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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <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
  28         are supported too - and when to use them</a>
  29 </h4></div></div></div>
  30 <p>
  31           Often you don't want the value of the CDF, but its complement, which is
  32           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>.
  33           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>
  34           is very close to <code class="computeroutput"><span class="number">1</span></code> then cancellation
  35           error will cause you to lose accuracy, perhaps totally.
  36         </p>
  37 <p>
  38           <a class="link" href="complements.html#why_complements">See below <span class="emphasis"><em>"Why and when
  39           to use complements?"</em></span></a>
  40         </p>
  41 <p>
  42           In this library, whenever you want to receive a complement, just wrap all
  43           the function arguments in a call to <code class="computeroutput"><span class="identifier">complement</span><span class="special">(...)</span></code>, for example:
  44         </p>
  45 <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>
  46 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"CDF at t = 1 is "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  47 <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Complement of CDF at t = 1 is "</span> <span class="special">&lt;&lt;</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">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  48 </pre>
  49 <p>
  50           But wait, now that we have a complement, we have to be able to use it as
  51           well. Any function that accepts a probability as an argument can also accept
  52           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>,
  53           for example:
  54         </p>
  55 <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>
  56
  57 <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">&lt;</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>
  58 <span class="special">{</span>
  59    <span class="comment">// Calculate the quantile for a 1 in i chance:</span>
  60    <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>
  61    <span class="comment">// Print it out:</span>
  62    <span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"Quantile of students-t with 5 degrees of freedom\n"</span>
  63            <span class="string">"for a 1 in "</span> <span class="special">&lt;&lt;</span> <span class="identifier">i</span> <span class="special">&lt;&lt;</span> <span class="string">" chance is "</span> <span class="special">&lt;&lt;</span> <span class="identifier">t</span> <span class="special">&lt;&lt;</span> <span class="identifier">endl</span><span class="special">;</span>
  64 <span class="special">}</span>
  65 </pre>
  66 <div class="tip"><table border="0" summary="Tip">
  67 <tr>
  68 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../../doc/src/images/tip.png"></td>
  69 <th align="left">Tip</th>
  70 </tr>
  71 <tr><td align="left" valign="top">
  72 <p>
  73             <span class="bold"><strong>Critical values are just quantiles</strong></span>
  74           </p>
  75 <p>
  76             Some texts talk about quantiles, or percentiles or fractiles, others
  77             about critical values, the basic rule is:
  78           </p>
  79 <p>
  80             <span class="emphasis"><em>Lower critical values</em></span> are the same as the quantile.
  81           </p>
  82 <p>
  83             <span class="emphasis"><em>Upper critical values</em></span> are the same as the quantile
  84             from the complement of the probability.
  85           </p>
  86 <p>
  87             For example, suppose we have a Bernoulli process, giving rise to a binomial
  88             distribution with success ratio 0.1 and 100 trials in total. The <span class="emphasis"><em>lower
  89             critical value</em></span> for a probability of 0.05 is given by:
  90           </p>
  91 <p>
  92             <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>
  93           </p>
  94 <p>
  95             and the <span class="emphasis"><em>upper critical value</em></span> is given by:
  96           </p>
  97 <p>
  98             <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>
  99           </p>
 100 <p>
 101             which return 4.82 and 14.63 respectively.
 102           </p>
 103 </td></tr>
 104 </table></div>
 105 <a name="why_complements"></a><div class="tip"><table border="0" summary="Tip">
 106 <tr>
 107 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../../doc/src/images/tip.png"></td>
 108 <th align="left">Tip</th>
 109 </tr>
 110 <tr><td align="left" valign="top">
 111 <p>
 112             <span class="bold"><strong>Why bother with complements anyway?</strong></span>
 113           </p>
 114 <p>
 115             It's very tempting to dispense with complements, and simply subtract
 116             the probability from 1 when required. However, consider what happens
 117             when the probability is very close to 1: let's say the probability expressed
 118             at float precision is <code class="computeroutput"><span class="number">0.999999940f</span></code>,
 119             then <code class="computeroutput"><span class="number">1</span> <span class="special">-</span>
 120             <span class="number">0.999999940f</span> <span class="special">=</span>
 121             <span class="number">5.96046448e-008</span></code>, but the result
 122             is actually accurate to just <span class="emphasis"><em>one single bit</em></span>: the
 123             only bit that didn't cancel out!
 124           </p>
 125 <p>
 126             Or to look at this another way: consider that we want the risk of falsely
 127             rejecting the null-hypothesis in the Student's t test to be 1 in 1 billion,
 128             for a sample size of 10,000. This gives a probability of 1 - 10<sup>-9</sup>, which
 129             is exactly 1 when calculated at float precision. In this case calculating
 130             the quantile from the complement neatly solves the problem, so for example:
 131           </p>
 132 <p>
 133             <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>
 134           </p>
 135 <p>
 136             returns the expected t-statistic <code class="computeroutput"><span class="number">6.00336</span></code>,
 137             where as:
 138           </p>
 139 <p>
 140             <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>
 141           </p>
 142 <p>
 143             raises an overflow error, since it is the same as:
 144           </p>
 145 <p>
 146             <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>
 147           </p>
 148 <p>
 149             Which has no finite result.
 150           </p>
 151 <p>
 152             With all distributions, even for more reasonable probability (unless
 153             the value of p can be represented exactly in the floating-point type)
 154             the loss of accuracy quickly becomes significant if you simply calculate
 155             probability from 1 - p (because it will be mostly garbage digits for
 156             p ~ 1).
 157           </p>
 158 <p>
 159             So always avoid, for example, using a probability near to unity like
 160             0.99999
 161           </p>
 162 <p>
 163             <code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">my_distribution</span><span class="special">,</span>
 164             <span class="number">0.99999</span><span class="special">)</span></code>
 165           </p>
 166 <p>
 167             and instead use
 168           </p>
 169 <p>
 170             <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>
 171             <span class="number">0.00001</span><span class="special">))</span></code>
 172           </p>
 173 <p>
 174             since 1 - 0.99999 is not exactly equal to 0.00001 when using floating-point
 175             arithmetic.
 176           </p>
 177 <p>
 178             This assumes that the 0.00001 value is either a constant, or can be computed
 179             by some manner other than subtracting 0.99999 from 1.
 180           </p>
 181 </td></tr>
 182 </table></div>
 183 </div>
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 185 <td align="left"></td>
 186 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 187       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 188       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 189       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 190       Daryle Walker and Xiaogang Zhang<p>
 191         Distributed under the Boost Software License, Version 1.0. (See accompanying
 192         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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