GSL-RANDIST

This module provides functionality to generate random variates from distributions.

Polymorphism and multimethods were used to simplify the original GSL library interface, reducing both the function count and the number of parameters per function. See <gsl-randist> for examples of this design.

The distributions available in this module are:

The <gsl-randist> interface

<gsl-randist> Abstract Uninstantiable Class

Base class for all distributions.

Init-Keywords:

  • rng – Random number generator. Used to generate random variates from the distribution. If not provided a new instance of <gsl-rng> is provided. This generator can be configured by environment variables (see Random number environment variables)

Operations:

Operations

gsl-randist-variate Generic function
Signature:

gsl-randist-variate distribution => (variate)

Generates a random variate from the distribution.

Parameters:
Result variate:

A random variate from the distribution. An instance of <float>.

gsl-randist-pdf Generic function

Evaluates the probability density function (PDF) for the distribution.

Signature:

gsl-randist-pdf (distribution, x) => (pd)

Parameters:

  • distribution – An instance of <gsl-randist>.

  • x – The value at which to evaluate the probability density function. An instance of <float>.

Result pd:

The probability density at x. An instance of <float>.

The Continuous Distribution

<gsl-randist-continuous> Abstract Uninstantiable Class
Superclasses:

<gsl-randist>

Base class for all continuous distributions.

Operations:

The Continuous distributions available are:

gsl-randist-cdf-p Generic function

Evaluates the cumulative distribution function (CDF) for the distribution.

Signature:

gsl-randist-cdf-p distribution x => (cd)

Parameters:

  • distribution – An instance of <gsl-randist-continuous>.

  • x – The value at which to evaluate the cumulative distribution function. An instance of <float>.

Result cd:

The cumulative probability at x. An instance of <float>.

gsl-randist-cdf-q Generic function
Signature:

gsl-randist-cdf-q distribution x => (cd)

Parameters:
Result cd:

The survival function at x. An instance of <float>.

gsl-randist-cdf-pinv Generic function

Evaluates the inverse cumulative distribution function (CDF) for the distribution.

Signature:

gsl-randist-cdf-pinv distribution x => (cd)

Parameters:

  • distribution – An instance of <gsl-randist-continuous>.

  • x – The value at which to evaluate the inverse cumulative distribution function. An instance of <float>.

Result cd:

The inverse cumulative distribution function at x. An instance of <float>.

gsl-randist-cdf-qinv Generic function

Evaluates the inverse cumulative distribution function (CDF) for the distribution.

Signature:

gsl-randist-cdf-qinv distribution x => (cd)

Parameters:
Result cd:

The inverse survival function at x. An instance of <float>.

The Gaussian Distribution

<gsl-randist-gaussian> Concrete Instantiable Class

A Gaussian distribution. This returns random values of Gaussian probability density function. See the original GSL documentation for more information at Gaussian Distribution. This distribution stores the type of algorithm used to generate random variates.

Superclasses:

<gsl-randist-continuous>

Init-Keywords:

  • sigma (required) – Standard deviation. An instance of <float>.

  • algorithm – The algorithm to use for generating random variates. See Gaussian algorithms for more information.

Conditions:

A <gsl-error-invalid-argument> condition is raised if the algorithm is not one of <gaussian-algorithm>.

Example:
let r = make(<gsl-rng>);
let d = make(<gsl-randist-gaussian>, sigma: 0.5d0, rng: r);
let k = d.gsl-randist-variate;
Example:
let d = make(<gsl-gaussian>,
             rng: make(<gsl-rng>),
             sigma: 0.5d0,
             algorithm: #"ratio");

let k = d.gsl-randist-variate;
seealso:

https://www.gnu.org/software/gsl/doc/html/randist.html#the-gaussian-distribution

The following operations are defined for <gsl-randist-gaussian>. See operations in <gsl-randist-continuous> for more information.

gsl-randist-variate(<gsl-randist-gaussian>) Method
gsl-randist-pdf(<gsl-randist-gaussian>) Method
gsl-randist-cdf-p(<gsl-randist-gaussian>) Method
gsl-randist-cdf-q(<gsl-randist-gaussian>) Method
gsl-randist-cdf-pinv(<gsl-randist-gaussian>) Method
gsl-randist-cdf-qinv(<gsl-randist-gaussian>) Method

The Unit Gaussian distribution

<gsl-randist-ugaussian> Concrete Instantiable Class

Unit Gaussian is a Gaussian distribution with sigma = 1.0d0.

Superclasses:

<gsl-randist-gaussian>

Key algorithm:

The algorithm to use for generating random variates. See Gaussian algorithms for more information.

Discussion:

The keyword sigma: must not be used since it is initialized by the class.

Example:
let d = make(<gsl-randist-ugaussian>, rng: make(<gsl-rng>));
let v = d.gsl-randist-variate;
See also:

Gaussian algorithms

<gaussian-algorithm> Class

This are the algorithms to generate a Gaussian variate value. This algorithm is passed as an optional keyword algorithm: to <gsl-randist-gaussian> class. The valid values are:

#"box-muller

Box-Muller algorithm. This is the default value.

#"ziggurat"

Computes a Gaussian random variate using the alternative Marsaglia-Tsang ziggurat method. The Ziggurat algorithm is the fastest available algorithm in most cases.

#"ratio-method"

Kinderman-Monahan-Leva ratio method.

The Gaussian tail distribution

<gsl-randist-gaussian-tail> Concrete Instantiable Class
Superclasses:

<gsl-randist-gaussian>

Init-Keywords:

  • rng (required)

  • sigma (required)

  • a (required) – An instance of <float>. The values returned are larger than this limit, which must be positive.

Conditions:

A <gsl-invalid-argument> is signaled if a keyword is not positive.

Summary:

Provides random variates from the upper tail of a Gaussian distribution with standard deviation sigma. The values returned are larger than the lower limit a, which must be positive.

See also:

The following operations are specialized for <gsl-randist-gaussian-tail>. See operations in <gsl-randist-continuous> for more information.

gsl-randist-variate(<gsl-randist-gaussian-tail>) Method
gsl-randist-pdf(<gsl-randist-gaussian-tail>) Method

The Ugaussian tail distribution

<gsl-randist-ugaussian-tail> Concrete Instantiable Class
Superclasses:

<gsl-randist-gaussian-tail>

Init-Keywords:

  • rng (required)

  • a (required) – An instance of <float>. The values returned are larger than this limit, which must be positive.

Summary:

Compute results for the tail of a unit Gaussian distribution. They are equivalent to the <gsl-randist-gaussian-tail> with a standard deviation of one, sigma = 1.0d0.

The Exponential Distribution

<gsl-randist-exponential> Concrete Instantiable Class
Superclasses:

<gsl-randist-continuous>`

Init-Keywords:

  • mu (required) – Mean mu.

See also:

The following operations are specialized for <gsl-randist-exponential>. See operations in <gsl-randist-continuous> for more information.

gsl-randist-variate(<gsl-randist-exponential>) Method
gsl-randist-pdf(<gsl-randist-exponential>) Method
gsl-randist-cdf-p(<gsl-randist-exponential>) Method
gsl-randist-cdf-q(<gsl-randist-exponential>) Method
gsl-randist-cdf-pinv(<gsl-randist-exponential>) Method
gsl-randist-cdf-qinv(<gsl-randist-exponential>) Method

The Laplace Distribution

<gsl-randist-laplace> Concrete Instantiable Class
Superclasses:

<gsl-randist-continuous>

Init-Keywords:

  • a (required) – Width

See also:

The following operations are specialized for <gsl-randist-laplace>. See operations in <gsl-randist-continuous> for more information.

gsl-randist-variate(<gsl-randist-laplace>) Method
gsl-randist-pdf(<gsl-randist-laplace>) Method
gsl-randist-cdf-p(<gsl-randist-laplace>) Method
gsl-randist-cdf-q(<gsl-randist-laplace>) Method
gsl-randist-cdf-pinv(<gsl-randist-laplace>) Method
gsl-randist-cdf-qinv(<gsl-randist-laplace>) Method

The Exponential Power Distribution

<gsl-randist-exppow> Concrete Instantiable Class
Superclasses:

<gsl-randist-continuous>

Init-Keywords:

  • a (required) – An instance of <float>

  • b (required) – An instance of <float>

See also:

The following operations are specialized for <gsl-randist-exppow>. See operations in <gsl-randist-continuous> for more information.

gsl-randist-variate(<gsl-randist-exppow>) Method
gsl-randist-pdf(<gsl-randist-exppow>) Method
gsl-randist-cdf-p(<gsl-randist-exppow>) Method
gsl-randist-cdf-q(<gsl-randist-exppow>) Method

The Cauchy Distribution

<gsl-randist-cauchy> Concrete Instantiable Class
Superclasses:

<gsl-randist-continuous>

Init-Keywords:

  • a (required) – Scale parameter. An instance of <float>

See also:

The following operations are specialized for <gsl-randist-cauchy>. See operations in <gsl-randist-continous> for more information.

gsl-randist-variate(<gsl-randist-cauchy>) Method
gsl-randist-pdf(<gsl-randist-cauchy>) Method
gsl-randist-cdf-p(<gsl-randist-cauchy>) Method
gsl-randist-cdf-q(<gsl-randist-cauchy>) Method
gsl-randist-cdf-pinv(<gsl-randist-cauchy>) Method
gsl-randist-cdf-qinv(<gsl-randist-cauchy>) Method

The Rayleigh Distribution

<gsl-randist-rayleigh> Concrete Instantiable Class
Superclasses:

<gsl-randist-continuous>

Init-Keywords:

  • sigma (required) – An instance of <float>

See also:

The following operations are specialized for <gsl-randist-rayleigh>. See operations in <gsl-randist-continuous> for more information.

gsl-randist-variate(<gsl-randist-rayleigh>) Method
gsl-randist-pdf(<gsl-randist-rayleigh>) Method
gsl-randist-cdf-p(<gsl-randist-rayleigh>) Method
gsl-randist-cdf-q(<gsl-randist-rayleigh>) Method

The Rayleigh Tail distribution

<gsl-randist-rayleigh-tail> Class
Superclasses:

<gsl-randist-rayleigh>

Init-Keywords:

  • rng (required)

  • sigma (required) – Scale parameter. An instance of <float>.

  • a (required) – An instance of <float>. Lower limit.

The following operations are specialized for <gsl-randist-rayleigh-tail>. See operations in <gsl-randist-continuous> for more information.

gsl-randist-variate(<gsl-randist-rayleigh-tail>) Method
gsl-randist-pdf(<gsl-randist-rayleigh-tail>) Method

The Landau distribution

<gsl-randist-landau> Concrete Instantiable Class
Superclasses:

<gsl-randist-continuous>

Init-Keywords:

  • rng (required)

The following operations are specialized for <gsl-randist-landau>. See operations in <gsl-randist-continuous> for more information.

gsl-randist-variate(<gsl-randist-landau>) Method
gsl-randist-pdf(<gsl-randist-landau>) Method

The Levy alpha-Stable Distribution

<gsl-randist-levy> Concrete Instantiable Class
Superclasses:

<gsl-randist-continuous>

Init-Keywords:

  • rng (required)

  • c (required) – Scale. An instance of <float>.

  • alpha (required) – Scale. An instance of <float>.

The following operations are specialized for <gsl-randist-levy>. See operations in <gsl-randist-continuous> for more information.

gsl-randist-variate(<gsl-randist-levy>) Method

The Levy skew alpha-Stable Distribution

<gsl-randist-levy-skew> Concrete Instantiable Class
Superclasses:

<gsl-randist-levy>

Init-Keywords:

  • rng (required)

  • c (required) – Scale. An instance of <float>.

  • alpha (required) – Scale. An instance of <float>.

  • beta (required) – Skew. An instance of <float>.

The following operations are specialized for <gsl-randist-levy-skew>. See operations in <gsl-randist-continuous> for more information.

gsl-randist-variate(<gsl-randist-levy-skew>) Method

The Gamma Distribution

<gsl-randist-gamma> Concrete Instantiable Class
Superclasses:

<gsl-randist-continuous>

Init-Keywords:

  • algorithm – The algorithm to use for generating random variates. See Gamma algorithms for more information.

  • a (required) – a > 0

  • b (required) – b > 0

See also:

The following operations are specialized for <gsl-randist-gamma>. See operations in <gsl-randist-continuous> for more information.

gsl-randist-variate(<gsl-randist-gamma>) Method
gsl-randist-pdf(<gsl-randist-gamma>) Method
gsl-randist-cdf-p(<gsl-randist-gamma>) Method
gsl-randist-cdf-q(<gsl-randist-gamma>) Method
gsl-randist-cdf-pinv(<gsl-randist-gamma>) Method
gsl-randist-cdf-qinv(<gsl-randist-gamma>) Method

Gamma algorithms

<gsl-gamma-algorithm> Class

This are the algorithms to generate a Gamma variate value. This algorithm is passed as an optional keyword algorithm: to <gsl-randist-gaussian> class. The valid values are:

#"default"

This is the default value.

#"knuth"

This function returns a gamma variate using the algorithms from Knuth (vol 2).

The Flat Distribution

<gsl-randist-flat> Concrete Instantiable Class
Superclasses:

<gsl-randist-continuous>

Init-Keywords:

  • a (required) – Lower limit. An instance of <float>

  • b (required) – Upper limit. An instance of <float>

See also:

Discussion:

Returns a random variate from the flat (uniform) distribution from a to b

The following operations are specialized for <gsl-randist-flat>. See operations in <gsl-randist-continuous> for more information.

gsl-randist-variate(<gsl-randist-flat>) Method
gsl-randist-pdf(<gsl-randist-flat>) Method
gsl-randist-cdf-p(<gsl-randist-flat>) Method
gsl-randist-cdf-q(<gsl-randist-flat>) Method
gsl-randist-cdf-pinv(<gsl-randist-flat>) Method
gsl-randist-cdf-qinv(<gsl-randist-flat>) Method

The Lognormal Distribution

<gsl-randist-lognormal> Concrete Instantiable Class
Superclasses:

<gsl-randist-continuous>

Init-Keywords:

  • zeta (required) – An instance of <float>

  • sigma (required) – An instance of <float>

See also:

The following operations are specialized for <gsl-randist-lognormal>. See operations in <gsl-randist-continuous> for more information.

gsl-randist-variate(<gsl-randist-lognormal>) Method
gsl-randist-pdf(<gsl-randist-lognormal>) Method
gsl-randist-cdf-p(<gsl-randist-lognormal>) Method
gsl-randist-cdf-q(<gsl-randist-lognormal>) Method
gsl-randist-cdf-pinv(<gsl-randist-lognormal>) Method
gsl-randist-cdf-qinv(<gsl-randist-lognormal>) Method

The Chi-Squared Distribution

<gsl-randist-chisq> Concrete Instantiable Class
Superclasses:

<gsl-randist-continuous>

Init-Keywords:

  • nu (required) – Degrees of freedom. An instance of <float>

Example:
let r = make(<gsl-rng>);
let d = make(<gsl-randist-chisq>, nu: 3.0d0, rng: r);
let k = d.gsl-randist-variate;
See also:

The following operations are specialized for <gsl-randist-chisq>. See operations in <gsl-randist-continuous> for more information.

gsl-randist-variate(<gsl-randist-chisq>) Method
gsl-randist-pdf(<gsl-randist-chisq>) Method
gsl-randist-cdf-p(<gsl-randist-chisq>) Method
gsl-randist-cdf-q(<gsl-randist-chisq>) Method
gsl-randist-cdf-pinv(<gsl-randist-chisq>) Method
gsl-randist-cdf-qinv(<gsl-randist-chisq>) Method