian (VSL_RNG_METHOD_GAUSSIAN_BOXMULLER2) Gaussian (VSL_RNG_METHOD_GAUSSIAN_ICDF) GaussianMV (VSL_RNG_METHOD_GAUSSIANMV_BOXMULLER) GaussianMV (VSL_RNG_METHOD_GAUSSIANMV_BOXMULLER2) GaussianMV  (VSL_RNG_METHOD_GAUSSIANMV_ICDF) Exponential (VSL_RNG_METHOD_EXPONENTIAL_ICDF/VSL_RNG_METHOD_EXPONENTIAL_ICDF_ACCURATE) Laplace (VSL_RNG_METHOD_LAPLACE_ICDF) Weibull (VSL_RNG_METHOD_WEIBULL_ICDF/ VSL_RNG_METHOD_WEIBULL_ICDF_ACCURATE) Cauchy (VSL_RNG_METHOD_CAUCHY_ICDF) Rayleigh (VSL_RNG_METHOD_RAYLEIGH_ICDF/ VSL_RNG_METHOD_RAYLEIGH_ICDF_ACCURATE) Lognormal (VSL_RNG_METHOD_LOGNORMAL_ BOXMULLER2/VSL_RNG_METHOD_LOGNORMAL_BOXMULLER2_ACCURATE) Lognormal (VSL_RNG_METHOD_LOGNORMAL_ICDF/VSL_RNG_METHOD_LOGNORMAL_ICDF_ACCURATE) Gumbel (VSL_RNG_METHOD_GUMBEL_ICDF) Gamma (VSL_RNG_METHOD_GAMMA_GNORM/ VSL_RNG_METHOD_GAMMA_GNORM_ACCURATE) Beta (VSL_RNG_METHOD_BETA_CJA/ VSL_RNG_METHOD_BETA_CJA_ACCURATE) ChiSquare (VSL_RNG_METHOD_CHISQUARE_CHI2GAMMA)
Discrete Distribution Random Number Generators x
Uniform (VSL_RNG_METHOD_UNIFORM_STD) UniformBits (VSL_RNG_METHOD_UNIFORMBITS_STD) UniformBits32 (VSL_RNG_METHOD_UNIFORMBITS32_STD) UniformBits64 (VSL_RNG_METHOD_UNIFORMBITS64_STD) Bernoulli (VSL_RNG_METHOD_BERNOULLI_ICDF) Geometric (VSL_RNG_METHOD_GEOMETRIC_ICDF) Binomial (VSL_RNG_METHOD_BINOMIAL_BTPE) Hypergeometric (VSL_RNG_METHOD_HYPERGEOMETRIC_H2PE) Poisson (VSL_RNG_METHOD_POISSON_PTPE) Poisson (VSL_RNG_METHOD_POISSON_POISNORM) PoissonV (VSL_RNG_METHOD_POISSONV_POISNORM) NegBinomial (VSL_RNG_METHOD_NEGBINOMIAL_NBAR) Multinomial (VSL_RNG_METHOD_MULTINOMIAL_MULTPOISSON)
Notes for Intel® oneAPI Math Kernel Library Vector Statistics

Bibliography

[Ant79]   Antonov, I.A., and Saleev, V.M. An economic method of computing LPt-sequences. USSR Comput. Math. Math. Phys., 19, 252-256, 1979.

[Atkin79] Atkinson A.C. A family of switching algorithms for the computer generation of beta random variables, Biometrika, 66, 1, 141-145, 1979.

[AVX] Intel® Advanced Vector Extensions Programming Reference, http://software.intel.com/file/36945

[BMT] Intel Bull Mountain Technology, Software Implementation Kit (SIK), http://software.intel.com/file/37157

[Box58] Box, G. E. P. and Muller, M. E. A Note on the Generation of Random Normal Deviates. Ann. Math. Stat. 28, 610-611, 1958.

[Brat87] Bratley, P., Fox, B.L., and Schrage, L.E.. A Guide to Simulation, 2nd Edition, Springer-Verlag, New York, 1987.

[Brat88] Bratley, P. and Fox, B.L. ALGORITHM 659: Implementing Sobol’s Quasirandom Sequence Generator. ACM Transactions on Modeling and Computer Simulation, Vol. 14, No. 1, 88-100, March 1988.

[Brat92] Bratley, P., Fox, B.L., and Niederreiter, H. Implementation and Tests of Low-Discrepancy Sequences. ACM Transactions on Modeling and Computer Simulation, Vol. 2, No. 3, 195-213, July 1992.

[Charles93] Charles S. Davis, The computer generation of multinomial random variates. Computational Statistics & Data Analysis, V. 16, 205-217, Aug. 1993.

[Cheng78] Cheng, R. C. H., Generating Beta variates with Nonintegral Shape Parameters, Communications of the ACM, 21, 4, 317-322, 1978.

[Cram46] Cramer, H. Mathematical Methods of Statistics. Cambridge, 1946.

[Dev86] Devroye, L. Non-Uniform Random Variate Generation, Springer-Verlag, New York, 1986.

[Ent98] Entacher, Karl. Bad Subsequences of Well-Known Linear Congruential Pseudorandom Number Generators. ACM Transactions on Modeling and Computer Simulation, Vol. 8, No. 1, 61-70, January 1998.

[FIPS-197] Federal Information Processing Standards Publication 197, ADVANCED ENCRYPTION STANDARD (AES)

[Haram08] Haramoto H., Matsumoto M., Nishimura T., Panneton F., and L’Ecuyer P. Efficient Jump Ahead for F2-Linear Random Number Generators, INFORMS Journal on Computing, Vol.20, No 3, Summer 2008, pp 385-390.

[IntelSWMan] Intel® 64 and IA-32 Architectures Software Developer’s Manual. 3 vols.

(http://www.intel.com/content/www/us/en/processors/architectures-softwaredeveloper-

manuals.html)

[Jöhnk64] Jöhnk, M.D. Erzeugung von Betaverteilten und Gammaverteilten Zufallszahlen, Metrika, 8, 5-15, 1964.

[Jun99] Jun, B., and Kocher, P. The Intel Random Number Generator. White paper prepared for Intel Corp., Cryptography Research, Inc., April 1999.

[Kach88] Kachitvichyanukul, V. and Schmeiser, B.W. Binomial random variate generation. Communications of the ACM, Volume 31, Issue 2, February 1988.

[Kach85] Kachitvichyanukul, V. and Schmeiser, B.W. Computer generation of hypergeometric random variates. J. Stat. Comput. Simul. 22, 1, 127-145, 1985.

[KIM04] Song-Ju Kim, Ken Umeno, and Akio Hasegawa, Corrections of the NIST Statistical Test Suite for Randomness, http://arxiv.org/PS_cache/nlin/pdf/0401/0401040v1.pdf

[Kirk81] Kirkpatrick, S., and E. Stoll. A Very Fast Shift-Register Sequence Random Number Generator. Journal of Computational Physics, V. 40, 517-526, 1981.

[Knuth81] Knuth, Donald E. The Art of Computer Programming, Volume 2, Seminumerical Algorithms, 2nd edition, Addison-Wesley Publishing Company, Reading, Massachusetts, 1981.

[L'Ecu94] L'Ecuyer, Pierre. Uniform Random Number Generators, Annals of Operations Research, 53, 77-120, 1994.

[L'Ecu99] L'Ecuyer, P. Good Parameter Sets for Combined Multiple Recursive Random Number Generators. Operations Research, 47, 1, 159-164, 1999.

[L'Ecuyer99] L'Ecuyer, Pierre. Tables of Linear Congruential Generators of Different Sizes and Good Lattice Structure. Mathematics of Computation, 68, 249-260, 1999.

[MacLaren89] MacLaren, N.M. The Generation of Multiple Independent Sequences of Pseudorandom Numbers. Applied Statistics, 38, 351-359, 1989.

[Mars95] Marsaglia, G. The Marsaglia Random Number CDROM, including the DIEHARD Battery of Tests of Randomness, Department of Statistics, Florida State University, Tallahassee, Florida, 1995.

[Mars2000] Marsaglia, G., and Tsang, W. W. A simple method for generating gamma variables, ACM Transactions on Mathematical Software, Vol. 26, No. 3, Pages 363-372, September 2000.

[Matsum92] Matsumoto, M., and Kurita, Y. Twisted GFSR generators, ACM Transactions on Modeling and Computer Simulation, Vol. 2, No. 3, Pages 179-194, July 1992.

[Matsum94] Matsumoto, M., and Kurita, Y. Twisted GFSR generators II, ACM Transactions on Modeling and Computer Simulation, Vol. 4, No. 3, Pages 254-266, July 1994.

[Matsum98] Matsumoto, M., and Nishumira T. Mersenne Twister: A 623-Dimensionally Equidistributed Uniform Pseudo-Random Number Generator, ACM Transactions on Modeling and Computer Simulation, Vol. 8, No. 1, Pages 3-30, January 1998.

[Matsum2000] Matsumoto, M., and Nishimura T. Dynamic Creation of Pseudorandom Number Generators, 56- 69, in: Monte Carlo and Quasi-Monte Carlo Methods 1998, Ed. Niederreiter, H. and Spanier, J., Springer 2000, http://www.math.sci.hiroshima-u.ac.jp/%7Em-mat/MT/DC/dc.html.

[Mikh2000] Mikhailov, G.A. Weight Monte Carlo Methods, Novosibirsk: SB RAS Publ., 2000 (In Russian).

[MT2002] http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/MT2002/emt19937ar.html

[NAG] Numerical Algorithms Group, www.nag.co.uk.

[NIST800-22] NIST Special Publication 800-22 A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications.

[NIST800-90] NIST Special Publication 800-90 Recommendation for Random Number Generation Using Deterministic Random Bit Generators (Revised), March 2007. http://csrc.nist.gov/publications/nistpubs/800-90/SP800-90revised_March2007.pdf

[Ripley87] Ripley, B.D. Stochastic Simulation, Wiley, New York, 1987.

[Saito08] Saito, M., and Matsumoto, M. SIMD-oriented Fast Mersenne Twister: a 128-bit Pseudorandom Number Generator,Monte Carlo and Quasi-Monte Carlo Methods 2006, Springer, pp. 607-622, 2008, http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/earticles.html

[Salmon2011] Salmon, J.K., Moraes M.A., Dror R. O., and Shaw, D.E. New York, 2011. http://www.thesalmons.org/john/random123/papers/random123sc11.pdf

[Schmeiser81] Schmeiser, Bruce, and Kachitvichyanukul, Voratas. Poisson Random Variate Generation. Research Memorandum 81-4, School of Industrial Engineering, Purdue University, 1981.

[Vad77] Vaduva, I. On computer generation of gamma random variables by rejection and composition procedures. Mathematische Operationsforschung und Statistik, Series Statistics, vol. 8, 545-576, 1977.

[Ziff98] Ziff, Robert M. Four-tap shift-register-sequence random-number generators. Computers in Physics, Vol. 12, No. 4, Jul/Aug 1998.

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