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

Bitstream Test

Test Purpose

The test uses simulation to check if it is possible to interpret the integer output of a BRNG as a sequence of random bits.

NOTE:

The bit precision of a BRNG defines the sequence of random bits formation. For example, only 59 lower bits take part in the bit stream formation for the MCG59 generator, and only 31 lower bits for the MCG31 generator.

First Level Test

The first level test initially forms the sequence of bits b0, b1, b2, ... from the integer output of the BRNG and then forms 20-bit overlapping words w0= b0b1... b19 , w1 = b1b2... b20, ... from the sequence. From the total number of 2021 formed words the test computes the quantity K of the missed 20-bit words. For the truly random sequence the K statistic distribution should be very close to normal with mean a = 141,909 and standard deviation s = 428. The test denotes the cumulative function of the normal distribution with these parameters as F(x). The result is that the distribution of the p-value p = F(K) should be uniform within the interval of (0, 1). 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:

0-31. 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 selects:

  1. NB of lower bits from each WS-bit integer to form a bit sequence

  2. NB of lower bits from each of four WS-bit elements for WH generator

Second Level Test

The second level test performs the first level test 20 times. The result of each first level test is the p-value pj, j = 1, 2, ..., 20. The test applies the Kolmogorov-Smirnov goodness-of-fit test with Anderson-Darling statistics to the obtained set of pj, j = 1, 2, ..., 20. If the resulting p-value is p < 0.05 or p > 0.95, the test fails.

Final Result Interpretation

The final result of the test is the FAIL percentage of the failed second level tests. The second level test performs ten times. The acceptable result is the value of FAIL < 50%.

Tested Generators

Function Name Application 
vsRngUniform
 not applicable 
vdRngUniform
 not applicable 
viRngUniform
 not applicable 
viRngUniformBits
 applicable

The lower bits are not random for multiplicative congruential generators where the module is the power of two (for example, MCG59), thus, the Bitstream Test fails for such generators.

Parent topic: BRNG Test Descriptions
Level Two Title