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<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.skew_normal_dist"></a><a class="link" href="skew_normal_dist.html" title="Skew Normal Distribution">Skew
        Normal Distribution</a>
</h4></div></div></div>
<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">skew_normal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<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="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>
<span class="keyword">class</span> <span class="identifier">skew_normal_distribution</span><span class="special">;</span>

<span class="keyword">typedef</span> <span class="identifier">skew_normal_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">normal</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">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">skew_normal_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="comment">// Constructor:</span>
   <span class="identifier">skew_normal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</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">shape</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span>
   <span class="comment">// Accessors:</span>
   <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// mean if normal.</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">// width, standard deviation if normal.</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">// The distribution is right skewed if shape &gt; 0 and is left skewed if shape &lt; 0.</span>
                          <span class="comment">// The distribution is normal if shape is zero.</span>
<span class="special">};</span>

<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
          The skew normal distribution is a variant of the most well known Gaussian
          statistical distribution.
        </p>
<p>
          The skew normal distribution with shape zero resembles the <a href="http://en.wikipedia.org/wiki/Normal_distribution" target="_top">Normal
          Distribution</a>, hence the latter can be regarded as a special case
          of the more generic skew normal distribution.
        </p>
<p>
          If the standard (mean = 0, scale = 1) normal distribution probability density
          function is
        </p>
<p>
          &#160; &#160;  <span class="inlinemediaobject"><img src="../../../../equations/normal01_pdf.svg"></span>
        </p>
<p>
          and the cumulative distribution function
        </p>
<p>
          &#160; &#160;  <span class="inlinemediaobject"><img src="../../../../equations/normal01_cdf.svg"></span>
        </p>
<p>
          then the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">PDF</a>
          of the <a href="http://en.wikipedia.org/wiki/Skew_normal_distribution" target="_top">skew
          normal distribution</a> with shape parameter &#945;, defined by O'Hagan and
          Leonhard (1976) is
        </p>
<p>
          &#160; &#160;  <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf0.svg"></span>
        </p>
<p>
          Given <a href="http://en.wikipedia.org/wiki/Location_parameter" target="_top">location</a>
          &#958;, <a href="http://en.wikipedia.org/wiki/Scale_parameter" target="_top">scale</a>
          &#969;, and <a href="http://en.wikipedia.org/wiki/Shape_parameter" target="_top">shape</a>
          &#945;, it can be <a href="http://en.wikipedia.org/wiki/Skew_normal_distribution" target="_top">transformed</a>,
          to the form:
        </p>
<p>
          &#160; &#160;  <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf.svg"></span>
        </p>
<p>
          and <a href="http://en.wikipedia.org/wiki/Cumulative_distribution_function" target="_top">CDF</a>:
        </p>
<p>
          &#160; &#160;  <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_cdf.svg"></span>
        </p>
<p>
          where <span class="emphasis"><em>T(h,a)</em></span> is Owen's T function, and <span class="emphasis"><em>&#934;(x)</em></span>
          is the normal distribution.
        </p>
<p>
          The variation the PDF and CDF with its parameters is illustrated in the
          following graphs:
        </p>
<p>
          <span class="inlinemediaobject"><img src="../../../../graphs/skew_normal_pdf.svg" align="middle"></span>
  <span class="inlinemediaobject"><img src="../../../../graphs/skew_normal_cdf.svg" align="middle"></span>
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.skew_normal_dist.h0"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.member_functions"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.member_functions">Member
          Functions</a>
        </h5>
<pre class="programlisting"><span class="identifier">skew_normal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</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">shape</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span>
</pre>
<p>
          Constructs a skew_normal distribution with location &#958;, scale &#969; and shape &#945;.
        </p>
<p>
          Requires scale &gt; 0, otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
          is called.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          returns the location &#958; 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 &#969; of this distribution,
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          returns the shape &#945; of this distribution.
        </p>
<p>
          (Location and scale function match other similar distributions, allowing
          the functions <code class="computeroutput"><span class="identifier">find_location</span></code>
          and <code class="computeroutput"><span class="identifier">find_scale</span></code> to be used
          generically).
        </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>
            While the shape parameter may be chosen arbitrarily (finite), the resulting
            <span class="bold"><strong>skewness</strong></span> of the distribution is in fact
            limited to about (-1, 1); strictly, the interval is (-0.9952717, 0.9952717).
          </p>
<p>
            A parameter &#948; is related to the shape &#945; by &#948; = &#945; / (1 + &#945;&#178;), and used in the expression
            for skewness <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_skewness.svg"></span>

          </p>
</td></tr>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.dists.skew_normal_dist.h1"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.references"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.references">References</a>
        </h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
              <a href="http://azzalini.stat.unipd.it/SN/" target="_top">Skew-Normal Probability
              Distribution</a> for many links and bibliography.
            </li>
<li class="listitem">
              <a href="http://azzalini.stat.unipd.it/SN/Intro/intro.html" target="_top">A very
              brief introduction to the skew-normal distribution</a> by Adelchi
              Azzalini (2005-11-2).
            </li>
<li class="listitem">
              See a <a href="http://www.tri.org.au/azzalini.html" target="_top">skew-normal
              function animation</a>.
            </li>
</ul></div>
<h5>
<a name="math_toolkit.dist_ref.dists.skew_normal_dist.h2"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.non_member_accessors"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_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_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
          <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>,
          <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>,
          <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>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
          <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>,
          <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>.
        </p>
<p>
          The domain of the random variable is <span class="emphasis"><em>-[max_value], +[min_value]</em></span>.
          Infinite values are not supported.
        </p>
<p>
          There are no <a href="http://en.wikipedia.org/wiki/Closed-form_expression" target="_top">closed-form
          expression</a> known for the mode and median, but these are computed
          for the
        </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
              mode - by finding the maximum of the PDF.
            </li>
<li class="listitem">
              median - by computing <code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="number">1</span><span class="special">/</span><span class="number">2</span><span class="special">)</span></code>.
            </li>
</ul></div>
<p>
          The maximum of the PDF is sought through searching the root of f'(x)=0.
        </p>
<p>
          Both involve iterative methods that will have lower accuracy than other
          estimates.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.skew_normal_dist.h3"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.testing"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.testing">Testing</a>
        </h5>
<p>
          <a href="http://www.r-project.org/" target="_top">The R Project for Statistical Computing</a>
          using library(sn) described at <a href="http://azzalini.stat.unipd.it/SN/" target="_top">Skew-Normal
          Probability Distribution</a>, and at <a href="http://cran.r-project.org/web/packages/sn/sn.pd" target="_top">R
          skew-normal(sn) package</a>.
        </p>
<p>
          Package sn provides functions related to the skew-normal (SN) and the skew-t
          (ST) probability distributions, both for the univariate and for the the
          multivariate case, including regression models.
        </p>
<p>
          <a href="http://www.wolfram.com/products/mathematica/index.html" target="_top">Wolfram
          Mathematica</a> was also used to generate some more accurate spot test
          data.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.skew_normal_dist.h4"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.accuracy"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.accuracy">Accuracy</a>
        </h5>
<p>
          The skew_normal distribution with shape = zero is implemented as a special
          case, equivalent to the normal distribution in terms of the <a class="link" href="../../sf_erf/error_function.html" title="Error Functions">error
          function</a>, and therefore should have excellent accuracy.
        </p>
<p>
          The PDF and mean, variance, skewness and kurtosis are also accurately evaluated
          using <a href="http://en.wikipedia.org/wiki/Analytical_expression" target="_top">analytical
          expressions</a>. The CDF requires <a href="http://en.wikipedia.org/wiki/Owen%27s_T_function" target="_top">Owen's
          T function</a> that is evaluated using a Boost C++ <a class="link" href="../../owens_t.html" title="Owen's T function">Owens
          T</a> implementation of the algorithms of M. Patefield and D. Tandy,
          Journal of Statistical Software, 5(5), 1-25 (2000); the complicated accuracy
          of this function is discussed in detail at <a class="link" href="../../owens_t.html" title="Owen's T function">Owens
          T</a>.
        </p>
<p>
          The median and mode are calculated by iterative root finding, and both
          will be less accurate.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.skew_normal_dist.h5"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.implementation"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.implementation">Implementation</a>
        </h5>
<p>
          In the following table, &#958; is the location of the distribution, and &#969; is its
          scale, and &#945; is its shape.
        </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>
                    Using: <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf.svg"></span>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf
                  </p>
                </td>
<td>
                  <p>
                    Using: <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_cdf.svg"></span><br> where <span class="emphasis"><em>T(h,a)</em></span>
                    is Owen's T function, and <span class="emphasis"><em>&#934;(x)</em></span> is the normal
                    distribution.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf complement
                  </p>
                </td>
<td>
                  <p>
                    Using: complement of normal distribution + 2 * Owens_t
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile
                  </p>
                </td>
<td>
                  <p>
                    Maximum of the pdf is sought through searching the root of f'(x)=0
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile from the complement
                  </p>
                </td>
<td>
                  <p>
                    -quantile(SN(-location &#958;, scale &#969;, -shape&#945;), p)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    location
                  </p>
                </td>
<td>
                  <p>
                    location &#958;
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    scale
                  </p>
                </td>
<td>
                  <p>
                    scale &#969;
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    shape
                  </p>
                </td>
<td>
                  <p>
                    shape &#945;
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    median
                  </p>
                </td>
<td>
                  <p>
                    quantile(1/2)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mean
                  </p>
                </td>
<td>
                  <p>
                    <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_mean.svg"></span>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mode
                  </p>
                </td>
<td>
                  <p>
                    Maximum of the pdf is sought through searching the root of f'(x)=0
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    variance
                  </p>
                </td>
<td>
                  <p>
                    <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_variance.svg"></span>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    skewness
                  </p>
                </td>
<td>
                  <p>
                    <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_skewness.svg"></span>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis
                  </p>
                </td>
<td>
                  <p>
                    kurtosis excess-3
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis excess
                  </p>
                </td>
<td>
                  <p>
                    <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_kurt_ex.svg"></span>
                  </p>
                </td>
</tr>
</tbody>
</table></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 &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
      Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
      Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
      Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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>
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