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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;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">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 <div class="blockquote"><blockquote class="blockquote"><p>
 110             <span class="serif_italic">f(x;&#956;, &#955;) = &#8730;(&#955;/2&#960;x<sup>3</sup>) e<sup>-&#955;(x-&#956;)&#178;/2&#956;&#178;x</sup> </span>
 111           </p></blockquote></div>
 112 <p>
 113           and Cumulative Density Function (CDF):
 114         </p>
 115 <div class="blockquote"><blockquote class="blockquote"><p>
 116             <span class="serif_italic">F(x;&#956;, &#955;) = &#934;{&#8730;(&#955;<span class="emphasis"><em>x) (x</em></span>&#956;-1)}
 117             + e<sup>2&#956;/&#955;</sup> &#934;{-&#8730;(&#955;/&#956;) (1+x/&#956;)} </span>
 118           </p></blockquote></div>
 119 <p>
 120           where &#934; is the standard normal distribution CDF.
 121         </p>
 122 <p>
 123           The following graphs illustrate how the PDF and CDF of the inverse_gaussian
 124           distribution varies for a few values of parameters &#956; and &#955;:
 125         </p>
 126 <div class="blockquote"><blockquote class="blockquote"><p>
 127             <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_pdf.svg" align="middle"></span>
 128
 129           </p></blockquote></div>
 130 <div class="blockquote"><blockquote class="blockquote"><p>
 131             <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_cdf.svg" align="middle"></span>
 132
 133           </p></blockquote></div>
 134 <p>
 135           Tweedie also provided 3 other parameterisations where (&#956; and &#955;) are replaced
 136           by their ratio &#966; = &#955;/&#956; and by 1/&#956;: these forms may be more suitable for Bayesian
 137           applications. These can be found on Seshadri, page 2 and are also discussed
 138           by Chhikara and Folks on page 105. Another related parameterisation, the
 139           __wald_distrib (where mean &#956; is unity) is also provided.
 140         </p>
 141 <h5>
 142 <a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h0"></a>
 143           <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
 144           Functions</a>
 145         </h5>
 146 <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>
 147 </pre>
 148 <p>
 149           Constructs an inverse_gaussian distribution with &#956; mean, and scale &#955;, with
 150           both default values 1.
 151         </p>
 152 <p>
 153           Requires that both the mean &#956; parameter and scale &#955; are greater than zero,
 154           otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 155         </p>
 156 <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>
 157 </pre>
 158 <p>
 159           Returns the mean &#956; parameter of this distribution.
 160         </p>
 161 <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>
 162 </pre>
 163 <p>
 164           Returns the scale &#955; parameter of this distribution.
 165         </p>
 166 <h5>
 167 <a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h1"></a>
 168           <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
 169           Accessors</a>
 170         </h5>
 171 <p>
 172           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 173           functions</a> that are generic to all distributions are supported:
 174           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 175           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 176           <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>,
 177           <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>,
 178           <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>,
 179           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 180           <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>,
 181           <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>.
 182         </p>
 183 <p>
 184           The domain of the random variate is [0,+&#8734;).
 185         </p>
 186 <div class="note"><table border="0" summary="Note">
 187 <tr>
 188 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 189 <th align="left">Note</th>
 190 </tr>
 191 <tr><td align="left" valign="top"><p>
 192             Unlike some definitions, this implementation supports a random variate
 193             equal to zero as a special case, returning zero for both pdf and cdf.
 194           </p></td></tr>
 195 </table></div>
 196 <h5>
 197 <a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h2"></a>
 198           <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>
 199         </h5>
 200 <p>
 201           The inverse_gaussian distribution is implemented in terms of the exponential
 202           function and standard normal distribution <span class="emphasis"><em>N</em></span>0,1 &#934; : refer
 203           to the accuracy data for those functions for more information. But in general,
 204           gamma (and thus inverse gamma) results are often accurate to a few epsilon,
 205           &gt;14 decimal digits accuracy for 64-bit double.
 206         </p>
 207 <h5>
 208 <a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h3"></a>
 209           <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>
 210         </h5>
 211 <p>
 212           In the following table &#956; is the mean parameter and &#955; is the scale parameter
 213           of the inverse_gaussian distribution, <span class="emphasis"><em>x</em></span> is the random
 214           variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>
 215           its complement. Parameters &#956; for shape and &#955; for scale are used for the inverse
 216           gaussian function.
 217         </p>
 218 <div class="informaltable"><table class="table">
 219 <colgroup>
 220 <col>
 221 <col>
 222 </colgroup>
 223 <thead><tr>
 224 <th>
 225                   <p>
 226                     Function
 227                   </p>
 228                 </th>
 229 <th>
 230                   <p>
 231                     Implementation Notes
 232                   </p>
 233                 </th>
 234 </tr></thead>
 235 <tbody>
 236 <tr>
 237 <td>
 238                   <p>
 239                     pdf
 240                   </p>
 241                 </td>
 242 <td>
 243                   <p>
 244                     &#8730;(&#955;/ 2&#960;x<sup>3</sup>) e<sup>-&#955;(x - &#956;)&#178;/ 2&#956;&#178;x</sup>
 245                   </p>
 246                 </td>
 247 </tr>
 248 <tr>
 249 <td>
 250                   <p>
 251                     cdf
 252                   </p>
 253                 </td>
 254 <td>
 255                   <p>
 256                     &#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;)}
 257                   </p>
 258                 </td>
 259 </tr>
 260 <tr>
 261 <td>
 262                   <p>
 263                     cdf complement
 264                   </p>
 265                 </td>
 266 <td>
 267                   <p>
 268                     using complement of &#934; above.
 269                   </p>
 270                 </td>
 271 </tr>
 272 <tr>
 273 <td>
 274                   <p>
 275                     quantile
 276                   </p>
 277                 </td>
 278 <td>
 279                   <p>
 280                     No closed form known. Estimated using a guess refined by Newton-Raphson
 281                     iteration.
 282                   </p>
 283                 </td>
 284 </tr>
 285 <tr>
 286 <td>
 287                   <p>
 288                     quantile from the complement
 289                   </p>
 290                 </td>
 291 <td>
 292                   <p>
 293                     No closed form known. Estimated using a guess refined by Newton-Raphson
 294                     iteration.
 295                   </p>
 296                 </td>
 297 </tr>
 298 <tr>
 299 <td>
 300                   <p>
 301                     mode
 302                   </p>
 303                 </td>
 304 <td>
 305                   <p>
 306                     &#956; {&#8730;(1+9&#956;&#178;/4&#955;&#178;)&#178; - 3&#956;/2&#955;}
 307                   </p>
 308                 </td>
 309 </tr>
 310 <tr>
 311 <td>
 312                   <p>
 313                     median
 314                   </p>
 315                 </td>
 316 <td>
 317                   <p>
 318                     No closed form analytic equation is known, but is evaluated as
 319                     quantile(0.5)
 320                   </p>
 321                 </td>
 322 </tr>
 323 <tr>
 324 <td>
 325                   <p>
 326                     mean
 327                   </p>
 328                 </td>
 329 <td>
 330                   <p>
 331                     &#956;
 332                   </p>
 333                 </td>
 334 </tr>
 335 <tr>
 336 <td>
 337                   <p>
 338                     variance
 339                   </p>
 340                 </td>
 341 <td>
 342                   <p>
 343                     &#956;&#179;/&#955;
 344                   </p>
 345                 </td>
 346 </tr>
 347 <tr>
 348 <td>
 349                   <p>
 350                     skewness
 351                   </p>
 352                 </td>
 353 <td>
 354                   <p>
 355                     3 &#8730; (&#956;/&#955;)
 356                   </p>
 357                 </td>
 358 </tr>
 359 <tr>
 360 <td>
 361                   <p>
 362                     kurtosis_excess
 363                   </p>
 364                 </td>
 365 <td>
 366                   <p>
 367                     15&#956;/&#955;
 368                   </p>
 369                 </td>
 370 </tr>
 371 <tr>
 372 <td>
 373                   <p>
 374                     kurtosis
 375                   </p>
 376                 </td>
 377 <td>
 378                   <p>
 379                     12&#956;/&#955;
 380                   </p>
 381                 </td>
 382 </tr>
 383 </tbody>
 384 </table></div>
 385 <h5>
 386 <a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h4"></a>
 387           <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>
 388         </h5>
 389 <div class="orderedlist"><ol class="orderedlist" type="1">
 390 <li class="listitem">
 391               Wald, A. (1947). Sequential analysis. Wiley, NY.
 392             </li>
 393 <li class="listitem">
 394               The Inverse Gaussian distribution : theory, methodology, and applications,
 395               Raj S. Chhikara, J. Leroy Folks. ISBN 0824779975 (1989).
 396             </li>
 397 <li class="listitem">
 398               The Inverse Gaussian distribution : statistical theory and applications,
 399               Seshadri, V , ISBN - 0387986189 (pbk) (Dewey 519.2) (1998).
 400             </li>
 401 <li class="listitem">
 402               <a href="http://docs.scipy.org/doc/numpy/reference/generated/numpy.random.wald.html" target="_top">Numpy
 403               and Scipy Documentation</a>.
 404             </li>
 405 <li class="listitem">
 406               <a href="http://bm2.genes.nig.ac.jp/RGM2/R_current/library/statmod/man/invgauss.html" target="_top">R
 407               statmod invgauss functions</a>.
 408             </li>
 409 <li class="listitem">
 410               <a href="http://cran.r-project.org/web/packages/SuppDists/index.html" target="_top">R
 411               SuppDists invGauss functions</a>. (Note that these R implementations
 412               names differ in case).
 413             </li>
 414 <li class="listitem">
 415               <a href="http://www.statsci.org/s/invgauss.html" target="_top">StatSci.org invgauss
 416               help</a>.
 417             </li>
 418 <li class="listitem">
 419               <a href="http://www.statsci.org/s/invgauss.statSci.org" target="_top">invgauss
 420               R source</a>.
 421             </li>
 422 <li class="listitem">
 423               <a href="http://www.biostat.wustl.edu/archives/html/s-news/2001-12/msg00144.html" target="_top">pwald,
 424               qwald</a>.
 425             </li>
 426 <li class="listitem">
 427               <a href="http://www.brighton-webs.co.uk/distributions/wald.asp" target="_top">Brighton
 428               Webs wald</a>.
 429             </li>
 430 </ol></div>
 431 </div>
 432 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 433 <td align="left"></td>
 434 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 435       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 436       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 437       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 438       Daryle Walker and Xiaogang Zhang<p>
 439         Distributed under the Boost Software License, Version 1.0. (See accompanying
 440         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>)
 441       </p>
 442 </div></td>
 443 </tr></table>
 444 <hr>
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