Notes for Intel® oneAPI Math Kernel Library Vector Statistics

Download PDF
ID 772987
Date 12/04/2020
Public
Document Table of Contents
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 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

Creating and Initializing Random Streams

VS assumes that at any moment during the program execution you may simultaneously use several random number subsequences generated by one or more BRNGs. Consider the following scenarios:

  1. The simulation system has several independent structural blocks of random number generation. For example, one block generates random numbers of normal distribution, another block generates uniformly distributed numbers, and so on. Each of the blocks should generate an independent random number sequence, that is, each block is assigned an individual stream that generates random numbers of a given distribution.

  2. You need to study correlation properties of the simulation output with different distribution parameters. In this case, it looks natural to assign an individual random number stream (subsequence) to each set of the parameters. For example, see [Mikh2000].

  3. Each parallel process (computational node) requires an independent random number subsequence of a given distribution, that is, a random number stream.

A random stream means a certain abstract source of random numbers. By linking such a stream to a specific BRNG and assigning specific initial values, you can predetermine the random number sequence produced by this particular stream. In VS, a universal stream state descriptor identifies every random number stream (in C language this is just a pointer to the structure). The descriptor specifies the dynamically allocated memory space that contains information on the respective BRNG and its current state, as well as some additional data necessary for the leapfrog and/or skip-ahead method.

VS has two stream creation and initialization functions:

vslNewStream( stream, brng, seed )
vslNewStreamEx( stream, brng, n, params )

Each of these subroutines allocates memory space to store information on the basic generator brng, its current state, etc., and then calls the initialization function of the basic generator brng that fills the fields of the generator current state with relevant initial values. The initial values are defined either by a single 32-bit value seed (for vslNewStream) or an array of n 32-bit initial values params (for vslNewStreamEx). The output of vslNewStream and vslNewStreamEx is a pointer to stream, that is, the stream state descriptor.

To initialize a non-deterministic random number generator, use the same NewStream-based mechanism as shown below:

 errstatus = vslNewStream( stream, VSL_BRNG_NONDETERM, VSL_BRNG_RDRAND );
 nretries = 5;
 params[0] = VSL_BRNG_RDRAND;
 params[1] = nretries;
 errstatus = vslNewStreamEx( stream, VSL_BRNG_NONDETERM, 2, params );

If the underlying hardware does not support this non-deterministic generator, creation and initialization function returns the corresponding error code.

You can create any number of streams through multiple calls of vslNewStream or vslNewStreamEx functions. For example, you can generate several thread-safe streams that are linked to the same BRNG.

The generated streams are further identified by their stream state descriptors. Although a random number stream is a source of random numbers produced by a BRNG, that is, a generator of uniform distribution, you can generate random numbers of non-uniform distribution using streams. To do this, the stream state descriptor is passed to the transformation function that generates random numbers of a given distribution. Each function uses the stream state descriptor to produce random numbers of a uniform distribution, which are further transformed into sequences of the required distribution. See section Generating Methods for Random Numbers of Non-Uniform Distribution for details.

When a given random number stream is no longer needed, delete it by calling vslDeleteStream function:

vslDeleteStream( stream )

This function frees the memory space related to the stream state descriptor stream. After that, you can no longer use the descriptor.

NOTE:

You can use non-deterministic random number generators only if the underlying hardware provides the respective support. For example, see Chapter 8 in [AVX] or Chapter 4 in [BMT] for instructions on how to determine whether an Intel® processor supports a non-deterministic random number generator.

Parent topic: Random Streams and RNGs in Parallel Computation
Level Two Title