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| <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"><</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">></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"><</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<></a> <span class="special">></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"><</span><span class="keyword">double</span><span class="special">></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"><></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 μ and scale (also called precision) parameter λ, and |
| random variate x, the inverse_gaussian distribution is defined by the |
| probability density function (PDF): |
| </p> |
| <p> |
|    f(x;μ, λ) = √(λ/2πx<sup>3</sup>) e<sup>-λ(x-μ)²/2μ²x</sup> |
| </p> |
| <p> |
| and Cumulative Density Function (CDF): |
| </p> |
| <p> |
|    F(x;μ, λ) = Φ{√(λ<span class="emphasis"><em>x) (x</em></span>μ-1)} + e<sup>2μ/λ</sup> Φ{-√(λ/μ) (1+x/μ)} |
| </p> |
| <p> |
| where Φ 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 μ and λ: |
| </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 (μ and λ) are replaced |
| by their ratio φ = λ/μ and by 1/μ: 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 μ is unity) is also provided. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.member_functions"></a><h5> |
| <a name="id1193328"></a> |
| <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 μ mean, and scale λ, with |
| both default values 1. |
| </p> |
| <p> |
| Requires that both the mean μ parameter and scale λ 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 μ 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 λ parameter of this distribution. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.non_member_accessors"></a><h5> |
| <a name="id1193470"></a> |
| <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,+∞). |
| </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> |
| <a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.accuracy"></a><h5> |
| <a name="id1193577"></a> |
| <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 Φ : |
| 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, >14 decimal digits accuracy for 64-bit double. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.implementation"></a><h5> |
| <a name="id1193600"></a> |
| <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 μ is the mean parameter and λ 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 μ for shape and λ 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> |
| √(λ/ 2πx<sup>3</sup>) e<sup>-λ(x - μ)²/ 2μ²x</sup> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf |
| </p> |
| </td> |
| <td> |
| <p> |
| Φ{√(λ<span class="emphasis"><em>x) (x</em></span>μ-1)} + e<sup>2μ/λ</sup> Φ{-√(λ/μ) (1+x/μ)} |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf complement |
| </p> |
| </td> |
| <td> |
| <p> |
| using complement of Φ 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> |
| μ {√(1+9μ²/4λ²)² - 3μ/2λ} |
| </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> |
| μ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| variance |
| </p> |
| </td> |
| <td> |
| <p> |
| μ³/λ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| skewness |
| </p> |
| </td> |
| <td> |
| <p> |
| 3 √ (μ/λ) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis_excess |
| </p> |
| </td> |
| <td> |
| <p> |
| 15μ/λ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis |
| </p> |
| </td> |
| <td> |
| <p> |
| 12μ/λ |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| <a name="math_toolkit.dist.dist_ref.dists.inverse_gaussian_dist.references"></a><h5> |
| <a name="id1193892"></a> |
| <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 type="1"> |
| <li> |
| Wald, A. (1947). Sequential analysis. Wiley, NY. |
| </li> |
| <li> |
| The Inverse Gaussian distribution : theory, methodology, and applications, |
| Raj S. Chhikara, J. Leroy Folks. ISBN 0824779975 (1989). |
| </li> |
| <li> |
| The Inverse Gaussian distribution : statistical theory and applications, |
| Seshadri, V , ISBN - 0387986189 (pbk) (Dewey 519.2) (1998). |
| </li> |
| <li> |
| <a href="http://docs.scipy.org/doc/numpy/reference/generated/numpy.random.wald.html" target="_top">Numpy |
| and Scipy Documentation</a>. |
| </li> |
| <li> |
| <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> |
| <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> |
| <a href="http://www.statsci.org/s/invgauss.html" target="_top">StatSci.org invgauss |
| help</a>. |
| </li> |
| <li> |
| <a href="http://www.statsci.org/s/invgauss.statSci.org" target="_top">invgauss |
| R source</a>. |
| </li> |
| <li> |
| <a href="http://www.biostat.wustl.edu/archives/html/s-news/2001-12/msg00144.html" target="_top">pwald, |
| qwald</a>. |
| </li> |
| <li> |
| <a href="http://www.brighton-webs.co.uk/distributions/wald.asp" target="_top">Brighton |
| Webs wald</a>. |
| </li> |
| </ol></div> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
| </p> |
| </div></td> |
| </tr></table> |
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