libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_gaussian_dist.html


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<html>
<head>
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<title>Inverse Gaussian (or Inverse Normal) Distribution</title>
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<div class="section math_toolkit_dist_dist_ref_dists_inverse_gaussian_dist">
<div class="titlepage"><div><div><h5 class="title">
<a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist"></a><a class="link" href="inverse_gaussian_dist.html" title="Inverse Gaussian (or Inverse Normal) Distribution">Inverse
          Gaussian (or Inverse Normal) Distribution</a>
</h5></div></div></div>
<p>
</p>
<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>
<p>
          </p>
<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>

<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>
          <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a>   <span class="special">=</span> <a class="link" href="../../../policy/pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">inverse_gaussian_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
   <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
   <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>

   <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>

   <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>
   <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>
   <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>
<span class="special">};</span>
<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>

<span class="special">}}</span> <span class="comment">// namespace boost // namespace math</span>
</pre>
<p>
            The Inverse Gaussian distribution distribution is a continuous probability
            distribution.
          </p>
<p>
            The distribution is also called 'normal-inverse Gaussian distribution',
            and 'normal Inverse' distribution.
          </p>
<p>
            It is also convenient to provide unity as default for both mean and scale.
            This is the Standard form for all distributions. The Inverse Gaussian
            distribution was first studied in relation to Brownian motion. In 1956
            M.C.K. Tweedie used the name Inverse Gaussian because there is an inverse
            relationship between the time to cover a unit distance and distance covered
            in unit time. The inverse Gaussian is one of family of distributions
            that have been called the <a href="http://en.wikipedia.org/wiki/Tweedie_distributions" target="_top">Tweedie
            distributions</a>.
          </p>
<p>
            (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).
          </p>
<p>
            The tails of the distribution decrease more slowly than the normal distribution.
            It is therefore suitable to model phenomena where numerically large values
            are more probable than is the case for the normal distribution. For stock
            market returns and prices, a key characteristic is that it models that
            extremely large variations from typical (crashes) can occur even when
            almost all (normal) variations are small.
          </p>
<p>
            Examples are returns from financial assets and turbulent wind speeds.
          </p>
<p>
            The normal-inverse Gaussian distributions form a subclass of the generalised
            hyperbolic distributions.
          </p>
<p>
            See <a href="http://en.wikipedia.org/wiki/Normal-inverse_Gaussian_distribution" target="_top">distribution</a>.
            <a href="http://mathworld.wolfram.com/InverseGaussianDistribution.html" target="_top">Weisstein,
            Eric W. "Inverse Gaussian Distribution." From MathWorld--A
            Wolfram Web Resource.</a>
          </p>
<p>
            If you want a <code class="computeroutput"><span class="keyword">double</span></code> precision
            inverse_gaussian distribution you can use
          </p>
<p>
</p>
<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>
<p>
          </p>
<p>
            or, more conveniently, you can write
          </p>
<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>
<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>
</pre>
<p>
            For mean parameters &#956; and scale (also called precision) parameter &#955;, and
            random variate x, the inverse_gaussian distribution is defined by the
            probability density function (PDF):
          </p>
<p>
            &#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>
          </p>
<p>
            and Cumulative Density Function (CDF):
          </p>
<p>
            &#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;)}
          </p>
<p>
            where &#934; is the standard normal distribution CDF.
          </p>
<p>
            The following graphs illustrate how the PDF and CDF of the inverse_gaussian
            distribution varies for a few values of parameters &#956; and &#955;:
          </p>
<p>
            <span class="inlinemediaobject"><img src="../../../../../graphs/inverse_gaussian_pdf.png" align="middle"></span>
          </p>
<p>
            <span class="inlinemediaobject"><img src="../../../../../graphs/inverse_gaussian_cdf.png" align="middle"></span>
          </p>
<p>
            Tweedie also provided 3 other parameterisations where (&#956; and &#955;) are replaced
            by their ratio &#966; = &#955;/&#956; and by 1/&#956;: these forms may be more suitable for Bayesian
            applications. These can be found on Seshadri, page 2 and are also discussed
            by Chhikara and Folks on page 105. Another related parameterisation,
            the __wald_distrib (where mean &#956; is unity) is also provided.
          </p>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.h0"></a>
            <span><a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.member_functions"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.member_functions">Member
            Functions</a>
          </h5>
<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>
</pre>
<p>
            Constructs an inverse_gaussian distribution with &#956; mean, and scale &#955;, with
            both default values 1.
          </p>
<p>
            Requires that both the mean &#956; parameter and scale &#955; are greater than zero,
            otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>.
          </p>
<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>
</pre>
<p>
            Returns the mean &#956; parameter of this distribution.
          </p>
<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>
</pre>
<p>
            Returns the scale &#955; parameter of this distribution.
          </p>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.h1"></a>
            <span><a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.non_member_accessors"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.non_member_accessors">Non-member
            Accessors</a>
          </h5>
<p>
            All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member
            accessor functions</a> that are generic to all distributions are supported:
            <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>,
            <a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>, <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>, <a class="link" href="../nmp.html#math.dist.hazard">Hazard
            Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>,
            <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>,
            <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>,
            <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>,
            <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>,
            <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>.
          </p>
<p>
            The domain of the random variate is [0,+&#8734;).
          </p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
              Unlike some definitions, this implementation supports a random variate
              equal to zero as a special case, returning zero for both pdf and cdf.
            </p></td></tr>
</table></div>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.h2"></a>
            <span><a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.accuracy"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.accuracy">Accuracy</a>
          </h5>
<p>
            The inverse_gaussian distribution is implemented in terms of the exponential
            function and standard normal distribution <span class="emphasis"><em>N</em></span>0,1 &#934; :
            refer to the accuracy data for those functions for more information.
            But in general, gamma (and thus inverse gamma) results are often accurate
            to a few epsilon, &gt;14 decimal digits accuracy for 64-bit double.
          </p>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.h3"></a>
            <span><a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.implementation"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.implementation">Implementation</a>
          </h5>
<p>
            In the following table &#956; is the mean parameter and &#955; is the scale parameter
            of the inverse_gaussian distribution, <span class="emphasis"><em>x</em></span> is the random
            variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q =
            1-p</em></span> its complement. Parameters &#956; for shape and &#955; for scale are
            used for the inverse gaussian function.
          </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                    <p>
                      Function
                    </p>
                  </th>
<th>
                    <p>
                      Implementation Notes
                    </p>
                  </th>
</tr></thead>
<tbody>
<tr>
<td>
                    <p>
                      pdf
                    </p>
                  </td>
<td>
                    <p>
                      &#8730;(&#955;/ 2&#960;x<sup>3</sup>) e<sup>-&#955;(x - &#956;)&#178;/ 2&#956;&#178;x</sup>
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      cdf
                    </p>
                  </td>
<td>
                    <p>
                      &#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;)}
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      cdf complement
                    </p>
                  </td>
<td>
                    <p>
                      using complement of &#934; above.
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      quantile
                    </p>
                  </td>
<td>
                    <p>
                      No closed form known. Estimated using a guess refined by Newton-Raphson
                      iteration.
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      quantile from the complement
                    </p>
                  </td>
<td>
                    <p>
                      No closed form known. Estimated using a guess refined by Newton-Raphson
                      iteration.
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      mode
                    </p>
                  </td>
<td>
                    <p>
                      &#956; {&#8730;(1+9&#956;&#178;/4&#955;&#178;)&#178; - 3&#956;/2&#955;}
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      median
                    </p>
                  </td>
<td>
                    <p>
                      No closed form analytic equation is known, but is evaluated
                      as quantile(0.5)
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      mean
                    </p>
                  </td>
<td>
                    <p>
                      &#956;
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      variance
                    </p>
                  </td>
<td>
                    <p>
                      &#956;&#179;/&#955;
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      skewness
                    </p>
                  </td>
<td>
                    <p>
                      3 &#8730; (&#956;/&#955;)
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      kurtosis_excess
                    </p>
                  </td>
<td>
                    <p>
                      15&#956;/&#955;
                    </p>
                  </td>
</tr>
<tr>
<td>
                    <p>
                      kurtosis
                    </p>
                  </td>
<td>
                    <p>
                      12&#956;/&#955;
                    </p>
                  </td>
</tr>
</tbody>
</table></div>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.h4"></a>
            <span><a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.references"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.references">References</a>
          </h5>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem">
                Wald, A. (1947). Sequential analysis. Wiley, NY.
              </li>
<li class="listitem">
                The Inverse Gaussian distribution : theory, methodology, and applications,
                Raj S. Chhikara, J. Leroy Folks. ISBN 0824779975 (1989).
              </li>
<li class="listitem">
                The Inverse Gaussian distribution : statistical theory and applications,
                Seshadri, V , ISBN - 0387986189 (pbk) (Dewey 519.2) (1998).
              </li>
<li class="listitem">
                <a href="http://docs.scipy.org/doc/numpy/reference/generated/numpy.random.wald.html" target="_top">Numpy
                and Scipy Documentation</a>.
              </li>
<li class="listitem">
                <a href="http://bm2.genes.nig.ac.jp/RGM2/R_current/library/statmod/man/invgauss.html" target="_top">R
                statmod invgauss functions</a>.
              </li>
<li class="listitem">
                <a href="http://cran.r-project.org/web/packages/SuppDists/index.html" target="_top">R
                SuppDists invGauss functions</a>. (Note that these R implementations
                names differ in case).
              </li>
<li class="listitem">
                <a href="http://www.statsci.org/s/invgauss.html" target="_top">StatSci.org invgauss
                help</a>.
              </li>
<li class="listitem">
                <a href="http://www.statsci.org/s/invgauss.statSci.org" target="_top">invgauss
                R source</a>.
              </li>
<li class="listitem">
                <a href="http://www.biostat.wustl.edu/archives/html/s-news/2001-12/msg00144.html" target="_top">pwald,
                qwald</a>.
              </li>
<li class="listitem">
                <a href="http://www.brighton-webs.co.uk/distributions/wald.asp" target="_top">Brighton
                Webs wald</a>.
              </li>
</ol></div>
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