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 Test Purpose Final Result Interpretation Tested Generators
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

Template Test

Test Purpose

The test evaluates the conformity of the generator output with the template sequence of random numbers. The test forms the specified output integer sequence x1, x2, ..., xk, xk+1, ... from the recurrence specifying initial conditions. The parameters of the recurrences are selected such that the output sequences possess "good" properties (good multidimensional uniformity, large period, etc.). If the test computes any member of sequence xk incorrectly, that results in incorrect computing of the other members xk+1, ... of the sequence. Moreover, if xk differs from the correct (template) sequence in one bit, the subsequent members of sequence may differ significantly from the template sequence. The quality of the obtained sequence is probably much worse than the quality of the template sequence. That is why all VS BRNGs undergo thorough tests for template sequences conformity.

The test also checks the BRNGs with the random output numbers u1, u2, ..., uk, uk+1, ..., uniformly distributed over the (a, b) interval for the template output conformity.

Obviously, the output sequences are different for real arithmetic of single and double precision. For the integer output every number should coincide bitwisely with the template number. For the real output numbers, such coincidence is not necessary. The upper bits of the mantissa of the real output determine the quality of the output sequence; the lower bits do not influence randomness. For example, the coincidence of the upper binary digits of the mantissa is sufficient for most applications. (See the chapter Spectral Test in [Knuth81]).

This test is also used to validate VS basic quasi-random number generators.

Final Result Interpretation

The final result is the number of the sequence members that do not coincide with the template members. The value should be equal to 0.

For real sequences the test assumes that the sequence member coincides with the template member, if at least 8 upper binary digits of the mantissa coincide.

Tested Generators

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