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 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

Special Properties

To improve the efficiency of the algorithms based on the general methods described above, you may have to use special properties of distributions. For example, use of polar coordinates for a pair of independent normal variates enables you to develop an efficient method of random number generation based on 2D inverse transformation known as the Box-Muller method:

Generating s-dimensional normally distributed quasi-random sequences with 2D inverse transformation (the VS name is the Box-Muller2 method) is problematic when s is odd, because quasi-random numbers are generated in pairs. One of the options is to generate (s+1)-dimensional normally distributed quasi-random numbers from (s+1)-dimensional quasi-random numbers produced by a basic quasi-random generator and then ignore the last dimension.

Another option is to use the method that produces one normally distributed number from two uniform ones (the VS name is the Box-Muller method). In this case, to generate s-dimensional normally distributed quasi-random numbers, use 2s-dimensional quasi-random numbers produced by a basic quasi-random generator.

For a binomial distribution with parameters m, p, the probability mass function is as follows:

For p > 0.5, it is convenient to use the fact that

Thus, you can convert a uniform distribution to a general distribution using a number of methods. Two different transformation techniques implemented for the same uniform distribution produce two different sequences of a general distribution, though possessing the same statistical properties.

Consider a simple example. If U1, U2 are two independent random values uniformly distributed over the interval (0, 1), that is, with the distribution function F(x) = x , 0 < x < 1, then the variate X = max(U1, U2) has the distribution F(x) ·F(x). On the one hand, the random number x1 with the maximum distribution from two independent uniform distributions may be derived either from a pair of uniformly distributed random numbers u1, u2 as x1 = max(u1, u2) or from one uniform random number u1 as x1 = sqrt(u1) by applying the inverse transformation method. It is obvious that applying two different methods to one and the same sequence u1, u2, u3, ... gives two absolutely different sequences xi.

You can perform transformations into non-uniform distribution sequences using various methods. The inverse transformation method may be preferable over the acceptance/rejection method for some applications and architectures, while the reverse preference is common for others. Taking this into account, the VS interface provides different options of random number generation for the same probability distribution. For example, a Poisson distribution may be transformed by two different methods: the first, known as PTPE [Schmeiser81], is based on acceptance/rejection and mixture of distributions techniques, while the second one is implemented through transformation of normally distributed random numbers.

The method number calls a method for a specified generator, for example: 

viRngPoisson( VSL_METHOD_IPOISSON_PTPE, stream, n, r, lambda )

- calling the PTPE method by passing the method number VSL_METHOD_IPOISSON_PTPE. 

viRngPoisson( VSL_METHOD_IPOISSON_POISNORM, stream, n, r, lambda )

- calling transformation from normally distributed random numbers by passing the method number VSL_METHOD_IPOISSON_POISNORM.

For details on methods to be used for specific distributions, see Continuous Distribution Random Number Generators and Discrete Distribution Random Number Generators sections.

Parent topic: Generating Methods for Random Numbers of Non-Uniform Distribution
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