Distributions Template Parameter Method

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Date 11/07/2023

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Distributions Template Parameter Method

Method Type	Distributions	Math Description
`uniform_method::standard` `uniform_method::accurate`	`uniform`	Standard method. Currently there is only one method for these functions. `uniform_method::accurate` checks for additional `s` and `d` data types. For `integer` data types, it uses `d` as a `BRNG` data type (`sBRNG` data type is used in `uniform_method::standard` method on GPU).
`gaussian_method::box_muller`	`gaussian`	Generates normally distributed random number x thru the pair of uniformly distributed numbers u1 and u2 according to the formula:
`gaussian_method::box_muller2`	`gaussian`	Generates normally distributed random numbers x1 and x2 thru the pair of uniformly distributed numbers u1 and u2 according to the formulas: Lognormal distribution: generated normally distributed random numbers x1 and x2 are converted to lognormal distribution.
`gaussian_method::icdfgeometric_method::icdf`	`gaussian` `geometric`	Inverse cumulative distribution function (ICDF) method.
`exponential_method::icdfexponential_method::icdf_accurate`	`exponential`	Inverse cumulative distribution function (ICDF) method.
`weibull_method::icdf` `weibull_method::icdf_accurate`	`weibull`	Inverse cumulative distribution function (ICDF) method.
`cauchy_method::icdf`	`cauchy`	Inverse cumulative distribution function (ICDF) method.
`rayleigh_method::icdf` `rayleigh_method::icdf_accurate`	`rayleigh`	Inverse cumulative distribution function (ICDF) method.
`lognormal_method::icdflognormal_method::icdf_accurate`	`lognormal`	Inverse cumulative distribution function (ICDF) method.
`lognormal_method::box_muller2lognormal_method::box_muller2_accurate`	`lognormal`	Normally distributed random numbers x1 and x2 are produced through the pair of uniformly distributed numbers u1 and u2 according to the formulas: Then x1 and x2 are converted to lognormal distribution.
`gumbel_method::icdf`	`gumbel`	Inverse cumulative distribution function (ICDF) method.
`bernoulli_method::icdf`	`bernoulli`	Inverse cumulative distribution function (ICDF) method.
`gamma_method::marsagliagamma_method::marsaglia_accurate`	`gamma`	For `α > 1`, a gamma distributed random number is generated as a cube of properly scaled normal random number; for `0.6 ≤α < 1`, a gamma distributed random number is generated using rejection from Weibull distribution; for `α < 0.6`, a gamma distributed random number is obtained using transformation of exponential power distribution; for `α = 1`, gamma distribution is reduced to exponential distribution.
`beta_method::cja` `beta_method::cja_accurate`	`beta`	Cheng-Johnk-Atkinson method. For `min(p, q) > 1`, Cheng method is used; for `min(p, q) < 1`, Johnk method is used, if `q + K·p2+ C≤ 0 (K = 0.852..., C=-0.956...)` otherwise, Atkinson switching algorithm is used; for `max(p, q) < 1`, method of Johnk is used; for `min(p, q) < 1, max(p, q)> 1`, Atkinson switching algorithm is used (CJA stands for Cheng, Johnk, Atkinson); for `p = 1or q = 1`, inverse cumulative distribution function method is used; for `p = 1` and `q = 1`, beta distribution is reduced to uniform distribution.
`chi_square_method::gamma_based`	`chi_square`	(most common): If `ν ≥ 17` or ν is odd and `5 ≤ ν ≤ 15`, a chi-square distribution is reduced to a Gamma distribution with these parameters: Shape `α = ν / 2` Offset a = 0 Scale factor `β = 2`. The random numbers of the Gamma distribution are generated.
`gaussian_mv_method::box_mullergaussian_mv_method::box_muller2gaussian_mv_method::icdf`	`gaussian_mv`	BoxMuller method for multivariate Gaussian distribution. BoxMuller_2 method for multivariate Gaussian distribution. Inverse cumulative distribution function (ICDF) method.
`binomial_method::btpe`	`binomial`	Acceptance/rejection method for `ntrial·min(p, 1p) ≥ 30` with decomposition into four regions: Two parallelograms Triangle Left exponential tail Right exponential tail
`poisson_method::ptpe`	`poisson`	Acceptance/rejection method for `λ≥ 27` with decomposition into four regions: Two parallelograms Triangle Left exponential tail Right exponenetial tail
`poisson_method::gaussian_icdf_basedpoisson_v_method::gaussian_icdf_based`	`poissonpoisson_v`	for `λ≥ 1`, method based on Poisson inverse CDF approximation by Gaussian inverse CDF; for `λ < 1`, table lookup method is used.
`hypergeometric_method::h2pe`	`hypergeometric`	Acceptance/rejection method for large mode of distribution with decomposition into three regions: Rectangular Left exponential tail Right exponential tail
`negative_binomial_method::nbar`	`negative_binomial`	Acceptance/rejection method for: with decomposition into five regions: Rectangular (2) trapezoid Left exponential tail Right exponential tail
`multinomial_method::poisson_icdf_based`	`multinomial`	Multinomial distribution with parameters `m`, `k`, and a probability vector `p`. Random numbers of the multinomial distribution are generated by Poisson Approximation method.

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Distributions Template Parameter Method