<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>F Distribution</title> <link rel="stylesheet" href="../../../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> <link rel="home" href="../../../../index.html" title="Math Toolkit"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="extreme_dist.html" title="Extreme Value Distribution"> <link rel="next" href="gamma_dist.html" title="Gamma (and Erlang) Distribution"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../../../../../libs/libraries.htm">Libraries</a></td> <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> <td align="center"><a href="../../../../../../../../../more/index.htm">More</a></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="extreme_dist.html"><img src="../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../../index.html"><img src="../../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="gamma_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section" lang="en"> <div class="titlepage"><div><div><h5 class="title"> <a name="math_toolkit.dist.dist_ref.dists.f_dist"></a><a class="link" href="f_dist.html" title="F Distribution"> F Distribution</a> </h5></div></div></div> <p> </p> <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">fisher_f</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> <p> </p> <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a> <span class="special">=</span> <a class="link" href="../../../policy/pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span> <span class="keyword">class</span> <span class="identifier">fisher_f_distribution</span><span class="special">;</span> <span class="keyword">typedef</span> <span class="identifier">fisher_f_distribution</span><span class="special"><></span> <span class="identifier">fisher_f</span><span class="special">;</span> <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a><span class="special">></span> <span class="keyword">class</span> <span class="identifier">fisher_f_distribution</span> <span class="special">{</span> <span class="keyword">public</span><span class="special">:</span> <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> <span class="comment">// Construct: </span> <span class="identifier">fisher_f_distribution</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">i</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">j</span><span class="special">);</span> <span class="comment">// Accessors: </span> <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom1</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom2</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="special">};</span> <span class="special">}}</span> <span class="comment">//namespaces </span></pre> <p> The F distribution is a continuous distribution that arises when testing whether two samples have the same variance. If χ<sup>2</sup><sub>m</sub> and χ<sup>2</sup><sub>n</sub> are independent variates each distributed as Chi-Squared with <span class="emphasis"><em>m</em></span> and <span class="emphasis"><em>n</em></span> degrees of freedom, then the test statistic: </p> <p> F<sub>n,m</sub> = (χ<sup>2</sup><sub>n</sub> / n) / (χ<sup>2</sup><sub>m</sub> / m) </p> <p> Is distributed over the range [0, ∞] with an F distribution, and has the PDF: </p> <p> <span class="inlinemediaobject"><img src="../../../../../equations/fisher_pdf.png"></span> </p> <p> The following graph illustrates how the PDF varies depending on the two degrees of freedom parameters. </p> <p> <span class="inlinemediaobject"><img src="../../../../../graphs/fisher_f_pdf.png" align="middle"></span> </p> <a name="math_toolkit.dist.dist_ref.dists.f_dist.member_functions"></a><h5> <a name="id1031861"></a> <a class="link" href="f_dist.html#math_toolkit.dist.dist_ref.dists.f_dist.member_functions">Member Functions</a> </h5> <pre class="programlisting"><span class="identifier">fisher_f_distribution</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">df1</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">df2</span><span class="special">);</span> </pre> <p> Constructs an F-distribution with numerator degrees of freedom <span class="emphasis"><em>df1</em></span> and denominator degrees of freedom <span class="emphasis"><em>df2</em></span>. </p> <p> Requires that <span class="emphasis"><em>df1</em></span> and <span class="emphasis"><em>df2</em></span> are both greater than zero, otherwise <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a> is called. </p> <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom1</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> </pre> <p> Returns the numerator degrees of freedom parameter of the distribution. </p> <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom2</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> </pre> <p> Returns the denominator degrees of freedom parameter of the distribution. </p> <a name="math_toolkit.dist.dist_ref.dists.f_dist.non_member_accessors"></a><h5> <a name="id1032014"></a> <a class="link" href="f_dist.html#math_toolkit.dist.dist_ref.dists.f_dist.non_member_accessors">Non-member Accessors</a> </h5> <p> All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor functions</a> that are generic to all distributions are supported: <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>, <a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>, <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>, <a class="link" href="../nmp.html#math.dist.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>, <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>, <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>, <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>, <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>, <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>. </p> <p> The domain of the random variable is [0, +∞]. </p> <a name="math_toolkit.dist.dist_ref.dists.f_dist.examples"></a><h5> <a name="id1032114"></a> <a class="link" href="f_dist.html#math_toolkit.dist.dist_ref.dists.f_dist.examples">Examples</a> </h5> <p> Various <a class="link" href="../../stat_tut/weg/f_eg.html" title="F Distribution Examples">worked examples</a> are available illustrating the use of the F Distribution. </p> <a name="math_toolkit.dist.dist_ref.dists.f_dist.accuracy"></a><h5> <a name="id1032136"></a> <a class="link" href="f_dist.html#math_toolkit.dist.dist_ref.dists.f_dist.accuracy">Accuracy</a> </h5> <p> The normal distribution is implemented in terms of the <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">incomplete beta function</a> and it's <a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">inverses</a>, refer to those functions for accuracy data. </p> <a name="math_toolkit.dist.dist_ref.dists.f_dist.implementation"></a><h5> <a name="id1032162"></a> <a class="link" href="f_dist.html#math_toolkit.dist.dist_ref.dists.f_dist.implementation">Implementation</a> </h5> <p> In the following table <span class="emphasis"><em>v1</em></span> and <span class="emphasis"><em>v2</em></span> are the first and second degrees of freedom parameters of the distribution, <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span> is the probability, and <span class="emphasis"><em>q = 1-p</em></span>. </p> <div class="informaltable"><table class="table"> <colgroup> <col> <col> </colgroup> <thead><tr> <th> <p> Function </p> </th> <th> <p> Implementation Notes </p> </th> </tr></thead> <tbody> <tr> <td> <p> pdf </p> </td> <td> <p> The usual form of the PDF is given by: </p> <p> <span class="inlinemediaobject"><img src="../../../../../equations/fisher_pdf.png"></span> </p> <p> However, that form is hard to evaluate directly without incurring problems with either accuracy or numeric overflow. </p> <p> Direct differentiation of the CDF expressed in terms of the incomplete beta function </p> <p> led to the following two formulas: </p> <p> f<sub>v1,v2</sub>(x) = y * <a class="link" href="../../../special/sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>(v2 / 2, v1 / 2, v2 / (v2 + v1 * x)) </p> <p> with y = (v2 * v1) / ((v2 + v1 * x) * (v2 + v1 * x)) </p> <p> and </p> <p> f<sub>v1,v2</sub>(x) = y * <a class="link" href="../../../special/sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>(v1 / 2, v2 / 2, v1 * x / (v2 + v1 * x)) </p> <p> with y = (z * v1 - x * v1 * v1) / z<sup>2</sup> </p> <p> and z = v2 + v1 * x </p> <p> The first of these is used for v1 * x > v2, otherwise the second is used. </p> <p> The aim is to keep the <span class="emphasis"><em>x</em></span> argument to <a class="link" href="../../../special/sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a> away from 1 to avoid rounding error. </p> </td> </tr> <tr> <td> <p> cdf </p> </td> <td> <p> Using the relations: </p> <p> p = <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(v1 / 2, v2 / 2, v1 * x / (v2 + v1 * x)) </p> <p> and </p> <p> p = <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(v2 / 2, v1 / 2, v2 / (v2 + v1 * x)) </p> <p> The first is used for v1 * x > v2, otherwise the second is used. </p> <p> The aim is to keep the <span class="emphasis"><em>x</em></span> argument to <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a> well away from 1 to avoid rounding error. </p> </td> </tr> <tr> <td> <p> cdf complement </p> </td> <td> <p> Using the relations: </p> <p> p = <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(v1 / 2, v2 / 2, v1 * x / (v2 + v1 * x)) </p> <p> and </p> <p> p = <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(v2 / 2, v1 / 2, v2 / (v2 + v1 * x)) </p> <p> The first is used for v1 * x < v2, otherwise the second is used. </p> <p> The aim is to keep the <span class="emphasis"><em>x</em></span> argument to <a class="link" href="../../../special/sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a> well away from 1 to avoid rounding error. </p> </td> </tr> <tr> <td> <p> quantile </p> </td> <td> <p> Using the relation: </p> <p> x = v2 * a / (v1 * b) </p> <p> where: </p> <p> a = <a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>(v1 / 2, v2 / 2, p) </p> <p> and </p> <p> b = 1 - a </p> <p> Quantities <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> are both computed by <a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a> without the subtraction implied above. </p> </td> </tr> <tr> <td> <p> quantile </p> <p> from the complement </p> </td> <td> <p> Using the relation: </p> <p> x = v2 * a / (v1 * b) </p> <p> where </p> <p> a = <a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>(v1 / 2, v2 / 2, p) </p> <p> and </p> <p> b = 1 - a </p> <p> Quantities <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> are both computed by <a class="link" href="../../../special/sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a> without the subtraction implied above. </p> </td> </tr> <tr> <td> <p> mean </p> </td> <td> <p> v2 / (v2 - 2) </p> </td> </tr> <tr> <td> <p> variance </p> </td> <td> <p> 2 * v2<sup>2 </sup> * (v1 + v2 - 2) / (v1 * (v2 - 2) * (v2 - 2) * (v2 - 4)) </p> </td> </tr> <tr> <td> <p> mode </p> </td> <td> <p> v2 * (v1 - 2) / (v1 * (v2 + 2)) </p> </td> </tr> <tr> <td> <p> skewness </p> </td> <td> <p> 2 * (v2 + 2 * v1 - 2) * sqrt((2 * v2 - 8) / (v1 * (v2 + v1 - 2))) / (v2 - 6) </p> </td> </tr> <tr> <td> <p> kurtosis and kurtosis excess </p> </td> <td> <p> Refer to, <a href="http://mathworld.wolfram.com/F-Distribution.html" target="_top">Weisstein, Eric W. "F-Distribution." From MathWorld--A Wolfram Web Resource.</a> </p> </td> </tr> </tbody> </table></div> </div> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> <td align="left"></td> <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and Thijs van den Berg<p> Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) </p> </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="extreme_dist.html"><img src="../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../../index.html"><img src="../../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="gamma_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>