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Chi

Chi distribution.

Installation

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

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

chi

Chi distribution.

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

The namespace contains the following distribution functions:

• cdf( x, k ): Chi distribution cumulative distribution function.
• logpdf( x, k ): evaluate the natural logarithm of the probability density function (PDF) for a chi distribution .
• pdf( x, k ): Chi distribution probability density function (PDF).
• quantile( p, k ): Chi distribution quantile function.

The namespace contains the following functions for calculating distribution properties:

• entropy( k ): Chi distribution differential entropy.
• kurtosis( k ): Chi distribution excess kurtosis.
• mean( k ): Chi distribution expected value.
• mode( k ): Chi distribution mode.
• skewness( k ): Chi distribution skewness.
• stdev( k ): Chi distribution standard deviation.
• variance( k ): Chi distribution variance.

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

• Chi( [k] ): Chi distribution constructor.

var Chi = require( '@stdlib/stats-base-dists-chi' ).Chi;

var dist = new Chi( 4.0 );

var mu = dist.mean;
// returns ~1.88

Examples

var chiRandomFactory = require( '@stdlib/random-base-chi' ).factory;
var filledarrayBy = require( '@stdlib/array-filled-by' );
var variance = require( '@stdlib/stats-base-variance' );
var linspace = require( '@stdlib/array-base-linspace' );
var rayleigh = require( '@stdlib/stats-base-dists-rayleigh' );
var absdiff = require( '@stdlib/math-base-utils-absolute-difference' );
var mean = require( '@stdlib/stats-base-mean' );
var abs = require( '@stdlib/math-base-special-abs' );
var max = require( '@stdlib/math-base-special-max' );
var chi = require( '@stdlib/stats-base-dists-chi' );

// Define the degrees of freedom parameter:
var k = 2;

// Generate an array of x values:
var x = linspace( 0, 10, 100 );

// Compute the PDF for each x:
var chiPDF = chi.pdf.factory( k );
var pdf = filledarrayBy( x.length, 'float64', chiPDF );

// Compute the CDF for each x:
var chiCDF = chi.cdf.factory( k );
var cdf = filledarrayBy( x.length, 'float64', chiCDF );

// Output the PDF and CDF values:
console.log( 'x values: ', x );
console.log( 'PDF values: ', pdf );
console.log( 'CDF values: ', cdf );

// Compute statistical properties:
var theoreticalMean = chi.mean( k );
var theoreticalVariance = chi.variance( k );
var theoreticalSkewness = chi.skewness( k );
var theoreticalKurtosis = chi.kurtosis( k );

console.log( 'Theoretical Mean: ', theoreticalMean );
console.log( 'Theoretical Variance: ', theoreticalVariance );
console.log( 'Skewness: ', theoreticalSkewness );
console.log( 'Kurtosis: ', theoreticalKurtosis );

// Generate random samples from the Chi distribution:
var rchi = chiRandomFactory( k );
var n = 1000;
var samples = filledarrayBy( n, 'float64', rchi );

// Compute sample mean and variance:
var sampleMean = mean( n, samples, 1 );
var sampleVariance = variance( n, 1, samples, 1 );

console.log( 'Sample Mean: ', sampleMean );
console.log( 'Sample Variance: ', sampleVariance );

// Compare sample statistics to theoretical values:
console.log( 'Difference in Mean: ', abs( theoreticalMean - sampleMean ) );
console.log( 'Difference in Variance: ', abs( theoreticalVariance - sampleVariance ) );

// Demonstrate the relationship with the Rayleigh distribution when k=2:
var rayleighPDF = rayleigh.pdf.factory( 1.0 );
var rayleighCDF = rayleigh.cdf.factory( 1.0 );

// Compute Rayleigh PDF and CDF for each x:
var rayleighPDFValues = filledarrayBy( x.length, 'float64', rayleighPDF );

var rayleighCDFValues = filledarrayBy( x.length, 'float64', rayleighCDF );

// Compare Chi and Rayleigh PDFs and CDFs:
var maxDiffPDF = 0.0;
var maxDiffCDF = 0.0;
var diffPDF;
var diffCDF;
var i;
for ( i = 0; i < x.length; i++ ) {
    diffPDF = absdiff( pdf[ i ], rayleighPDFValues[ i ] );
    maxDiffPDF = max( maxDiffPDF, diffPDF );
    diffCDF = absdiff( cdf[ i ], rayleighCDFValues[ i ] );
    maxDiffCDF = max( maxDiffCDF, diffCDF );
}
console.log( 'Maximum difference between Chi(k=2) PDF and Rayleigh PDF: ', maxDiffPDF );
console.log( 'Maximum difference between Chi(k=2) CDF and Rayleigh CDF: ', maxDiffCDF );

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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.