libs/math/doc/sf_and_dist/html/math_toolkit/dist/stat_tut/overview/complements.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   4 <title>Complements are supported too - and when to use them</title>
   5 <link rel="stylesheet" href="../../../../../../../../../doc/src/boostbook.css" type="text/css">
   6 <meta name="generator" content="DocBook XSL Stylesheets V1.76.1">
   7 <link rel="home" href="../../../../index.html" title="Math Toolkit">
   8 <link rel="up" href="../overview.html" title="Overview of Distributions">
   9 <link rel="prev" href="generic.html" title="Generic operations common to all distributions are non-member functions">
  10 <link rel="next" href="parameters.html" title="Parameters can be calculated">
  11 </head>
  12 <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
  13 <table cellpadding="2" width="100%"><tr>
  14 <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../../boost.png"></td>
  15 <td align="center"><a href="../../../../../../../../../index.html">Home</a></td>
  16 <td align="center"><a href="../../../../../../../../../libs/libraries.htm">Libraries</a></td>
  17 <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
  18 <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
  19 <td align="center"><a href="../../../../../../../../../more/index.htm">More</a></td>
  20 </tr></table>
  21 <hr>
  22 <div class="spirit-nav">
  23 <a accesskey="p" href="generic.html"><img src="../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../overview.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="parameters.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a>
  24 </div>
  25 <div class="section math_toolkit_dist_stat_tut_overview_complements">
  26 <div class="titlepage"><div><div><h5 class="title">
  27 <a name="math_toolkit.dist.stat_tut.overview.complements"></a><a name="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 </h5></div></div></div>
  30 <p>
  31             Often you don't want the value of the CDF, but its complement, which
  32             is 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
  43             all 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
  51             as well. Any function that accepts a probability as an argument can also
  52             accept 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
  88               binomial distribution with success ratio 0.1 and 100 trials in total.
  89               The <span class="emphasis"><em>lower critical value</em></span> for a probability of
  90               0.05 is given by:
  91             </p>
  92 <p>
  93               <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>
  94             </p>
  95 <p>
  96               and the <span class="emphasis"><em>upper critical value</em></span> is given by:
  97             </p>
  98 <p>
  99               <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>
 100             </p>
 101 <p>
 102               which return 4.82 and 14.63 respectively.
 103             </p>
 104 </td></tr>
 105 </table></div>
 106 <a name="why_complements"></a><div class="tip"><table border="0" summary="Tip">
 107 <tr>
 108 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../../../../../doc/src/images/tip.png"></td>
 109 <th align="left">Tip</th>
 110 </tr>
 111 <tr><td align="left" valign="top">
 112 <p>
 113               <span class="bold"><strong>Why bother with complements anyway?</strong></span>
 114             </p>
 115 <p>
 116               It's very tempting to dispense with complements, and simply subtract
 117               the probability from 1 when required. However, consider what happens
 118               when the probability is very close to 1: let's say the probability
 119               expressed at float precision is <code class="computeroutput"><span class="number">0.999999940f</span></code>,
 120               then <code class="computeroutput"><span class="number">1</span> <span class="special">-</span>
 121               <span class="number">0.999999940f</span> <span class="special">=</span>
 122               <span class="number">5.96046448e-008</span></code>, but the result
 123               is actually accurate to just <span class="emphasis"><em>one single bit</em></span>: the
 124               only bit that didn't cancel out!
 125             </p>
 126 <p>
 127               Or to look at this another way: consider that we want the risk of falsely
 128               rejecting the null-hypothesis in the Student's t test to be 1 in 1
 129               billion, for a sample size of 10,000. This gives a probability of 1
 130               - 10<sup>-9</sup>, which is exactly 1 when calculated at float precision. In this
 131               case calculating the quantile from the complement neatly solves the
 132               problem, so for example:
 133             </p>
 134 <p>
 135               <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>
 136             </p>
 137 <p>
 138               returns the expected t-statistic <code class="computeroutput"><span class="number">6.00336</span></code>,
 139               where as:
 140             </p>
 141 <p>
 142               <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>
 143             </p>
 144 <p>
 145               raises an overflow error, since it is the same as:
 146             </p>
 147 <p>
 148               <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>
 149             </p>
 150 <p>
 151               Which has no finite result.
 152             </p>
 153 <p>
 154               With all distributions, even for more reasonable probability (unless
 155               the value of p can be represented exactly in the floating-point type)
 156               the loss of accuracy quickly becomes significant if you simply calculate
 157               probability from 1 - p (because it will be mostly garbage digits for
 158               p ~ 1).
 159             </p>
 160 <p>
 161               So always avoid, for example, using a probability near to unity like
 162               0.99999
 163             </p>
 164 <p>
 165               <code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="identifier">my_distribution</span><span class="special">,</span>
 166               <span class="number">0.99999</span><span class="special">)</span></code>
 167             </p>
 168 <p>
 169               and instead use
 170             </p>
 171 <p>
 172               <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>
 173               <span class="number">0.00001</span><span class="special">))</span></code>
 174             </p>
 175 <p>
 176               since 1 - 0.99999 is not exactly equal to 0.00001 when using floating-point
 177               arithmetic.
 178             </p>
 179 <p>
 180               This assumes that the 0.00001 value is either a constant, or can be
 181               computed by some manner other than subtracting 0.99999 from 1.
 182             </p>
 183 </td></tr>
 184 </table></div>
 185 </div>
 186 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 187 <td align="left"></td>
 188 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno
 189       Lalande, Johan R&#229;de, Gautam Sewani, Thijs van den Berg and Benjamin Sobotta<p>
 190         Distributed under the Boost Software License, Version 1.0. (See accompanying
 191         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>)
 192       </p>
 193 </div></td>
 194 </tr></table>
 195 <hr>
 196 <div class="spirit-nav">
 197 <a accesskey="p" href="generic.html"><img src="../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../overview.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="parameters.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a>
 198 </div>
 199 </body>
 200 </html>