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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist"></a><a class="link" href="inverse_gaussian_dist.html" title="Inverse Gaussian (or Inverse Normal) Distribution">Inverse
  28         Gaussian (or Inverse Normal) 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">inverse_gaussian</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;15.&#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">inverse_gaussian_distribution</span>
  36 <span class="special">{</span>
  37 <span class="keyword">public</span><span class="special">:</span>
  38    <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
  39    <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
  40
  41    <span class="identifier">inverse_gaussian_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
  42
  43    <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// mean default 1.</span>
  44    <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Optional scale, default 1 (unscaled).</span>
  45    <span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Shape = scale/mean.</span>
  46 <span class="special">};</span>
  47 <span class="keyword">typedef</span> <span class="identifier">inverse_gaussian_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">inverse_gaussian</span><span class="special">;</span>
  48
  49 <span class="special">}}</span> <span class="comment">// namespace boost // namespace math</span>
  50 </pre>
  51 <p>
  52           The Inverse Gaussian distribution distribution is a continuous probability
  53           distribution.
  54         </p>
  55 <p>
  56           The distribution is also called 'normal-inverse Gaussian distribution',
  57           and 'normal Inverse' distribution.
  58         </p>
  59 <p>
  60           It is also convenient to provide unity as default for both mean and scale.
  61           This is the Standard form for all distributions. The Inverse Gaussian distribution
  62           was first studied in relation to Brownian motion. In 1956 M.C.K. Tweedie
  63           used the name Inverse Gaussian because there is an inverse relationship
  64           between the time to cover a unit distance and distance covered in unit
  65           time. The inverse Gaussian is one of family of distributions that have
  66           been called the <a href="http://en.wikipedia.org/wiki/Tweedie_distributions" target="_top">Tweedie
  67           distributions</a>.
  68         </p>
  69 <p>
  70           (So <span class="emphasis"><em>inverse</em></span> in the name may mislead: it does <span class="bold"><strong>not</strong></span> relate to the inverse of a distribution).
  71         </p>
  72 <p>
  73           The tails of the distribution decrease more slowly than the normal distribution.
  74           It is therefore suitable to model phenomena where numerically large values
  75           are more probable than is the case for the normal distribution. For stock
  76           market returns and prices, a key characteristic is that it models that
  77           extremely large variations from typical (crashes) can occur even when almost
  78           all (normal) variations are small.
  79         </p>
  80 <p>
  81           Examples are returns from financial assets and turbulent wind speeds.
  82         </p>
  83 <p>
  84           The normal-inverse Gaussian distributions form a subclass of the generalised
  85           hyperbolic distributions.
  86         </p>
  87 <p>
  88           See <a href="http://en.wikipedia.org/wiki/Normal-inverse_Gaussian_distribution" target="_top">distribution</a>.
  89           <a href="http://mathworld.wolfram.com/InverseGaussianDistribution.html" target="_top">Weisstein,
  90           Eric W. "Inverse Gaussian Distribution." From MathWorld--A Wolfram
  91           Web Resource.</a>
  92         </p>
  93 <p>
  94           If you want a <code class="computeroutput"><span class="keyword">double</span></code> precision
  95           inverse_gaussian distribution you can use
  96         </p>
  97 <pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">inverse_gaussian_distribution</span><span class="special">&lt;&gt;</span></pre>
  98 <p>
  99           or, more conveniently, you can write
 100         </p>
 101 <pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">inverse_gaussian</span><span class="special">;</span>
 102 <span class="identifier">inverse_gaussian</span> <span class="identifier">my_ig</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">3</span><span class="special">);</span>
 103 </pre>
 104 <p>
 105           For mean parameters &#956; and scale (also called precision) parameter &#955;, and random
 106           variate x, the inverse_gaussian distribution is defined by the probability
 107           density function (PDF):
 108         </p>
 109 <p>
 110           &#8192;&#8192; f(x;&#956;, &#955;) = &#8730;(&#955;/2&#960;x<sup>3</sup>) e<sup>-&#955;(x-&#956;)&#178;/2&#956;&#178;x</sup>
 111         </p>
 112 <p>
 113           and Cumulative Density Function (CDF):
 114         </p>
 115 <p>
 116           &#8192;&#8192;  F(x;&#956;, &#955;) = &#934;{&#8730;(&#955;<span class="emphasis"><em>x) (x</em></span>&#956;-1)} + e<sup>2&#956;/&#955;</sup> &#934;{-&#8730;(&#955;/&#956;) (1+x/&#956;)}
 117         </p>
 118 <p>
 119           where &#934; is the standard normal distribution CDF.
 120         </p>
 121 <p>
 122           The following graphs illustrate how the PDF and CDF of the inverse_gaussian
 123           distribution varies for a few values of parameters &#956; and &#955;:
 124         </p>
 125 <p>
 126           <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_pdf.svg" align="middle"></span>
 127         </p>
 128 <p>
 129           <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_cdf.svg" align="middle"></span>
 130         </p>
 131 <p>
 132           Tweedie also provided 3 other parameterisations where (&#956; and &#955;) are replaced
 133           by their ratio &#966; = &#955;/&#956; and by 1/&#956;: these forms may be more suitable for Bayesian
 134           applications. These can be found on Seshadri, page 2 and are also discussed
 135           by Chhikara and Folks on page 105. Another related parameterisation, the
 136           __wald_distrib (where mean &#956; is unity) is also provided.
 137         </p>
 138 <h5>
 139 <a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h0"></a>
 140           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.member_functions"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist_ref.dists.inverse_gaussian_dist.member_functions">Member
 141           Functions</a>
 142         </h5>
 143 <pre class="programlisting"><span class="identifier">inverse_gaussian_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// optionally scaled.</span>
 144 </pre>
 145 <p>
 146           Constructs an inverse_gaussian distribution with &#956; mean, and scale &#955;, with
 147           both default values 1.
 148         </p>
 149 <p>
 150           Requires that both the mean &#956; parameter and scale &#955; are greater than zero,
 151           otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 152         </p>
 153 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 154 </pre>
 155 <p>
 156           Returns the mean &#956; parameter of this distribution.
 157         </p>
 158 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 159 </pre>
 160 <p>
 161           Returns the scale &#955; parameter of this distribution.
 162         </p>
 163 <h5>
 164 <a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h1"></a>
 165           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.non_member_accessors"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist_ref.dists.inverse_gaussian_dist.non_member_accessors">Non-member
 166           Accessors</a>
 167         </h5>
 168 <p>
 169           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 170           functions</a> that are generic to all distributions are supported:
 171           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 172           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 173           <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>,
 174           <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>,
 175           <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>,
 176           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 177           <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>,
 178           <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>.
 179         </p>
 180 <p>
 181           The domain of the random variate is [0,+&#8734;).
 182         </p>
 183 <div class="note"><table border="0" summary="Note">
 184 <tr>
 185 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 186 <th align="left">Note</th>
 187 </tr>
 188 <tr><td align="left" valign="top"><p>
 189             Unlike some definitions, this implementation supports a random variate
 190             equal to zero as a special case, returning zero for both pdf and cdf.
 191           </p></td></tr>
 192 </table></div>
 193 <h5>
 194 <a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h2"></a>
 195           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.accuracy"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist_ref.dists.inverse_gaussian_dist.accuracy">Accuracy</a>
 196         </h5>
 197 <p>
 198           The inverse_gaussian distribution is implemented in terms of the exponential
 199           function and standard normal distribution <span class="emphasis"><em>N</em></span>0,1 &#934; : refer
 200           to the accuracy data for those functions for more information. But in general,
 201           gamma (and thus inverse gamma) results are often accurate to a few epsilon,
 202           &gt;14 decimal digits accuracy for 64-bit double.
 203         </p>
 204 <h5>
 205 <a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h3"></a>
 206           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.implementation"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist_ref.dists.inverse_gaussian_dist.implementation">Implementation</a>
 207         </h5>
 208 <p>
 209           In the following table &#956; is the mean parameter and &#955; is the scale parameter
 210           of the inverse_gaussian distribution, <span class="emphasis"><em>x</em></span> is the random
 211           variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>
 212           its complement. Parameters &#956; for shape and &#955; for scale are used for the inverse
 213           gaussian function.
 214         </p>
 215 <div class="informaltable"><table class="table">
 216 <colgroup>
 217 <col>
 218 <col>
 219 </colgroup>
 220 <thead><tr>
 221 <th>
 222                   <p>
 223                     Function
 224                   </p>
 225                 </th>
 226 <th>
 227                   <p>
 228                     Implementation Notes
 229                   </p>
 230                 </th>
 231 </tr></thead>
 232 <tbody>
 233 <tr>
 234 <td>
 235                   <p>
 236                     pdf
 237                   </p>
 238                 </td>
 239 <td>
 240                   <p>
 241                     &#8730;(&#955;/ 2&#960;x<sup>3</sup>) e<sup>-&#955;(x - &#956;)&#178;/ 2&#956;&#178;x</sup>
 242                   </p>
 243                 </td>
 244 </tr>
 245 <tr>
 246 <td>
 247                   <p>
 248                     cdf
 249                   </p>
 250                 </td>
 251 <td>
 252                   <p>
 253                     &#934;{&#8730;(&#955;<span class="emphasis"><em>x) (x</em></span>&#956;-1)} + e<sup>2&#956;/&#955;</sup> &#934;{-&#8730;(&#955;/&#956;) (1+x/&#956;)}
 254                   </p>
 255                 </td>
 256 </tr>
 257 <tr>
 258 <td>
 259                   <p>
 260                     cdf complement
 261                   </p>
 262                 </td>
 263 <td>
 264                   <p>
 265                     using complement of &#934; above.
 266                   </p>
 267                 </td>
 268 </tr>
 269 <tr>
 270 <td>
 271                   <p>
 272                     quantile
 273                   </p>
 274                 </td>
 275 <td>
 276                   <p>
 277                     No closed form known. Estimated using a guess refined by Newton-Raphson
 278                     iteration.
 279                   </p>
 280                 </td>
 281 </tr>
 282 <tr>
 283 <td>
 284                   <p>
 285                     quantile from the complement
 286                   </p>
 287                 </td>
 288 <td>
 289                   <p>
 290                     No closed form known. Estimated using a guess refined by Newton-Raphson
 291                     iteration.
 292                   </p>
 293                 </td>
 294 </tr>
 295 <tr>
 296 <td>
 297                   <p>
 298                     mode
 299                   </p>
 300                 </td>
 301 <td>
 302                   <p>
 303                     &#956; {&#8730;(1+9&#956;&#178;/4&#955;&#178;)&#178; - 3&#956;/2&#955;}
 304                   </p>
 305                 </td>
 306 </tr>
 307 <tr>
 308 <td>
 309                   <p>
 310                     median
 311                   </p>
 312                 </td>
 313 <td>
 314                   <p>
 315                     No closed form analytic equation is known, but is evaluated as
 316                     quantile(0.5)
 317                   </p>
 318                 </td>
 319 </tr>
 320 <tr>
 321 <td>
 322                   <p>
 323                     mean
 324                   </p>
 325                 </td>
 326 <td>
 327                   <p>
 328                     &#956;
 329                   </p>
 330                 </td>
 331 </tr>
 332 <tr>
 333 <td>
 334                   <p>
 335                     variance
 336                   </p>
 337                 </td>
 338 <td>
 339                   <p>
 340                     &#956;&#179;/&#955;
 341                   </p>
 342                 </td>
 343 </tr>
 344 <tr>
 345 <td>
 346                   <p>
 347                     skewness
 348                   </p>
 349                 </td>
 350 <td>
 351                   <p>
 352                     3 &#8730; (&#956;/&#955;)
 353                   </p>
 354                 </td>
 355 </tr>
 356 <tr>
 357 <td>
 358                   <p>
 359                     kurtosis_excess
 360                   </p>
 361                 </td>
 362 <td>
 363                   <p>
 364                     15&#956;/&#955;
 365                   </p>
 366                 </td>
 367 </tr>
 368 <tr>
 369 <td>
 370                   <p>
 371                     kurtosis
 372                   </p>
 373                 </td>
 374 <td>
 375                   <p>
 376                     12&#956;/&#955;
 377                   </p>
 378                 </td>
 379 </tr>
 380 </tbody>
 381 </table></div>
 382 <h5>
 383 <a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h4"></a>
 384           <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.references"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist_ref.dists.inverse_gaussian_dist.references">References</a>
 385         </h5>
 386 <div class="orderedlist"><ol class="orderedlist" type="1">
 387 <li class="listitem">
 388               Wald, A. (1947). Sequential analysis. Wiley, NY.
 389             </li>
 390 <li class="listitem">
 391               The Inverse Gaussian distribution : theory, methodology, and applications,
 392               Raj S. Chhikara, J. Leroy Folks. ISBN 0824779975 (1989).
 393             </li>
 394 <li class="listitem">
 395               The Inverse Gaussian distribution : statistical theory and applications,
 396               Seshadri, V , ISBN - 0387986189 (pbk) (Dewey 519.2) (1998).
 397             </li>
 398 <li class="listitem">
 399               <a href="http://docs.scipy.org/doc/numpy/reference/generated/numpy.random.wald.html" target="_top">Numpy
 400               and Scipy Documentation</a>.
 401             </li>
 402 <li class="listitem">
 403               <a href="http://bm2.genes.nig.ac.jp/RGM2/R_current/library/statmod/man/invgauss.html" target="_top">R
 404               statmod invgauss functions</a>.
 405             </li>
 406 <li class="listitem">
 407               <a href="http://cran.r-project.org/web/packages/SuppDists/index.html" target="_top">R
 408               SuppDists invGauss functions</a>. (Note that these R implementations
 409               names differ in case).
 410             </li>
 411 <li class="listitem">
 412               <a href="http://www.statsci.org/s/invgauss.html" target="_top">StatSci.org invgauss
 413               help</a>.
 414             </li>
 415 <li class="listitem">
 416               <a href="http://www.statsci.org/s/invgauss.statSci.org" target="_top">invgauss
 417               R source</a>.
 418             </li>
 419 <li class="listitem">
 420               <a href="http://www.biostat.wustl.edu/archives/html/s-news/2001-12/msg00144.html" target="_top">pwald,
 421               qwald</a>.
 422             </li>
 423 <li class="listitem">
 424               <a href="http://www.brighton-webs.co.uk/distributions/wald.asp" target="_top">Brighton
 425               Webs wald</a>.
 426             </li>
 427 </ol></div>
 428 </div>
 429 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 430 <td align="left"></td>
 431 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 432       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 433       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 434       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 435         Distributed under the Boost Software License, Version 1.0. (See accompanying
 436         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>)
 437       </p>
 438 </div></td>
 439 </tr></table>
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