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| <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="id1023494"></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="id1023648"></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="id1023747"></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="id1023769"></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="id1023795"></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, 2010 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> |
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