| <html> |
| <head> |
| <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> |
| <title>Discrete Quantile Policies</title> |
| <link rel="stylesheet" href="../../math.css" type="text/css"> |
| <meta name="generator" content="DocBook XSL Stylesheets V1.77.1"> |
| <link rel="home" href="../../index.html" title="Math Toolkit 2.2.0"> |
| <link rel="up" href="../pol_ref.html" title="Policy Reference"> |
| <link rel="prev" href="assert_undefined.html" title="Mathematically Undefined Function Policies"> |
| <link rel="next" href="precision_pol.html" title="Precision Policies"> |
| </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="assert_undefined.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../pol_ref.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="precision_pol.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| <div class="section"> |
| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.pol_ref.discrete_quant_ref"></a><a class="link" href="discrete_quant_ref.html" title="Discrete Quantile Policies">Discrete Quantile |
| Policies</a> |
| </h3></div></div></div> |
| <p> |
| If a statistical distribution is <span class="emphasis"><em>discrete</em></span> then the random |
| variable can only have integer values - this leaves us with a problem when |
| calculating quantiles - we can either ignore the discreteness of the distribution |
| and return a real value, or we can round to an integer. As it happens, computing |
| integer values can be substantially faster than calculating a real value, |
| so there are definite advantages to returning an integer, but we do then |
| need to decide how best to round the result. The <code class="computeroutput"><span class="identifier">discrete_quantile</span></code> |
| policy defines how discrete quantiles work, and how integer results are rounded: |
| </p> |
| <pre class="programlisting"><span class="keyword">enum</span> <span class="identifier">discrete_quantile_policy_type</span> |
| <span class="special">{</span> |
| <span class="identifier">real</span><span class="special">,</span> |
| <span class="identifier">integer_round_outwards</span><span class="special">,</span> <span class="comment">// default</span> |
| <span class="identifier">integer_round_inwards</span><span class="special">,</span> |
| <span class="identifier">integer_round_down</span><span class="special">,</span> |
| <span class="identifier">integer_round_up</span><span class="special">,</span> |
| <span class="identifier">integer_round_nearest</span> |
| <span class="special">};</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="identifier">discrete_quantile_policy_type</span><span class="special">></span> |
| <span class="keyword">struct</span> <span class="identifier">discrete_quantile</span><span class="special">;</span> |
| </pre> |
| <p> |
| The values that <code class="computeroutput"><span class="identifier">discrete_quantile</span></code> |
| can take have the following meanings: |
| </p> |
| <h6> |
| <a name="math_toolkit.pol_ref.discrete_quant_ref.h0"></a> |
| <span class="phrase"><a name="math_toolkit.pol_ref.discrete_quant_ref.real"></a></span><a class="link" href="discrete_quant_ref.html#math_toolkit.pol_ref.discrete_quant_ref.real">real</a> |
| </h6> |
| <p> |
| Ignores the discreteness of the distribution, and returns a real-valued result. |
| For example: |
| </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">negative_binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial_distribution</span><span class="special">;</span> |
| |
| <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">negative_binomial_distribution</span><span class="special"><</span> |
| <span class="keyword">double</span><span class="special">,</span> |
| <span class="identifier">policy</span><span class="special"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">real</span><span class="special">></span> <span class="special">></span> |
| <span class="special">></span> <span class="identifier">dist_type</span><span class="special">;</span> |
| |
| <span class="comment">// Lower 5% quantile:</span> |
| <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist_type</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">0.3</span><span class="special">),</span> <span class="number">0.05</span><span class="special">);</span> |
| <span class="comment">// Upper 95% quantile:</span> |
| <span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist_type</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">0.3</span><span class="special">),</span> <span class="number">0.05</span><span class="special">));</span> |
| </pre> |
| <p> |
| Results in <code class="computeroutput"><span class="identifier">x</span> <span class="special">=</span> |
| <span class="number">27.3898</span></code> and <code class="computeroutput"><span class="identifier">y</span> |
| <span class="special">=</span> <span class="number">68.1584</span></code>. |
| </p> |
| <h6> |
| <a name="math_toolkit.pol_ref.discrete_quant_ref.h1"></a> |
| <span class="phrase"><a name="math_toolkit.pol_ref.discrete_quant_ref.integer_round_outwards"></a></span><a class="link" href="discrete_quant_ref.html#math_toolkit.pol_ref.discrete_quant_ref.integer_round_outwards">integer_round_outwards</a> |
| </h6> |
| <p> |
| This is the default policy: an integer value is returned so that: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| Lower quantiles (where the probability is less than 0.5) are rounded |
| down. |
| </li> |
| <li class="listitem"> |
| Upper quantiles (where the probability is greater than 0.5) are rounded |
| up. |
| </li> |
| </ul></div> |
| <p> |
| This is normally the safest rounding policy, since it ensures that both one |
| and two sided intervals are guaranteed to have <span class="emphasis"><em>at least</em></span> |
| the requested coverage. For example: |
| </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">negative_binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial</span><span class="special">;</span> |
| |
| <span class="comment">// Use the default rounding policy integer_round_outwards.</span> |
| <span class="comment">// Lower quantile rounded down:</span> |
| <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">0.3</span><span class="special">),</span> <span class="number">0.05</span><span class="special">);</span> <span class="comment">// rounded up 27 from 27.3898</span> |
| <span class="comment">// Upper quantile rounded up:</span> |
| <span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">0.3</span><span class="special">),</span> <span class="number">0.05</span><span class="special">));</span> <span class="comment">// rounded down to 69 from 68.1584</span> |
| </pre> |
| <p> |
| Results in <code class="computeroutput"><span class="identifier">x</span> <span class="special">=</span> |
| <span class="number">27</span></code> (rounded down from 27.3898) and |
| <code class="computeroutput"><span class="identifier">y</span> <span class="special">=</span> |
| <span class="number">69</span></code> (rounded up from 68.1584). |
| </p> |
| <p> |
| The variables x and y are now defined so that: |
| </p> |
| <pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="number">20</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special"><=</span> <span class="number">0.05</span> |
| <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="number">20</span><span class="special">),</span> <span class="identifier">y</span><span class="special">)</span> <span class="special">>=</span> <span class="number">0.95</span> |
| </pre> |
| <p> |
| In other words we guarantee <span class="emphasis"><em>at least 90% coverage in the central |
| region overall</em></span>, and also <span class="emphasis"><em>no more than 5% coverage in |
| each tail</em></span>. |
| </p> |
| <h6> |
| <a name="math_toolkit.pol_ref.discrete_quant_ref.h2"></a> |
| <span class="phrase"><a name="math_toolkit.pol_ref.discrete_quant_ref.integer_round_inwards"></a></span><a class="link" href="discrete_quant_ref.html#math_toolkit.pol_ref.discrete_quant_ref.integer_round_inwards">integer_round_inwards</a> |
| </h6> |
| <p> |
| This is the opposite of <span class="emphasis"><em>integer_round_outwards</em></span>: an integer |
| value is returned so that: |
| </p> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| Lower quantiles (where the probability is less than 0.5) are rounded |
| <span class="emphasis"><em>up</em></span>. |
| </li> |
| <li class="listitem"> |
| Upper quantiles (where the probability is greater than 0.5) are rounded |
| <span class="emphasis"><em>down</em></span>. |
| </li> |
| </ul></div> |
| <p> |
| For example: |
| </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">negative_binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial_distribution</span><span class="special">;</span> |
| |
| <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">negative_binomial_distribution</span><span class="special"><</span> |
| <span class="keyword">double</span><span class="special">,</span> |
| <span class="identifier">policy</span><span class="special"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">integer_round_inwards</span><span class="special">></span> <span class="special">></span> |
| <span class="special">></span> <span class="identifier">dist_type</span><span class="special">;</span> |
| |
| <span class="comment">// Lower quantile rounded up:</span> |
| <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist_type</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">0.3</span><span class="special">),</span> <span class="number">0.05</span><span class="special">);</span> <span class="comment">// 28 rounded up from 27.3898</span> |
| <span class="comment">// Upper quantile rounded down:</span> |
| <span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist_type</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">0.3</span><span class="special">),</span> <span class="number">0.05</span><span class="special">));</span> <span class="comment">// 68 rounded down from 68.1584</span> |
| </pre> |
| <p> |
| Results in <code class="computeroutput"><span class="identifier">x</span> <span class="special">=</span> |
| <span class="number">28</span></code> (rounded up from 27.3898) and <code class="computeroutput"><span class="identifier">y</span> <span class="special">=</span> <span class="number">68</span></code> |
| (rounded down from 68.1584). |
| </p> |
| <p> |
| The variables x and y are now defined so that: |
| </p> |
| <pre class="programlisting"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="number">20</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">>=</span> <span class="number">0.05</span> |
| <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">negative_binomial</span><span class="special">(</span><span class="number">20</span><span class="special">),</span> <span class="identifier">y</span><span class="special">)</span> <span class="special"><=</span> <span class="number">0.95</span> |
| </pre> |
| <p> |
| In other words we guarantee <span class="emphasis"><em>at no more than 90% coverage in the |
| central region overall</em></span>, and also <span class="emphasis"><em>at least 5% coverage |
| in each tail</em></span>. |
| </p> |
| <h6> |
| <a name="math_toolkit.pol_ref.discrete_quant_ref.h3"></a> |
| <span class="phrase"><a name="math_toolkit.pol_ref.discrete_quant_ref.integer_round_down"></a></span><a class="link" href="discrete_quant_ref.html#math_toolkit.pol_ref.discrete_quant_ref.integer_round_down">integer_round_down</a> |
| </h6> |
| <p> |
| Always rounds down to an integer value, no matter whether it's an upper or |
| a lower quantile. |
| </p> |
| <h6> |
| <a name="math_toolkit.pol_ref.discrete_quant_ref.h4"></a> |
| <span class="phrase"><a name="math_toolkit.pol_ref.discrete_quant_ref.integer_round_up"></a></span><a class="link" href="discrete_quant_ref.html#math_toolkit.pol_ref.discrete_quant_ref.integer_round_up">integer_round_up</a> |
| </h6> |
| <p> |
| Always rounds up to an integer value, no matter whether it's an upper or |
| a lower quantile. |
| </p> |
| <h6> |
| <a name="math_toolkit.pol_ref.discrete_quant_ref.h5"></a> |
| <span class="phrase"><a name="math_toolkit.pol_ref.discrete_quant_ref.integer_round_nearest"></a></span><a class="link" href="discrete_quant_ref.html#math_toolkit.pol_ref.discrete_quant_ref.integer_round_nearest">integer_round_nearest</a> |
| </h6> |
| <p> |
| Always rounds to the nearest integer value, no matter whether it's an upper |
| or a lower quantile. This will produce the requested coverage <span class="emphasis"><em>in |
| the average case</em></span>, but for any specific example may results in |
| either significantly more or less coverage than the requested amount. For |
| example: |
| </p> |
| <p> |
| For example: |
| </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">negative_binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">negative_binomial_distribution</span><span class="special">;</span> |
| |
| <span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">policies</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">negative_binomial_distribution</span><span class="special"><</span> |
| <span class="keyword">double</span><span class="special">,</span> |
| <span class="identifier">policy</span><span class="special"><</span><span class="identifier">discrete_quantile</span><span class="special"><</span><span class="identifier">integer_round_nearest</span><span class="special">></span> <span class="special">></span> |
| <span class="special">></span> <span class="identifier">dist_type</span><span class="special">;</span> |
| |
| <span class="comment">// Lower quantile rounded (down) to nearest:</span> |
| <span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">dist_type</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">0.3</span><span class="special">),</span> <span class="number">0.05</span><span class="special">);</span> <span class="comment">// 27</span> |
| <span class="comment">// Upper quantile rounded (down) to nearest:</span> |
| <span class="keyword">double</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist_type</span><span class="special">(</span><span class="number">20</span><span class="special">,</span> <span class="number">0.3</span><span class="special">),</span> <span class="number">0.05</span><span class="special">));</span> <span class="comment">// 68</span> |
| </pre> |
| <p> |
| Results in <code class="computeroutput"><span class="identifier">x</span> <span class="special">=</span> |
| <span class="number">27</span></code> (rounded from 27.3898) and <code class="computeroutput"><span class="identifier">y</span> <span class="special">=</span> <span class="number">68</span></code> |
| (rounded from 68.1584). |
| </p> |
| </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> |
| <hr> |
| <div class="spirit-nav"> |
| <a accesskey="p" href="assert_undefined.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../pol_ref.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="precision_pol.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| </body> |
| </html> |
