Notes for Intel® oneAPI Math Kernel Library Vector Statistics

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Date 12/04/2020
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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 Real Implementation (Single and Double Precision) Integer Implementation Stream Initialization by Function vslNewStream Stream Initialization by Function vslNewStreamEx Subsequences Selection Methods Generator Period Empirical Testing Results Summary
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

ARS5

This is a keyed family of counter-based BRNGs. The state consists of 128-bit integer counter c and a 128-bit key k. The BRNG is based on the AES encryption algorithm [FIPS-197]. The 32-bit output is obtained in the following way [Salmon2011]:

  1. The i-th number is defined by the following formula: ri = (f(i/4) >> ((i mod 4) * 32) mod 232
  2. Function f(c) takes 128-bit input and produces 128-bit result obtained in the following way:
    1. Put c0 = c xor k and k0 = k.
    2. The following recurrence is calculated N times:
      1. ci+1 = SubBytes(c)
      2. ci+1 = ShiftRows(ci+1)
      3. ci+1 = MixColumns(ci+1), this step is omitted if i + 1 = N
      4. ci+1 = AddRoundKey(ci+1, kj)

      5. Lo(ki+1) = Lo(k) + 0x9E3779B97F4A7C15

        Hi(ki+1) = Hi(k) + 0xBB67AE8584CAA73B

    3. Put f(c) = cN, where N = 5
  3. Real output: un= (int)rn/232+ ½

Specification for the SubBytes, ShiftRows, MixColumns and AddRoundKey functions can be found in [FIPS-197].

Real Implementation (Single and Double Precision)

The output vector is the sequence of the floating-point values u0, u1, ...

Integer Implementation

The output vector of 32-bit integers r0, r1, ...

Stream Initialization by Function vslNewStream

ARS5 generates the stream and initializes it specifying the 32-bit input integer parameter seed. The stream state is two 128-bit numbers c and k initialized in the following way:

  1. Assume k = seed.
  2. Assume c = 0.

Stream Initialization by Function vslNewStreamEx

ARS5 generates the stream and initializes it specifying the array params[] of n 32-bit integers:

  1. If n = 0, assume c = k = 0.
  2. If n = 1, assume k = params[0], c = 0.
  3. If n = 2, assume k = params[0] + params[1]*232, c = 0.
  4. If n = 3, assume k = params[0] + params[1]*232 + params[2]*264, c = 0.
  5. If n = 4, assume k = params[0] + params[1]*232 + params[2]*264 + params[3]*296, c = 0.
  6. If n = 5, assume k = params[0] + params[1]*232 + params[2]*264 + params[3]*296, c = params[4].
  7. If n = 6, assume k = params[0] + params[1]*232 + params[2]*264 + params[3]*296, c = params[4] + params[5]*232.
  8. If n = 7, assume k = params[0] + params[1]*232 + params[2]*264 + params[3]*296, c = params[4] + params[5]*232 + params[6]*264.
  9. If n >= 8, assume k = params[0] + params[1]*232 + params[2]*264 + params[3]*296, c = params[4] + params[5]*232 + params[6]*264 + params[7]*296.

Subsequences Selection Methods

vslSkipAheadStream
 Supported 
vslSkipAheadStreamEx
 Supported 
vslLeapfrogStream
 Not supported

Generator Period

Empirical Testing Results Summary

Test Name 
vsRngUniform
 
vdRngUniform
 
viRngUniform
 
viRngUniformBits
3D Spheres Test OK (20% errors) OK (20% errors) Not applicable OK (20% errors)
Birthday Spacing Test Not applicable Not applicable Not applicable OK (0% errors)
Bitstream Test Not applicable Not applicable Not applicable OK (15% errors)
Rank of 31x31 Binary Matrices Test Not applicable Not applicable Not applicable OK (0% errors)
Rank of 32x32 Binary Matrices Test Not applicable Not applicable Not applicable OK (0% errors)
Rank of 6x8 Binary Matrices Test Not applicable Not applicable Not applicable OK (0% errors)
Counts-the-1’s Test (stream of bits) Not applicable Not applicable Not applicable OK (0% errors)
Counts-the-1’s Test (stream of specific bytes) Not applicable Not applicable Not applicable OK (0% errors)
Craps Test OK (30% errors) OK (30% errors) OK (30% errors) OK (30% errors)
Parking Lot Test OK (10% errors) OK (10% errors) Not applicable OK (10% errors)
2D Self-Avoiding Random Walk Test OK (20% errors) OK (10% errors) Not applicable OK (10% errors)
NOTE:

  1. The tabulated data is obtained using the one-level (threshold) testing technique. The OK result indicates FAIL < 50%. The run fails when p-value falls outside the interval [0.05, 0.95].

  2. The stream tested is generated by calling the function vslNewStream with seed=7,777,777.

Parent topic: BRNG Properties and Testing Results
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