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<title>Complements are supported too - and when to use them</title>
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<div class="section">
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<div class="titlepage"><div><div><h4 class="title">
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<a name="math_toolkit.stat_tut.overview.complements"></a><a class="link" href="complements.html" title="Complements are supported too - and when to use them">Complements
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are supported too - and when to use them</a>
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</h4></div></div></div>
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<p>
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Often you don't want the value of the CDF, but its complement, which is
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to say <code class="computeroutput"><span class="number">1</span><span class="special">-</span><span class="identifier">p</span></code> rather than <code class="computeroutput"><span class="identifier">p</span></code>.
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It is tempting to calculate the CDF and subtract it from <code class="computeroutput"><span class="number">1</span></code>, but if <code class="computeroutput"><span class="identifier">p</span></code>
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is very close to <code class="computeroutput"><span class="number">1</span></code> then cancellation
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error will cause you to lose accuracy, perhaps totally.
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</p>
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<p>
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<a class="link" href="complements.html#why_complements">See below <span class="emphasis"><em>"Why and when
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to use complements?"</em></span></a>
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</p>
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<p>
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In this library, whenever you want to receive a complement, just wrap all
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the function arguments in a call to <code class="computeroutput"><span class="identifier">complement</span><span class="special">(...)</span></code>, for example:
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</p>
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<pre class="programlisting"><span class="identifier">students_t</span> <span class="identifier">dist</span><span class="special">(</span><span class="number">5</span><span class="special">);</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"CDF at t = 1 is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="number">1.0</span><span class="special">)</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Complement of CDF at t = 1 is "</span> <span class="special"><<</span> <span class="identifier">cdf</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">dist</span><span class="special">,</span> <span class="number">1.0</span><span class="special">))</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span>
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</pre>
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<p>
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But wait, now that we have a complement, we have to be able to use it as
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well. Any function that accepts a probability as an argument can also accept
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a complement by wrapping all of its arguments in a call to <code class="computeroutput"><span class="identifier">complement</span><span class="special">(...)</span></code>,
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for example:
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</p>
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<pre class="programlisting"><span class="identifier">students_t</span> <span class="identifier">dist</span><span class="special">(</span><span class="number">5</span><span class="special">);</span>
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<span class="keyword">for</span><span class="special">(</span><span class="keyword">double</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">10</span><span class="special">;</span> <span class="identifier">i</span> <span class="special"><</span> <span class="number">1e10</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">*=</span> <span class="number">10</span><span class="special">)</span>
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<span class="special">{</span>
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<span class="comment">// Calculate the quantile for a 1 in i chance:</span>
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<span class="keyword">double</span> <span class="identifier">t</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</span><span class="special">,</span> <span class="number">1</span><span class="special">/</span><span class="identifier">i</span><span class="special">));</span>
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<span class="comment">// Print it out:</span>
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<span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Quantile of students-t with 5 degrees of freedom\n"</span>
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<span class="string">"for a 1 in "</span> <span class="special"><<</span> <span class="identifier">i</span> <span class="special"><<</span> <span class="string">" chance is "</span> <span class="special"><<</span> <span class="identifier">t</span> <span class="special"><<</span> <span class="identifier">endl</span><span class="special">;</span>
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<span class="special">}</span>
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</pre>
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<div class="tip"><table border="0" summary="Tip">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../../doc/src/images/tip.png"></td>
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<th align="left">Tip</th>
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</tr>
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<tr><td align="left" valign="top">
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<p>
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<span class="bold"><strong>Critical values are just quantiles</strong></span>
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</p>
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<p>
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Some texts talk about quantiles, or percentiles or fractiles, others
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about critical values, the basic rule is:
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</p>
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<p>
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<span class="emphasis"><em>Lower critical values</em></span> are the same as the quantile.
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</p>
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<p>
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<span class="emphasis"><em>Upper critical values</em></span> are the same as the quantile
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from the complement of the probability.
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</p>
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<p>
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For example, suppose we have a Bernoulli process, giving rise to a binomial
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distribution with success ratio 0.1 and 100 trials in total. The <span class="emphasis"><em>lower
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critical value</em></span> for a probability of 0.05 is given by:
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</p>
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<p>
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<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="number">100</span><span class="special">,</span> <span class="number">0.1</span><span class="special">),</span> <span class="number">0.05</span><span class="special">)</span></code>
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</p>
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<p>
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and the <span class="emphasis"><em>upper critical value</em></span> is given by:
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</p>
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<p>
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<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">binomial</span><span class="special">(</span><span class="number">100</span><span class="special">,</span> <span class="number">0.1</span><span class="special">),</span> <span class="number">0.05</span><span class="special">))</span></code>
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</p>
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<p>
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which return 4.82 and 14.63 respectively.
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</p>
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</td></tr>
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</table></div>
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<a name="why_complements"></a><div class="tip"><table border="0" summary="Tip">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../../doc/src/images/tip.png"></td>
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<th align="left">Tip</th>
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</tr>
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<tr><td align="left" valign="top">
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<p>
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<span class="bold"><strong>Why bother with complements anyway?</strong></span>
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</p>
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<p>
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It's very tempting to dispense with complements, and simply subtract
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the probability from 1 when required. However, consider what happens
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when the probability is very close to 1: let's say the probability expressed
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at float precision is <code class="computeroutput"><span class="number">0.999999940f</span></code>,
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then <code class="computeroutput"><span class="number">1</span> <span class="special">-</span>
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<span class="number">0.999999940f</span> <span class="special">=</span>
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<span class="number">5.96046448e-008</span></code>, but the result
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is actually accurate to just <span class="emphasis"><em>one single bit</em></span>: the
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only bit that didn't cancel out!
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</p>
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<p>
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Or to look at this another way: consider that we want the risk of falsely
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rejecting the null-hypothesis in the Student's t test to be 1 in 1 billion,
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for a sample size of 10,000. This gives a probability of 1 - 10<sup>-9</sup>, which
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is exactly 1 when calculated at float precision. In this case calculating
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the quantile from the complement neatly solves the problem, so for example:
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</p>
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<p>
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<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">students_t</span><span class="special">(</span><span class="number">10000</span><span class="special">),</span> <span class="number">1e-9</span><span class="special">))</span></code>
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</p>
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<p>
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returns the expected t-statistic <code class="computeroutput"><span class="number">6.00336</span></code>,
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where as:
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</p>
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<p>
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<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">students_t</span><span class="special">(</span><span class="number">10000</span><span class="special">),</span> <span class="number">1</span><span class="special">-</span><span class="number">1e-9f</span><span class="special">)</span></code>
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</p>
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<p>
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raises an overflow error, since it is the same as:
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</p>
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<p>
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<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">students_t</span><span class="special">(</span><span class="number">10000</span><span class="special">),</span> <span class="number">1</span><span class="special">)</span></code>
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</p>
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<p>
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Which has no finite result.
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</p>
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<p>
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With all distributions, even for more reasonable probability (unless
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the value of p can be represented exactly in the floating-point type)
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the loss of accuracy quickly becomes significant if you simply calculate
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probability from 1 - p (because it will be mostly garbage digits for
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p ~ 1).
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</p>
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<p>
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So always avoid, for example, using a probability near to unity like
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0.99999
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</p>
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<p>
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<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">my_distribution</span><span class="special">,</span>
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<span class="number">0.99999</span><span class="special">)</span></code>
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</p>
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<p>
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and instead use
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</p>
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<p>
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<code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">complement</span><span class="special">(</span><span class="identifier">my_distribution</span><span class="special">,</span>
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<span class="number">0.00001</span><span class="special">))</span></code>
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</p>
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<p>
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since 1 - 0.99999 is not exactly equal to 0.00001 when using floating-point
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arithmetic.
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</p>
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<p>
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This assumes that the 0.00001 value is either a constant, or can be computed
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by some manner other than subtracting 0.99999 from 1.
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</p>
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<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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>)
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