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<title>F Distribution</title>
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<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.f_dist"></a><a class="link" href="f_dist.html" title="F Distribution">F Distribution</a>
</h4></div></div></div>
<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>
<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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../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="Chapter 14. Policies: Controlling Precision, Error Handling etc">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>
<h5>
<a name="math_toolkit.dist_ref.dists.f_dist.h0"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.member_functions"></a></span><a class="link" href="f_dist.html#math_toolkit.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="../../error_handling.html#math_toolkit.error_handling.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>
<h5>
<a name="math_toolkit.dist_ref.dists.f_dist.h1"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.non_member_accessors"></a></span><a class="link" href="f_dist.html#math_toolkit.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_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
</p>
<p>
The domain of the random variable is [0, +∞].
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.f_dist.h2"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.examples"></a></span><a class="link" href="f_dist.html#math_toolkit.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>
<h5>
<a name="math_toolkit.dist_ref.dists.f_dist.h3"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.accuracy"></a></span><a class="link" href="f_dist.html#math_toolkit.dist_ref.dists.f_dist.accuracy">Accuracy</a>
</h5>
<p>
The normal distribution is implemented in terms of the <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">incomplete
beta function</a> and its <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">inverses</a>,
refer to those functions for accuracy data.
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.f_dist.h4"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.implementation"></a></span><a class="link" href="f_dist.html#math_toolkit.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="../../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="../../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="../../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="../../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="../../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="../../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="../../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="../../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="../../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="../../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="../../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="../../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="../../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 © 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta,
Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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>
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