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Gumbel

Gumbel distribution.

Installation

npm install @stdlib/stats-base-dists-gumbel

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var gumbel = require( '@stdlib/stats-base-dists-gumbel' );

gumbel

Gumbel distribution.

var dist = gumbel;
// returns {...}

The namespace contains the following distribution functions:

• cdf( x, mu, beta ): Gumbel distribution cumulative distribution function.
• logcdf( x, mu, beta ): Gumbel distribution logarithm of cumulative distribution function.
• logpdf( x, mu, beta ): Gumbel distribution logarithm of probability density function (PDF).
• mgf( t, mu, beta ): Gumbel distribution moment-generating function (MGF).
• pdf( x, mu, beta ): Gumbel distribution probability density function (PDF).
• quantile( p, mu, beta ): Gumbel distribution quantile function.

The namespace contains the following functions for calculating distribution properties:

• entropy( mu, beta ): Gumbel distribution differential entropy.
• kurtosis( mu, beta ): Gumbel distribution excess kurtosis.
• mean( mu, beta ): Gumbel distribution expected value.
• median( mu, beta ): Gumbel distribution median.
• mode( mu, beta ): Gumbel distribution mode.
• skewness( mu, beta ): Gumbel distribution skewness.
• stdev( mu, beta ): Gumbel distribution standard deviation.
• variance( mu, beta ): Gumbel distribution variance.

The namespace contains a constructor function for creating a Gumbel distribution object.

• Gumbel( [mu, beta] ): Gumbel distribution constructor.

var Gumbel = require( '@stdlib/stats-base-dists-gumbel' ).Gumbel;

var dist = new Gumbel( 2.0, 4.0 );

var y = dist.pdf( 2.0 );
// returns ~0.092

Examples

var Float64Array = require( '@stdlib/array-float64' );
var filledarrayBy = require( '@stdlib/array-filled-by' );
var mean = require( '@stdlib/stats-base-mean' );
var variance = require( '@stdlib/stats-base-variance' );
var stdev = require( '@stdlib/stats-base-stdev' );
var randGumbel = require( '@stdlib/random-base-gumbel' ).factory;
var gumbel = require( '@stdlib/stats-base-dists-gumbel' );

// Set the parameters of the Gumbel distribution:
var mu = 30.0;   // Location parameter (e.g., average annual maximum temperature in °C)
var beta = 5.0;  // Scale parameter

// Simulate annual maximum daily temperatures over 1000 years:
var N = 1000;
var rgumbel = randGumbel( mu, beta );
var maxTemperatures = filledarrayBy( N, 'float64', rgumbel );

// Compute theoretical statistics of the Gumbel distribution:
var theoreticalMean = gumbel.mean( mu, beta);
var theoreticalVariance = gumbel.variance( mu, beta );
var theoreticalStdev = gumbel.stdev( mu, beta );

// Compute sample statistics of the simulated data:
var sampleMean = mean( N, maxTemperatures, 1 );
var sampleVariance = variance( N, 1, maxTemperatures, 1 ); // with Bessel's correction
var sampleStdev = stdev( N, 1, maxTemperatures, 1 ); // with Bessel's correction

// Display theoretical and sample statistics:
console.log( '--- Statistical Comparison ---\n' );
console.log( 'Mean:');
console.log( '  Theoretical: %d°C', theoreticalMean.toFixed(2) );
console.log( '  Sample:      %d°C\n', sampleMean.toFixed(2) );
console.log( 'Variance:');
console.log( '  Theoretical: %d°C²', theoreticalVariance.toFixed(2) );
console.log( '  Sample:      %d°C²\n', sampleVariance.toFixed(2) );
console.log( 'Standard Deviation:' );
console.log( '  Theoretical: %d°C', theoreticalStdev.toFixed(2) );
console.log( '  Sample:      %d°C\n', sampleStdev.toFixed(2) );

// Define quantile probabilities:
var p = new Float64Array( [ 0.25, 0.5, 0.75 ] );
var label = [ 'First Quartile', 'Median', 'Third Quartile' ];
var theoreticalQuantiles = new Float64Array([
    gumbel.quantile( p[0], mu, beta ),
    gumbel.quantile( p[1], mu, beta ),
    gumbel.quantile( p[2], mu, beta )
]);

console.log( 'Quantiles:' );
var i;
for ( i = 0; i < p.length; i++ ) {
    console.log( label[i] + ': %d°C', theoreticalQuantiles[i].toFixed(2) );
}

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Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

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License

See LICENSE.

Copyright

Copyright © 2016-2024. The Stdlib Authors.