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

Craps Test

Test Purpose

The test evaluates the randomness of the output sequence of random numbers of the uniform distribution that imitates the process of dice tossing when gambling Craps. The stable response is the number of tosses of the pair of dice necessary to complete the game and the frequency of wins in the game.

First Level Test

The test forms a sequence of random numbers equiprobably taking the values from 1 to 6 from the output sequence of random numbers. The test treats every number as a number of spots on the face of a die. Thus the test regards a pair of numbers as the result of a toss of two dice. If on the first throw of dice the sum of the spots on the faces of dice equals to 7 or 11, it is a win; if the sum equals 2, 3 or 12, it is a loss. In other cases it is necessary to make additional throws to define the result of the game.

The test performs additional throws until the sum of the spots equals to 7 or coincides with the sum thrown on the first throw. If the sum equals to 7, it is a loss, otherwise, it is a win.

The theoretical probability of the win is 244/495, that is, a little less than 0.5. Further, the frequency of wins with the K-multiple repeats of the game, when K = 200,000, has a very close to normal distribution with mean a = K*244/495 and standard deviation s = a*251/495.

The number of throws necessary to complete the game can take the 1,2, ... values. On K-multiple iterations of the game, the test computes the frequencies of getting c = 1, c = 2, ..., c = 20, c > 20. Based on these frequencies, the test makes the chi-square statistics V with the chi-square distribution with 20 degrees of freedom.

The result of the first level test is the pair of p-values p and q for the number of tosses and the frequency of wins respectively.

Second Level Test

The test performs the first level test ten times. The result of each iteration of the first level test is the pair of p-values pj and qj, j = 1, 2, ..., 10. 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. Similarly, the test applies the Kolmogorov-Smirnov goodness-of-fit test with Anderson-Darling statistics to the obtained p-values of qj, j = 1, 2, ..., 10. If the resulting p-value is q < 0.05 or q > 0.95, the s test fails. The test passes in all other cases.

Final Result Interpretation

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

Tested Generators

Function Name Application 
vsRngUniform
 applicable 
vdRngUniform
 applicable 
viRngUniform
 applicable 
viRngUniformBits
 applicable
Parent topic: BRNG Test Descriptions
Level Two Title