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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.policy.pol_ref.discrete_quant_ref"></a><a class="link" href="discrete_quant_ref.html" title="Discrete Quantile Policies"> Discrete |
| Quantile Policies</a> |
| </h4></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> |
| <a name="math_toolkit.policy.pol_ref.discrete_quant_ref.real"></a><h6> |
| <a name="id1263626"></a> |
| <a class="link" href="discrete_quant_ref.html#math_toolkit.policy.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> |
| <p> |
| </p> |
| <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">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> |
| </p> |
| <p> |
| </p> |
| <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> |
| <a name="math_toolkit.policy.pol_ref.discrete_quant_ref.integer_round_outwards"></a><h6> |
| <a name="id1263981"></a> |
| <a class="link" href="discrete_quant_ref.html#math_toolkit.policy.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 type="disc"> |
| <li> |
| Lower quantiles (where the probability is less than 0.5) are rounded |
| down. |
| </li> |
| <li> |
| 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> |
| <p> |
| </p> |
| <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">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> |
| </p> |
| <p> |
| </p> |
| <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> |
| <a name="math_toolkit.policy.pol_ref.discrete_quant_ref.integer_round_inwards"></a><h6> |
| <a name="id1264937"></a> |
| <a class="link" href="discrete_quant_ref.html#math_toolkit.policy.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 type="disc"> |
| <li> |
| Lower quantiles (where the probability is less than 0.5) are rounded |
| <span class="emphasis"><em>up</em></span>. |
| </li> |
| <li> |
| 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> |
| <p> |
| </p> |
| <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">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> |
| </p> |
| <p> |
| </p> |
| <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> |
| <a name="math_toolkit.policy.pol_ref.discrete_quant_ref.integer_round_down"></a><h6> |
| <a name="id1265434"></a> |
| <a class="link" href="discrete_quant_ref.html#math_toolkit.policy.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> |
| <a name="math_toolkit.policy.pol_ref.discrete_quant_ref.integer_round_up"></a><h6> |
| <a name="id1265453"></a> |
| <a class="link" href="discrete_quant_ref.html#math_toolkit.policy.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> |
| <a name="math_toolkit.policy.pol_ref.discrete_quant_ref.integer_round_nearest"></a><h6> |
| <a name="id1265472"></a> |
| <a class="link" href="discrete_quant_ref.html#math_toolkit.policy.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> |
| <p> |
| </p> |
| <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">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 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">// 27 |
| </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 |
| </span> |
| </pre> |
| <p> |
| </p> |
| <p> |
| </p> |
| <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 , 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> |
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