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| <div class="titlepage"><div><div><h2 class="title" style="clear: both"> |
| <a name="document_to_test_formatting.lists_and_tables"></a><a class="link" href="lists_and_tables.html" title="Lists and Tables">Lists and |
| Tables</a> |
| </h2></div></div></div> |
| <div class="toc"><dl> |
| <dt><span class="section"><a href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.lists">Lists</a></span></dt> |
| <dt><span class="section"><a href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.variable_lists">Variable |
| Lists</a></span></dt> |
| <dt><span class="section"><a href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.tables">Tables</a></span></dt> |
| </dl></div> |
| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="document_to_test_formatting.lists_and_tables.lists"></a><a class="link" href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.lists" title="Lists">Lists</a> |
| </h3></div></div></div> |
| <p> |
| A numbered list: |
| </p> |
| <div class="orderedlist"><ol class="orderedlist" type="1"> |
| <li class="listitem"> |
| One |
| </li> |
| <li class="listitem"> |
| Two |
| </li> |
| <li class="listitem"> |
| Three |
| <div class="orderedlist"><ol class="orderedlist" type="a"> |
| <li class="listitem"> |
| Three.a |
| </li> |
| <li class="listitem"> |
| Three.b |
| </li> |
| <li class="listitem"> |
| Three.c |
| </li> |
| </ol></div> |
| </li> |
| <li class="listitem"> |
| Four |
| <div class="orderedlist"><ol class="orderedlist" type="a"><li class="listitem"> |
| Four.a |
| <div class="orderedlist"><ol class="orderedlist" type="i"> |
| <li class="listitem"> |
| Four.a.i |
| </li> |
| <li class="listitem"> |
| Four.a.ii |
| </li> |
| </ol></div> |
| </li></ol></div> |
| </li> |
| <li class="listitem"> |
| Five |
| </li> |
| </ol></div> |
| <p> |
| An unordered list: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" type="disc"> |
| <li class="listitem"> |
| First |
| </li> |
| <li class="listitem"> |
| Second |
| </li> |
| <li class="listitem"> |
| Third |
| </li> |
| </ul></div> |
| <p> |
| A mixture of the two: |
| </p> |
| <div class="orderedlist"><ol class="orderedlist" type="1"> |
| <li class="listitem"> |
| 1 |
| <div class="itemizedlist"><ul class="itemizedlist" type="disc"> |
| <li class="listitem"> |
| 1.a |
| <div class="orderedlist"><ol class="orderedlist" type="a"> |
| <li class="listitem"> |
| 1.a.1 |
| </li> |
| <li class="listitem"> |
| 1.a.2 |
| </li> |
| </ol></div> |
| </li> |
| <li class="listitem"> |
| 1.b |
| </li> |
| </ul></div> |
| </li> |
| <li class="listitem"> |
| 2 |
| <div class="itemizedlist"><ul class="itemizedlist" type="disc"> |
| <li class="listitem"> |
| 2.a |
| </li> |
| <li class="listitem"> |
| 2.b |
| <div class="orderedlist"><ol class="orderedlist" type="a"> |
| <li class="listitem"> |
| 2.b.1 |
| </li> |
| <li class="listitem"> |
| 2.b.2 |
| <div class="itemizedlist"><ul class="itemizedlist" type="circle"> |
| <li class="listitem"> |
| 2.b.2.a |
| </li> |
| <li class="listitem"> |
| 2.b.2.b |
| </li> |
| </ul></div> |
| </li> |
| </ol></div> |
| </li> |
| </ul></div> |
| </li> |
| </ol></div> |
| </div> |
| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="document_to_test_formatting.lists_and_tables.variable_lists"></a><a class="link" href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.variable_lists" title="Variable Lists">Variable |
| Lists</a> |
| </h3></div></div></div> |
| <div class="variablelist"> |
| <p class="title"><b>A Variable List</b></p> |
| <dl> |
| <dt><span class="term">term 1</span></dt> |
| <dd><p> |
| The definition of term 1 |
| </p></dd> |
| <dt><span class="term">term 2</span></dt> |
| <dd><p> |
| The definition of term 2 |
| </p></dd> |
| <dt><span class="term">term 3</span></dt> |
| <dd><p> |
| The definition of term 3 |
| </p></dd> |
| </dl> |
| </div> |
| </div> |
| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="document_to_test_formatting.lists_and_tables.tables"></a><a class="link" href="lists_and_tables.html#document_to_test_formatting.lists_and_tables.tables" title="Tables">Tables</a> |
| </h3></div></div></div> |
| <p> |
| Here's a big table with code and other tricky things: |
| </p> |
| <div class="table"> |
| <a name="id769721"></a><p class="title"><b>Table 1. Notes on the Implementation of the Beta Distribution</b></p> |
| <div class="table-contents"><table class="table" summary="Notes on the Implementation of the Beta Distribution"> |
| <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> |
| f(x;α,β) = x<sup>α - 1</sup> (1 - x)<sup>β -1</sup> / B(α, β) |
| </p> |
| <p> |
| Implemented using ibeta_derivative(a, b, x). |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the incomplete beta function ibeta(a, b, x) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf complement |
| </p> |
| </td> |
| <td> |
| <p> |
| ibetac(a, b, x) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the inverse incomplete beta function ibeta_inv(a, b, p) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile from the complement |
| </p> |
| </td> |
| <td> |
| <p> |
| ibetac_inv(a, b, q) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mean |
| </p> |
| </td> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">a</span><span class="special">/(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">)</span></code> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| variance |
| </p> |
| </td> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">a</span> <span class="special">*</span> |
| <span class="identifier">b</span> <span class="special">/</span> |
| <span class="special">(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">)^</span><span class="number">2</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> |
| <span class="identifier">b</span> <span class="special">+</span> |
| <span class="number">1</span><span class="special">)</span></code> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mode |
| </p> |
| </td> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="special">(</span><span class="identifier">a</span><span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">a</span> |
| <span class="special">+</span> <span class="identifier">b</span> |
| <span class="special">+</span> <span class="number">2</span><span class="special">)</span></code> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| skewness |
| </p> |
| </td> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="number">2</span> <span class="special">(</span><span class="identifier">b</span><span class="special">-</span><span class="identifier">a</span><span class="special">)</span> |
| <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">+</span><span class="number">1</span><span class="special">)/(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">+</span><span class="number">2</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">a</span> |
| <span class="special">*</span> <span class="identifier">b</span><span class="special">)</span></code> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis excess |
| </p> |
| </td> |
| <td> |
| <p> |
| <span class="inlinemediaobject"><img src="../images/beta_dist_kurtosis.png" alt="beta_dist_kurtosis"></span> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis |
| </p> |
| </td> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">kurtosis</span> <span class="special">+</span> |
| <span class="number">3</span></code> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| parameter estimation |
| </p> |
| </td> |
| <td> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| alpha |
| </p> |
| <p> |
| from mean and variance |
| </p> |
| </td> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">mean</span> <span class="special">*</span> |
| <span class="special">((</span> <span class="special">(</span><span class="identifier">mean</span> <span class="special">*</span> |
| <span class="special">(</span><span class="number">1</span> |
| <span class="special">-</span> <span class="identifier">mean</span><span class="special">))</span> <span class="special">/</span> <span class="identifier">variance</span><span class="special">)-</span> |
| <span class="number">1</span><span class="special">)</span></code> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| beta |
| </p> |
| <p> |
| from mean and variance |
| </p> |
| </td> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="special">(</span><span class="number">1</span> |
| <span class="special">-</span> <span class="identifier">mean</span><span class="special">)</span> <span class="special">*</span> <span class="special">(((</span><span class="identifier">mean</span> |
| <span class="special">*</span> <span class="special">(</span><span class="number">1</span> <span class="special">-</span> <span class="identifier">mean</span><span class="special">))</span> |
| <span class="special">/</span><span class="identifier">variance</span><span class="special">)-</span><span class="number">1</span><span class="special">)</span></code> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| The member functions <code class="computeroutput"><span class="identifier">estimate_alpha</span></code> |
| and <code class="computeroutput"><span class="identifier">estimate_beta</span></code> |
| </p> |
| <p> |
| from cdf and probability x |
| </p> |
| <p> |
| and <span class="bold"><strong>either</strong></span> <code class="computeroutput"><span class="identifier">alpha</span></code> |
| or <code class="computeroutput"><span class="identifier">beta</span></code> |
| </p> |
| </td> |
| <td> |
| <p> |
| Implemented in terms of the inverse incomplete beta functions |
| </p> |
| <p> |
| ibeta_inva, and ibeta_invb respectively. |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">estimate_alpha</span></code> |
| </p> |
| </td> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">ibeta_inva</span><span class="special">(</span><span class="identifier">beta</span><span class="special">,</span> |
| <span class="identifier">x</span><span class="special">,</span> |
| <span class="identifier">probability</span><span class="special">)</span></code> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">estimate_beta</span></code> |
| </p> |
| </td> |
| <td> |
| <p> |
| <code class="computeroutput"><span class="identifier">ibeta_invb</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">,</span> |
| <span class="identifier">x</span><span class="special">,</span> |
| <span class="identifier">probability</span><span class="special">)</span></code> |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"> |
| </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 © 2007 John Maddock, Joel de Guzman, Eric Niebler and Matias |
| Capeletto<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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