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<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="id1267605"></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="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</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:
</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 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="id1267958"></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="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">;</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">// 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>
</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="id1268343"></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="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</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">// 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>
</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="id1268838"></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="id1268857"></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="id1268876"></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="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</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">// 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>
</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>
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<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 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>
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