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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.hypergeometric_dist"></a><a class="link" href="hypergeometric_dist.html" title="Hypergeometric Distribution">Hypergeometric
  28         Distribution</a>
  29 </h4></div></div></div>
  30 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">hypergeometric</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
  31 <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
  32
  33 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
  34           <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
  35 <span class="keyword">class</span> <span class="identifier">hypergeometric_distribution</span><span class="special">;</span>
  36
  37 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">&gt;</span>
  38 <span class="keyword">class</span> <span class="identifier">hypergeometric_distribution</span>
  39 <span class="special">{</span>
  40 <span class="keyword">public</span><span class="special">:</span>
  41    <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
  42    <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
  43    <span class="comment">// Construct:</span>
  44    <span class="identifier">hypergeometric_distribution</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">r</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">N</span><span class="special">);</span>
  45    <span class="comment">// Accessors:</span>
  46    <span class="keyword">unsigned</span> <span class="identifier">total</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  47    <span class="keyword">unsigned</span> <span class="identifier">defective</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  48    <span class="keyword">unsigned</span> <span class="identifier">sample_count</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  49 <span class="special">};</span>
  50
  51 <span class="keyword">typedef</span> <span class="identifier">hypergeometric_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">hypergeometric</span><span class="special">;</span>
  52
  53 <span class="special">}}</span> <span class="comment">// namespaces</span>
  54 </pre>
  55 <p>
  56           The hypergeometric distribution describes the number of "events"
  57           <span class="emphasis"><em>k</em></span> from a sample <span class="emphasis"><em>n</em></span> drawn from
  58           a total population <span class="emphasis"><em>N</em></span> <span class="emphasis"><em>without replacement</em></span>.
  59         </p>
  60 <p>
  61           Imagine we have a sample of <span class="emphasis"><em>N</em></span> objects of which <span class="emphasis"><em>r</em></span>
  62           are "defective" and N-r are "not defective" (the terms
  63           "success/failure" or "red/blue" are also used). If
  64           we sample <span class="emphasis"><em>n</em></span> items <span class="emphasis"><em>without replacement</em></span>
  65           then what is the probability that exactly <span class="emphasis"><em>k</em></span> items
  66           in the sample are defective? The answer is given by the pdf of the hypergeometric
  67           distribution <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">k</span><span class="special">;</span> <span class="identifier">r</span><span class="special">,</span> <span class="identifier">n</span><span class="special">,</span>
  68           <span class="identifier">N</span><span class="special">)</span></code>,
  69           whilst the probability of <span class="emphasis"><em>k</em></span> defectives or fewer is
  70           given by F(k; r, n, N), where F(k) is the CDF of the hypergeometric distribution.
  71         </p>
  72 <div class="note"><table border="0" summary="Note">
  73 <tr>
  74 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
  75 <th align="left">Note</th>
  76 </tr>
  77 <tr><td align="left" valign="top"><p>
  78             Unlike almost all of the other distributions in this library, the hypergeometric
  79             distribution is strictly discrete: it can not be extended to real valued
  80             arguments of its parameters or random variable.
  81           </p></td></tr>
  82 </table></div>
  83 <p>
  84           The following graph shows how the distribution changes as the proportion
  85           of "defective" items changes, while keeping the population and
  86           sample sizes constant:
  87         </p>
  88 <div class="blockquote"><blockquote class="blockquote"><p>
  89             <span class="inlinemediaobject"><img src="../../../../graphs/hypergeometric_pdf_1.svg" align="middle"></span>
  90
  91           </p></blockquote></div>
  92 <p>
  93           Note that since the distribution is symmetrical in parameters <span class="emphasis"><em>n</em></span>
  94           and <span class="emphasis"><em>r</em></span>, if we change the sample size and keep the population
  95           and proportion "defective" the same then we obtain basically
  96           the same graphs:
  97         </p>
  98 <div class="blockquote"><blockquote class="blockquote"><p>
  99             <span class="inlinemediaobject"><img src="../../../../graphs/hypergeometric_pdf_2.svg" align="middle"></span>
 100
 101           </p></blockquote></div>
 102 <h5>
 103 <a name="math_toolkit.dist_ref.dists.hypergeometric_dist.h0"></a>
 104           <span class="phrase"><a name="math_toolkit.dist_ref.dists.hypergeometric_dist.member_functions"></a></span><a class="link" href="hypergeometric_dist.html#math_toolkit.dist_ref.dists.hypergeometric_dist.member_functions">Member
 105           Functions</a>
 106         </h5>
 107 <pre class="programlisting"><span class="identifier">hypergeometric_distribution</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">r</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">N</span><span class="special">);</span>
 108 </pre>
 109 <p>
 110           Constructs a hypergeometric distribution with a population of <span class="emphasis"><em>N</em></span>
 111           objects, of which <span class="emphasis"><em>r</em></span> are defective, and from which
 112           <span class="emphasis"><em>n</em></span> are sampled.
 113         </p>
 114 <pre class="programlisting"><span class="keyword">unsigned</span> <span class="identifier">total</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 115 </pre>
 116 <p>
 117           Returns the total number of objects <span class="emphasis"><em>N</em></span>.
 118         </p>
 119 <pre class="programlisting"><span class="keyword">unsigned</span> <span class="identifier">defective</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 120 </pre>
 121 <p>
 122           Returns the number of objects <span class="emphasis"><em>r</em></span> in population <span class="emphasis"><em>N</em></span>
 123           which are defective.
 124         </p>
 125 <pre class="programlisting"><span class="keyword">unsigned</span> <span class="identifier">sample_count</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 126 </pre>
 127 <p>
 128           Returns the number of objects <span class="emphasis"><em>n</em></span> which are sampled
 129           from the population <span class="emphasis"><em>N</em></span>.
 130         </p>
 131 <h5>
 132 <a name="math_toolkit.dist_ref.dists.hypergeometric_dist.h1"></a>
 133           <span class="phrase"><a name="math_toolkit.dist_ref.dists.hypergeometric_dist.non_member_accessors"></a></span><a class="link" href="hypergeometric_dist.html#math_toolkit.dist_ref.dists.hypergeometric_dist.non_member_accessors">Non-member
 134           Accessors</a>
 135         </h5>
 136 <p>
 137           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 138           functions</a> that are generic to all distributions are supported:
 139           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 140           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 141           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
 142           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
 143           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
 144           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 145           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
 146           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
 147         </p>
 148 <p>
 149           The domain of the random variable is the unsigned integers in the range
 150           [max(0, n + r - N), min(n, r)]. A <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 151           is raised if the random variable is outside this range, or is not an integral
 152           value.
 153         </p>
 154 <div class="caution"><table border="0" summary="Caution">
 155 <tr>
 156 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Caution]" src="../../../../../../../doc/src/images/caution.png"></td>
 157 <th align="left">Caution</th>
 158 </tr>
 159 <tr><td align="left" valign="top">
 160 <p>
 161             The quantile function will by default return an integer result that has
 162             been <span class="emphasis"><em>rounded outwards</em></span>. That is to say lower quantiles
 163             (where the probability is less than 0.5) are rounded downward, and upper
 164             quantiles (where the probability is greater than 0.5) are rounded upwards.
 165             This behaviour ensures that if an X% quantile is requested, then <span class="emphasis"><em>at
 166             least</em></span> the requested coverage will be present in the central
 167             region, and <span class="emphasis"><em>no more than</em></span> the requested coverage
 168             will be present in the tails.
 169           </p>
 170 <p>
 171             This behaviour can be changed so that the quantile functions are rounded
 172             differently using <a class="link" href="../../pol_overview.html" title="Policy Overview">Policies</a>.
 173             It is strongly recommended that you read the tutorial <a class="link" href="../../pol_tutorial/understand_dis_quant.html" title="Understanding Quantiles of Discrete Distributions">Understanding
 174             Quantiles of Discrete Distributions</a> before using the quantile
 175             function on the Hypergeometric distribution. The <a class="link" href="../../pol_ref/discrete_quant_ref.html" title="Discrete Quantile Policies">reference
 176             docs</a> describe how to change the rounding policy for these distributions.
 177           </p>
 178 <p>
 179             However, note that the implementation method of the quantile function
 180             always returns an integral value, therefore attempting to use a <a class="link" href="../../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> that requires (or produces) a real valued
 181             result will result in a compile time error.
 182           </p>
 183 </td></tr>
 184 </table></div>
 185 <h5>
 186 <a name="math_toolkit.dist_ref.dists.hypergeometric_dist.h2"></a>
 187           <span class="phrase"><a name="math_toolkit.dist_ref.dists.hypergeometric_dist.accuracy"></a></span><a class="link" href="hypergeometric_dist.html#math_toolkit.dist_ref.dists.hypergeometric_dist.accuracy">Accuracy</a>
 188         </h5>
 189 <p>
 190           For small N such that <code class="computeroutput"><span class="identifier">N</span> <span class="special">&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">max_factorial</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">value</span></code>
 191           then table based lookup of the results gives an accuracy to a few epsilon.
 192           <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">max_factorial</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">value</span></code> is 170 at double or long double
 193           precision.
 194         </p>
 195 <p>
 196           For larger N such that <code class="computeroutput"><span class="identifier">N</span> <span class="special">&lt;</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">prime</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">max_prime</span><span class="special">)</span></code> then only basic arithmetic is required
 197           for the calculation and the accuracy is typically &lt; 20 epsilon. This
 198           takes care of N up to 104729.
 199         </p>
 200 <p>
 201           For <code class="computeroutput"><span class="identifier">N</span> <span class="special">&gt;</span>
 202           <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">prime</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">max_prime</span><span class="special">)</span></code>
 203           then accuracy quickly degrades, with 5 or 6 decimal digits being lost for
 204           N = 110000.
 205         </p>
 206 <p>
 207           In general for very large N, the user should expect to lose log<sub>10</sub>N decimal
 208           digits of precision during the calculation, with the results becoming meaningless
 209           for N &gt;= 10<sup>15</sup>.
 210         </p>
 211 <h5>
 212 <a name="math_toolkit.dist_ref.dists.hypergeometric_dist.h3"></a>
 213           <span class="phrase"><a name="math_toolkit.dist_ref.dists.hypergeometric_dist.testing"></a></span><a class="link" href="hypergeometric_dist.html#math_toolkit.dist_ref.dists.hypergeometric_dist.testing">Testing</a>
 214         </h5>
 215 <p>
 216           There are three sets of tests: our implementation is tested against a table
 217           of values produced by Mathematica's implementation of this distribution.
 218           We also sanity check our implementation against some spot values computed
 219           using the online calculator here <a href="http://stattrek.com/Tables/Hypergeometric.aspx" target="_top">http://stattrek.com/Tables/Hypergeometric.aspx</a>.
 220           Finally we test accuracy against some high precision test data using this
 221           implementation and NTL::RR.
 222         </p>
 223 <h5>
 224 <a name="math_toolkit.dist_ref.dists.hypergeometric_dist.h4"></a>
 225           <span class="phrase"><a name="math_toolkit.dist_ref.dists.hypergeometric_dist.implementation"></a></span><a class="link" href="hypergeometric_dist.html#math_toolkit.dist_ref.dists.hypergeometric_dist.implementation">Implementation</a>
 226         </h5>
 227 <p>
 228           The PDF can be calculated directly using the formula:
 229         </p>
 230 <div class="blockquote"><blockquote class="blockquote"><p>
 231             <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric1.svg"></span>
 232
 233           </p></blockquote></div>
 234 <p>
 235           However, this can only be used directly when the largest of the factorials
 236           is guaranteed not to overflow the floating point representation used. This
 237           formula is used directly when <code class="computeroutput"><span class="identifier">N</span>
 238           <span class="special">&lt;</span> <span class="identifier">max_factorial</span><span class="special">&lt;</span><span class="identifier">RealType</span><span class="special">&gt;::</span><span class="identifier">value</span></code>
 239           in which case table lookup of the factorials gives a rapid and accurate
 240           implementation method.
 241         </p>
 242 <p>
 243           For larger <span class="emphasis"><em>N</em></span> the method described in "An Accurate
 244           Computation of the Hypergeometric Distribution Function", Trong Wu,
 245           ACM Transactions on Mathematical Software, Vol. 19, No. 1, March 1993,
 246           Pages 33-43 is used. The method relies on the fact that there is an easy
 247           method for factorising a factorial into the product of prime numbers:
 248         </p>
 249 <div class="blockquote"><blockquote class="blockquote"><p>
 250             <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric2.svg"></span>
 251
 252           </p></blockquote></div>
 253 <p>
 254           Where p<sub>i</sub> is the i'th prime number, and e<sub>i</sub> is a small positive integer or
 255           zero, which can be calculated via:
 256         </p>
 257 <div class="blockquote"><blockquote class="blockquote"><p>
 258             <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric3.svg"></span>
 259
 260           </p></blockquote></div>
 261 <p>
 262           Further we can combine the factorials in the expression for the PDF to
 263           yield the PDF directly as the product of prime numbers:
 264         </p>
 265 <div class="blockquote"><blockquote class="blockquote"><p>
 266             <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric4.svg"></span>
 267
 268           </p></blockquote></div>
 269 <p>
 270           With this time the exponents e<sub>i</sub> being either positive, negative or zero.
 271           Indeed such a degree of cancellation occurs in the calculation of the e<sub>i</sub> that
 272           many are zero, and typically most have a magnitude or no more than 1 or
 273           2.
 274         </p>
 275 <p>
 276           Calculation of the product of the primes requires some care to prevent
 277           numerical overflow, we use a novel recursive method which splits the calculation
 278           into a series of sub-products, with a new sub-product started each time
 279           the next multiplication would cause either overflow or underflow. The sub-products
 280           are stored in a linked list on the program stack, and combined in an order
 281           that will guarantee no overflow or unnecessary-underflow once the last
 282           sub-product has been calculated.
 283         </p>
 284 <p>
 285           This method can be used as long as N is smaller than the largest prime
 286           number we have stored in our table of primes (currently 104729). The method
 287           is relatively slow (calculating the exponents requires the most time),
 288           but requires only a small number of arithmetic operations to calculate
 289           the result (indeed there is no shorter method involving only basic arithmetic
 290           once the exponents have been found), the method is therefore much more
 291           accurate than the alternatives.
 292         </p>
 293 <p>
 294           For much larger N, we can calculate the PDF from the factorials using either
 295           lgamma, or by directly combining lanczos approximations to avoid calculating
 296           via logarithms. We use the latter method, as it is usually 1 or 2 decimal
 297           digits more accurate than computing via logarithms with lgamma. However,
 298           in this area where N &gt; 104729, the user should expect to lose around
 299           log<sub>10</sub>N decimal digits during the calculation in the worst case.
 300         </p>
 301 <p>
 302           The CDF and its complement is calculated by directly summing the PDF's.
 303           We start by deciding whether the CDF, or its complement, is likely to be
 304           the smaller of the two and then calculate the PDF at <span class="emphasis"><em>k</em></span>
 305           (or <span class="emphasis"><em>k+1</em></span> if we're calculating the complement) and calculate
 306           successive PDF values via the recurrence relations:
 307         </p>
 308 <div class="blockquote"><blockquote class="blockquote"><p>
 309             <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric5.svg"></span>
 310
 311           </p></blockquote></div>
 312 <p>
 313           Until we either reach the end of the distributions domain, or the next
 314           PDF value to be summed would be too small to affect the result.
 315         </p>
 316 <p>
 317           The quantile is calculated in a similar manner to the CDF: we first guess
 318           which end of the distribution we're nearer to, and then sum PDFs starting
 319           from the end of the distribution this time, until we have some value <span class="emphasis"><em>k</em></span>
 320           that gives the required CDF.
 321         </p>
 322 <p>
 323           The median is simply the quantile at 0.5, and the remaining properties
 324           are calculated via:
 325         </p>
 326 <div class="blockquote"><blockquote class="blockquote"><p>
 327             <span class="inlinemediaobject"><img src="../../../../equations/hypergeometric6.svg"></span>
 328
 329           </p></blockquote></div>
 330 </div>
 331 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 332 <td align="left"></td>
 333 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 334       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 335       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 336       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 337       Daryle Walker and Xiaogang Zhang<p>
 338         Distributed under the Boost Software License, Version 1.0. (See accompanying
 339         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>)
 340       </p>
 341 </div></td>
 342 </tr></table>
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