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

Count-the-1's Test (Stream of Bits)

Test Purpose

The test evaluates the randomness of the overlapping random five-letter words sequence. The five-letter words have the specified distribution of the probabilities of obtaining the specified letter. The test forms the random letters from the integer output of a BRNG. The test treats the integer output as a sequence of bits.

First Level Test

The first level test assumes that the integer output is a sequence of random bits. The test interprets this bit sequence as a sequence of bytes, that is, a sequence of 8-bit integer numbers. The number of 1’s in every random byte should have a binominal distribution with m = 8, p = 1/2 parameters. Therefore, the probability of getting k 1’s in a byte is equal to 2-8C8k. The first level test regards a random variable c that takes five possible values:

c = 0, if the number of 1’s in a random byte is less than three,

c = 1, if the number of 1’s in a random byte is three,

c = 2, if the number of 1’s in a random byte is four,

c = 3, if the number of 1’s in a random byte is five,

c = 4, if the number of 1’s in a random byte is more than five.

The probability distribution of c is the following:

The test interprets c as a selection of a random letter from the alphabet {a, b, c, d, e} with the probabilities q0, q1, q3, q4 respectively. Thus, the sequence of random bytes b0, b1, b2, ... corresponds with the defined sequence of random letters l0, l1, l2, ... . The test forms overlapping words of length four: v1= l1l2l3l4, v2= l2l3l4l5, ... and length five: w1= l1l2l3l4l5, w2= l2l3l4l5l6, ... from this sequence. The test computes the frequencies of getting each of 625 of possible four-letter words and of 3,125 of possible five-letter words for 2,560,000 of the obtained words. According to these frequencies, the test makes the chi-square statistics V1 and V2 for the four- and five-letter words respectively. The test takes into account the covariance of the frequencies of the fallouts of four-letter and five-letter words and performs the chi-square test for the V2 - V1 statistic. The V2 -V1 statistic is asymptotically normal with a mean a = 2500 and standard deviation s = 70.71. The result of the first level test is the p-value.

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

Second Level Test

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

Final Result Interpretation

The second level test performs ten times. The test computes the FAIL percentage of the failed second level tests. 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 WH and SFMT19937 generators use all the four elements to form a bit sequence.

Parent topic: BRNG Test Descriptions
Level Two Title