@stdlib/stats/base/dists/chi/README.md

Standard library statistical functions.

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# Chi

> Chi distribution.

<section class="usage">

## Usage

```javascript
var chi = require( '@stdlib/stats/base/dists/chi' );
```

#### chi

Chi distribution.

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

The namespace contains the following distribution functions:

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<div class="namespace-toc">

-   <span class="signature">[`cdf( x, k )`][@stdlib/stats/base/dists/chi/cdf]</span><span class="delimiter">: </span><span class="description">Chi distribution cumulative distribution function.</span>
-   <span class="signature">[`logpdf( x, k )`][@stdlib/stats/base/dists/chi/logpdf]</span><span class="delimiter">: </span><span class="description">evaluate the natural logarithm of the probability density function (PDF) for a chi distribution .</span>
-   <span class="signature">[`pdf( x, k )`][@stdlib/stats/base/dists/chi/pdf]</span><span class="delimiter">: </span><span class="description">Chi distribution probability density function (PDF).</span>
-   <span class="signature">[`quantile( p, k )`][@stdlib/stats/base/dists/chi/quantile]</span><span class="delimiter">: </span><span class="description">Chi distribution quantile function.</span>

</div>

<!-- </toc> -->

The namespace contains the following functions for calculating distribution properties:

<!-- <toc pattern="*+(entropy|kurtosis|mean|median|mode|skewness|stdev|variance)*"> -->

<div class="namespace-toc">

-   <span class="signature">[`entropy( k )`][@stdlib/stats/base/dists/chi/entropy]</span><span class="delimiter">: </span><span class="description">Chi distribution differential entropy.</span>
-   <span class="signature">[`kurtosis( k )`][@stdlib/stats/base/dists/chi/kurtosis]</span><span class="delimiter">: </span><span class="description">Chi distribution excess kurtosis.</span>
-   <span class="signature">[`mean( k )`][@stdlib/stats/base/dists/chi/mean]</span><span class="delimiter">: </span><span class="description">Chi distribution expected value.</span>
-   <span class="signature">[`mode( k )`][@stdlib/stats/base/dists/chi/mode]</span><span class="delimiter">: </span><span class="description">Chi distribution mode.</span>
-   <span class="signature">[`skewness( k )`][@stdlib/stats/base/dists/chi/skewness]</span><span class="delimiter">: </span><span class="description">Chi distribution skewness.</span>
-   <span class="signature">[`stdev( k )`][@stdlib/stats/base/dists/chi/stdev]</span><span class="delimiter">: </span><span class="description">Chi distribution standard deviation.</span>
-   <span class="signature">[`variance( k )`][@stdlib/stats/base/dists/chi/variance]</span><span class="delimiter">: </span><span class="description">Chi distribution variance.</span>

</div>

<!-- </toc> -->

The namespace contains a constructor function for creating a [Chi][chi-distribution] distribution object.

<!-- <toc pattern="*ctor*"> -->

<div class="namespace-toc">

-   <span class="signature">[`Chi( [k] )`][@stdlib/stats/base/dists/chi/ctor]</span><span class="delimiter">: </span><span class="description">Chi distribution constructor.</span>

</div>

<!-- </toc> -->

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

var dist = new Chi( 4.0 );

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

</section>

<!-- /.usage -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
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 );
```

</section>

<!-- /.examples -->

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<section class="related">

</section>

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<section class="links">

[chi-distribution]: https://en.wikipedia.org/wiki/Chi_distribution

<!-- <toc-links> -->

[@stdlib/stats/base/dists/chi/ctor]: https://github.com/stdlib-js/stats/tree/main/base/dists/chi/ctor

[@stdlib/stats/base/dists/chi/entropy]: https://github.com/stdlib-js/stats/tree/main/base/dists/chi/entropy

[@stdlib/stats/base/dists/chi/kurtosis]: https://github.com/stdlib-js/stats/tree/main/base/dists/chi/kurtosis

[@stdlib/stats/base/dists/chi/mean]: https://github.com/stdlib-js/stats/tree/main/base/dists/chi/mean

[@stdlib/stats/base/dists/chi/mode]: https://github.com/stdlib-js/stats/tree/main/base/dists/chi/mode

[@stdlib/stats/base/dists/chi/skewness]: https://github.com/stdlib-js/stats/tree/main/base/dists/chi/skewness

[@stdlib/stats/base/dists/chi/stdev]: https://github.com/stdlib-js/stats/tree/main/base/dists/chi/stdev

[@stdlib/stats/base/dists/chi/variance]: https://github.com/stdlib-js/stats/tree/main/base/dists/chi/variance

[@stdlib/stats/base/dists/chi/cdf]: https://github.com/stdlib-js/stats/tree/main/base/dists/chi/cdf

[@stdlib/stats/base/dists/chi/logpdf]: https://github.com/stdlib-js/stats/tree/main/base/dists/chi/logpdf

[@stdlib/stats/base/dists/chi/pdf]: https://github.com/stdlib-js/stats/tree/main/base/dists/chi/pdf

[@stdlib/stats/base/dists/chi/quantile]: https://github.com/stdlib-js/stats/tree/main/base/dists/chi/quantile

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</section>

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