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) MCG31m1 R250 MRG32k3a MCG59 WH MT19937 MT2203 SFMT19937 SOBOL NIEDERR PHILOX4X32X10 ARS5 NON-DETERMINISTIC 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

UniformBits (VSL_RNG_METHOD_UNIFORMBITS_STD)

A random number generator of uniform distribution that produces an integer (non-normalized to the interval (0, 1)) sequence. You can identify the underlying BRNG by passing the random stream descriptor stream as a parameter. Then UniformBits function calls integer implementation of this basic generator.

Basic generators differ in bit capacity and structure of the integer output, therefore you should interpret the output integer array of the function viRngUniformBits correctly. The following list provides rules for interpreting 32-bit integer output r[i] for each VS BRNG.

MCG31m1

Integer Recurrence

Interpretation of 32-bit integer output array r[i] after calling viRngUniformBits

R250

Integer Recurrence

Interpretation of 32-bit integer output array r[i] after calling viRngUniformBits

MRG32k3a

Integer Recurrence

Interpretation of 32-bit integer output array r[i] after calling viRngUniformBits

MCG59

Integer Recurrence

Interpretation of 32-bit integer output array r[i] after calling viRngUniformBits

WH

Integer Recurrence

Interpretation of 32-bit integer output array r[i] after calling viRngUniformBits

MT19937

Integer Recurrence

,

,

,

,

where

,

with .

Interpretation of 32-bit integer output array r[i] after calling viRngUniformBits

MT2203

Integer Recurrence

,

where

,

with , .

Interpretation of 32-bit integer output array r[i] after calling viRngUniformBits

SFMT19937

Integer Recurrence

Interpretation of 32-bit integer output array r[i] after calling viRngUniformBits

r[i] = wi/4(i % 4)

SOBOL

Integer Recurrence

,

where

,

and s is the dimension of quasi-random vector.

Interpretation of 32-bit integer output array r[i] after calling viRngUniformBits

NIEDERR

Integer Recurrence

,

where

,

and s is the dimension of quasi-random vector.

Interpretation of 32-bit integer output array r[i] after calling viRngUniformBits

PHILOX4X32X10

Integer Recurrence

cn = cn-1 + 1

wn = f(cn)

f(c) is computed as follows:

k00 = k0

k10 = k1

f(c)= , N= 10

Interpretation of 32-bit integer output array r[i] after calling viRngUniformBits

r[i] = wi/4(i % 4)

wi(k) is the k-th 32-bit integer in quadruple wn, k= 0, 1, 2, 3

ARS5

Integer Recurrence

cn = cn-1 + 1

wn = f(cn)

f(c) is computed as follows:

c0 = c xor kand k0 = k

ci+1= SubBytes(c)

ci+1= ShiftRows(ci+1)

ci+1= MixColumns(ci+1), omitted if i + 1 = N

ci+1= AddRoundKey(ci+1,kj)

Lo(ki+1)=Lo(k)+ 0x9E3779B97F4A7C15

Hi(ki+1)= Hi(k)+ 0xBB67AE8584CAA73B

f(c) = cN, N= 5

Interpretation of 32-bit integer output array r[i] after calling viRngUniformBits

r[i] = wi/4(i % 4)

wi(k) is the k-th 32-bit integer in quadruple wn, k= 0, 1, 2, 3

NON-DETERMINISTIC

Integer Recurrence

Non-deterministic random generator [BMT] available in the latest Intel® CPUs [AVX].

Interpretation of 32-bit integer output array r[i] after calling viRngUniformBits

NOTE:

  1. Lo(x) means obtaining lower 32 bits of the 64-bit unsigned integer x, that is, Lo(x) = xmod232.

  2. Hi(x) means obtaining upper 32 bits of the 64-bit unsigned integer x, that is, Hi(x) = ⌊x/232⌋.

  3. ABCD means a 128-bit number composed of four 32-bit numbers A, B, C and D, that is

So, when you generate an integer sequence of n elements, the output array r[i] of the function viRngUniformBits comprises:

  1. n elements for the basic generators MCG31m1, R250, MRG32k3a, MT19937, MT2203, SOBOL, NIEDERR, Philox4x32-10 and ARS5.

  2. 2n elements for the basic generator MCG59.

  3. 4n elements for the basic generators WH and SFMT19937.

You may use the integer output, in particular, for fast generation of bit vectors. However, in this case some bits (or groups of them) might be non-random. For example, lower bits produced by linear congruential generators are less random than their higher bits. Note that quasi-random numbers are not random at all. Thoroughly check the integer output bits and bit groups for randomness before forming bit vectors from r[i] array.

See Intel® oneAPI Math Kernel Library Vector Statistics Random Number Generator Performance Data for test results summary.

Parent topic: Discrete Distribution Random Number Generators
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