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
BRNG Properties and Testing Results x
MCG31m1 R250 MRG32k3a MCG59 WH Real Implementation (Single and Double Precision) Integer Implementation Stream Initialization by Function vslNewStream Stream Initialization of Function vslNewStreamEx Subsequences Selection Methods Generator Period Empirical Testing Results Summary 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

WH

This is a set of 273 Wichmann-Hill’s combined multiplicative congruential generators (j = 1, 2, ..., 273):

WH is a set of 273 different basic generators. This generator is the second basic generator in the NAG libraries. The constants ai,j  range from 112 to 127, the constants mi,j are prime numbers ranging from 16,718,909 to 16,776,971, close to 224. These constants should show good results in the spectral test (see Knuth [Knuth81] and MacLaren [MacLaren89]). The period of each Wichmann-Hill generator may be equal to 292 if not for common factors between (m1,j-1), (m2,j-1), (m3,j-1) and (m4,j-1). However, each generator should still have a period of at least 280. The generated pseudo-random sequences are essentially independent of one another according to the spectral test (for detailed information about properties of these generators see [MacLaren89]).

Real Implementation (Single and Double Precision)

The output vector is the sequence of the floating-point values u0, u1, ...

Integer Implementation

The output vector of 32-bit integers x0, y0, z0, w0, x1, y1, z1, w1

Thus, the output vector stores practically every quadruple (x, y, z, w) of members of the integer output as four 32-bit integers. For example, to get a vector from n quadruples (x, y, z, w), the size of the output array should be large enough to store 4n 32-bit numbers.

Stream Initialization by Function vslNewStream

WH generates the stream and initializes it specifying the 32-bit input integer parameter seed :

  1. Assume x0 = seed mod m1. If x0 = 0, assume x0 = 1.

  2. Assume y0 = 1, z0 = 1, w0 = 1.

The test selects a WH generator adding an offset to the named constant VSL_BRNG_WH: VSL_BRNG_WH+0, VSL_BRNG_WH+1, ... , VSL_BRNG_WH+272. The following example illustrates the initialization of the seventh (of 273) WH generator: 

vslNewStream (&stream, VSL_BRNG_WH+6, seed);

Stream Initialization of Function vslNewStreamEx

WH generates the stream and initializes it specifying the the array params[] of n 32-bit integers:

  1. If n = 0, assume x0 = 1, y0 = 1, z0 = 1, w0 = 1.

  2. If n = 1, assume x0 = params[0] mod m1, y0 = 1, z0 = 1, w0 = 1.

    If x0 = 0, assume x0 =1.

  3. If n = 2, assume x0 = params[0] mod m1, y0 = params[1] mod m2, z0 = 1, w0 = 1.

    If x0 = 0, assume x0 = 1.

    If y0 = 0, assume y0 = 1.

  4. If n = 3, assume x0 = params[0] mod m1, y0 = params[1] mod m2, z0 = params[2] mod m3, w0 = 1.

    If x0 = 0, assume x0 = 1.

    If y0 = 0, assume y0 = 1.

    If z0 = 0, assume z0 = 1.

  5. If n ≥ 4, assume x0 = params[0] mod m1, y0 = params[1] mod m2, z0= params[2] mod m3, w0 = params[3] mod m4.

    If x0 = 0, assume x0 = 1.

    If y0 = 0, assume y0 = 1.

    If z0 = 0, assume z0 = 1.

    If w0 = 0, assume w0 = 1.

Subsequences Selection Methods

vslSkipAheadStream
 Supported 
vslSkipAheadStreamEx
 Not supported 
vslLeapfrogStream
 Supported

Generator Period

Empirical Testing Results Summary

Test Name 
vsRngUniform
 
vdRngUniform
 
viRngUniform
 
viRngUniformBits
3D Spheres Test OK (0% errors) OK (0% errors) Not applicable OK (0% errors)
Birthday Spacing Test Not applicable Not applicable Not applicable FAIL (60% errors)
Bitstream Test Not applicable Not applicable Not applicable OK (10% errors)
Rank of 31x31 Binary Matrices Test Not applicable Not applicable Not applicable N/A
Rank of 32x32 Binary Matrices Test Not applicable Not applicable Not applicable N/A
Rank of 6x8 Binary Matrices Test Not applicable Not applicable Not applicable OK (0% errors)[6]
Counts-the-1’s Test (stream of bits) Not applicable Not applicable Not applicable OK (10% errors)
Counts-the-1’s Test (stream of specific bytes) Not applicable Not applicable Not applicable OK (0% errors)
Craps Test OK (20% errors) OK (20% errors) OK (20% errors) OK (10% errors)
Parking Lot Test OK (10% errors) OK (10% errors) Not applicable OK (10% errors)
2D Self-Avoiding Random Walk Test OK (10% errors) OK (0% errors) Not applicable OK (20% errors)
NOTE:

  1. The tabulated data is obtained using the one-level (threshold) testing technique. The OK result indicates FAIL < 50%. The run fails when p-value falls outside the interval [0.05, 0.95].

  2. The stream tested is generated by calling the function vslNewStream with seed=7,777,777.

Parent topic: BRNG Properties and Testing Results
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