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 Test Purpose First Level Test Second Level Test Final Result Interpretation Tested Generators 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

Birthday Spacing Test

Test Purpose

The test uses simulation to evaluate the randomness of groups of 24 sequential bits of the integer output of a BRNG. The test analyzes all possible groups of the kind, that is, for example, from 0 to 23 bit, from 1 to 24 bit, and so on.

First Level Test

The first level test selects at random m = 210 ”birthdays” from a ”year” of n = 224 days. Then the test computes the spacing between the birthdays for each pair of sequential birthdays. The test then uses the spacings to determine the K value, that is, the number of pairs of sequential birthdays with the spacing of more than one day. In this case K should have the distribution close to the Poisson distribution with the l = 16 parameter. The first level test determines 200 values of Kj (j = 1, 2, ..., 200). To obtain the p-value p, the test applies the chi-square goodness-of-fit test to the determined values.

The integer output lists different interpretations for each BRNG. In the table below, NB stands for the number of bits required to represent a random number in integer arithmetic, WS stands for the machine word size, in bits, used in integer random number generation.

BRNG Integer Output Interpretation
MCG31m1

Array of 32-bit integers. Each 32-bit integer uses the following bits:

 0-30. NB=31, WS=32.

R250

Array of 32-bit integers. Each 32-bit integer uses the following bits:

 0-31. NB=32, WS=32.

MRG32k3a

Array of 32-bit integers. Each 32-bit integer uses the following bits:

 0-31. NB=32, WS=32.

MCG59

Array of 64-bit integers. Each 64-bit integer uses the following bits:

 0-58. NB=59, WS=64.

WH

Array of quadruples of 32-bit integers. Each 32-bit integer uses the following bits:

0-23. NB=24, WS=32.

MT19937

Array of 32-bit integers. Each 32-bit integer uses the following bits:

0-31. NB=32, WS=32.

MT2203

Array of 32-bit integers. Each 32-bit integer uses the following bits:

0-31. NB=32, WS=32.

SFMT19937

Array of quadruples of 32-bit integers. Each 32-bit integer uses the following bits:

0-32. NB=32, WS=32.

PHILOX4X32X10

Array of 32-bit integers. Each 32-bit integer uses the following bits:

0-31. NB=32, WS=32.

ARS5

Array of 32-bit integers. Each 32-bit integer uses the following bits:

0-31. NB=32, WS=32.

The test generates the dates of the birthdays in the following way:

  1. Selects the bs, bs +1, ..., bs +23 bits from the next WS-bit integer of the integer output of viRngUniformBits.

  2. Treats the selected bits as a 24-bit integer, that is, the number of the date on which the next birthday takes place and thus generates a birthday.

  3. The test performs the steps 1 and 2 m times to generate m birthdays taken that the year consists of n days. The legitimate values s are different for each base generator (see the table above):  0 £ s £ NB - 24.

Second Level Test

The second level test performs the first level test ten times for the fixed s. The test applies the Kolmogorov-Smirnov goodness-of-fit test with Anderson-Darling statistics to the obtained set of pj(j = 1, 2 , ..., 10). If the resulting p-value is p < 0.05 or p > 0.95, the test fails for the given s.

Final Result Interpretation

The second level test performs ten times for each 0 £ s £ NB - 24. The test computes the FAILs percentage for the failed second level tests. The final result is the minimal percentage of the failed tests FAIL = min(FAIL0, FAIL1, ..., FAILNB-24) for 0 £ s £ NB - 24. The applicable result is the value of FAIL < 50%. Thus, the test determines if it is possible to select 24 random bits from every element of the integer output of the generator.

  1. The integer output for the WH generator is the quadruples of the 32-bits values (xi, yi, zi, wi). In each 32-bit value only the lower 24 bits are significant.

  2. The second level test performs ten times for the xi element. Then the test computes the FAILx percentage the failed second level tests.

  3. The second level test performs ten times for the yi. Then the test computes the FAILy percentage for the failed second level tests.

  4. The test performs the same procedure to compute the FAILz and FAILw values.

The final result is the minimal percentage of the failed tests FAIL = min(FAILx, FAILy, FAILz, FAILw). The acceptable result is the value of FAIL < 50%.

The test determines if it is possible to select 24 random bits from the fixed element x, y, z or w for each element of the integer output of the generator.

Tested Generators

Function Name Application 
vsRngUniform
 not applicable 
vdRngUniform
 not applicable 
viRngUniform
 not applicable 
viRngUniformBits
 applicable
Parent topic: BRNG Test Descriptions
Level Two Title