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<div class="titlepage"><div><div><h4 class="title">
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<a name="math_toolkit.dist_ref.dists.beta_dist"></a><a class="link" href="beta_dist.html" title="Beta Distribution">Beta Distribution</a>
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</h4></div></div></div>
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<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">beta</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre>
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<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>
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<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>
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<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. 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<></a> <span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">beta_distribution</span><span class="special">;</span>
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<span class="comment">// typedef beta_distribution<double> beta;</span>
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<span class="comment">// Note that this is deliberately NOT provided,</span>
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<span class="comment">// to avoid a clash with the function name beta.</span>
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<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">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">beta_distribution</span>
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<span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
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<span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
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<span class="comment">// Constructor from two shape parameters, alpha & beta:</span>
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<span class="identifier">beta_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">b</span><span class="special">);</span>
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<span class="comment">// Parameter accessors:</span>
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<span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
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<span class="comment">// Parameter estimators of alpha or beta from mean and variance.</span>
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<span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
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<span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.</span>
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<span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.</span>
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<span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
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<span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.</span>
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<span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.</span>
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<span class="comment">// Parameter estimators from</span>
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<span class="comment">// either alpha or beta, and x and probability.</span>
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<span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
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<span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> <span class="comment">// from beta.</span>
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<span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">// x.</span>
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<span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// cdf</span>
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<span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
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<span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="comment">// alpha.</span>
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<span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">// probability x.</span>
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<span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// probability cdf.</span>
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<span class="special">};</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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The class type <code class="computeroutput"><span class="identifier">beta_distribution</span></code>
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represents a <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">beta
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</a> <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability
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distribution function</a>.
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</p>
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<p>
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The <a href="http://mathworld.wolfram.com/BetaDistribution.htm" target="_top">beta
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distribution </a> is used as a <a href="http://en.wikipedia.org/wiki/Prior_distribution" target="_top">prior
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distribution</a> for binomial proportions in <a href="http://mathworld.wolfram.com/BayesianAnalysis.html" target="_top">Bayesian
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analysis</a>.
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</p>
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<p>
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||
See also: <a href="http://documents.wolfram.com/calculationcenter/v2/Functions/ListsMatrices/Statistics/BetaDistribution.html" target="_top">beta
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distribution</a> and <a href="http://en.wikipedia.org/wiki/Bayesian_statistics" target="_top">Bayesian
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||
statistics</a>.
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||
</p>
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||
<p>
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How the beta distribution is used for <a href="http://home.uchicago.edu/~grynav/bayes/ABSLec5.ppt" target="_top">Bayesian
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||
analysis of one parameter models</a> is discussed by Jeff Grynaviski.
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||
</p>
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<p>
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The <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
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density function PDF</a> for the <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">beta
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||
distribution</a> defined on the interval [0,1] is given by:
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||
</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">f(x;α,β) = x<sup>α - 1</sup> (1 - x)<sup>β -1</sup> / B(α, β)</span>
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</p></blockquote></div>
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<p>
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where <span class="serif_italic">B(α, β)</span> is the <a href="http://en.wikipedia.org/wiki/Beta_function" target="_top">beta
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function</a>, implemented in this library as <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta</a>.
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Division by the beta function ensures that the pdf is normalized to the
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range zero to unity.
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</p>
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<p>
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The following graph illustrates examples of the pdf for various values
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of the shape parameters. Note the <span class="emphasis"><em>α = β = 2</em></span> (blue line)
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is dome-shaped, and might be approximated by a symmetrical triangular distribution.
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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||
<span class="inlinemediaobject"><img src="../../../../graphs/beta_pdf.svg" align="middle"></span>
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</p></blockquote></div>
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<p>
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If α = β = 1, then it is a
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||
<a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform
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||
distribution</a>, equal to unity in the entire interval x = 0 to 1.
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||
If α and β are < 1, then the pdf is U-shaped. If α != β, then the shape is
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||
asymmetric and could be approximated by a triangle whose apex is away from
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the centre (where x = half).
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||
</p>
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||
<h5>
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<a name="math_toolkit.dist_ref.dists.beta_dist.h0"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.member_functions"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.member_functions">Member
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Functions</a>
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||
</h5>
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||
<h6>
|
||
<a name="math_toolkit.dist_ref.dists.beta_dist.h1"></a>
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<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.constructor"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.constructor">Constructor</a>
|
||
</h6>
|
||
<pre class="programlisting"><span class="identifier">beta_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">);</span>
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||
</pre>
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||
<p>
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||
Constructs a beta distribution with shape parameters <span class="emphasis"><em>alpha</em></span>
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||
and <span class="emphasis"><em>beta</em></span>.
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||
</p>
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||
<p>
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||
Requires alpha,beta > 0,otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
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||
is called. Note that technically the beta distribution is defined for alpha,beta
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||
>= 0, but it's not clear whether any program can actually make use of
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||
that latitude or how many of the non-member functions can be usefully defined
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||
in that case. Therefore for now, we regard it as an error if alpha or beta
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||
is zero.
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||
</p>
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||
<p>
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||
For example:
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||
</p>
|
||
<pre class="programlisting"><span class="identifier">beta_distribution</span><span class="special"><></span> <span class="identifier">mybeta</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">5</span><span class="special">);</span>
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||
</pre>
|
||
<p>
|
||
Constructs a the beta distribution with alpha=2 and beta=5 (shown in yellow
|
||
in the graph above).
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||
</p>
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||
<h6>
|
||
<a name="math_toolkit.dist_ref.dists.beta_dist.h2"></a>
|
||
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.parameter_accessors"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.parameter_accessors">Parameter
|
||
Accessors</a>
|
||
</h6>
|
||
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
|
||
</pre>
|
||
<p>
|
||
Returns the parameter <span class="emphasis"><em>alpha</em></span> from which this distribution
|
||
was constructed.
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||
</p>
|
||
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">()</span> <span class="keyword">const</span><span class="special">;</span>
|
||
</pre>
|
||
<p>
|
||
Returns the parameter <span class="emphasis"><em>beta</em></span> from which this distribution
|
||
was constructed.
|
||
</p>
|
||
<p>
|
||
So for example:
|
||
</p>
|
||
<pre class="programlisting"><span class="identifier">beta_distribution</span><span class="special"><></span> <span class="identifier">mybeta</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">5</span><span class="special">);</span>
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||
<span class="identifier">assert</span><span class="special">(</span><span class="identifier">mybeta</span><span class="special">.</span><span class="identifier">alpha</span><span class="special">()</span> <span class="special">==</span> <span class="number">2.</span><span class="special">);</span> <span class="comment">// mybeta.alpha() returns 2</span>
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||
<span class="identifier">assert</span><span class="special">(</span><span class="identifier">mybeta</span><span class="special">.</span><span class="identifier">beta</span><span class="special">()</span> <span class="special">==</span> <span class="number">5.</span><span class="special">);</span> <span class="comment">// mybeta.beta() returns 5</span>
|
||
</pre>
|
||
<h5>
|
||
<a name="math_toolkit.dist_ref.dists.beta_dist.h3"></a>
|
||
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.parameter_estimators"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.parameter_estimators">Parameter
|
||
Estimators</a>
|
||
</h5>
|
||
<p>
|
||
Two pairs of parameter estimators are provided.
|
||
</p>
|
||
<p>
|
||
One estimates either α or β
|
||
from presumed-known mean and variance.
|
||
</p>
|
||
<p>
|
||
The other pair estimates either α or β from the cdf and x.
|
||
</p>
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<p>
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It is also possible to estimate α and β from 'known' mode & quantile. For
|
||
example, calculators are provided by the <a href="http://www.ausvet.com.au/pprev/content.php?page=PPscript" target="_top">Pooled
|
||
Prevalence Calculator</a> and <a href="http://www.epi.ucdavis.edu/diagnostictests/betabuster.html" target="_top">Beta
|
||
Buster</a> but this is not yet implemented here.
|
||
</p>
|
||
<pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
|
||
<span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.</span>
|
||
<span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.</span>
|
||
</pre>
|
||
<p>
|
||
Returns the unique value of α that corresponds to a beta distribution with
|
||
mean <span class="emphasis"><em>mean</em></span> and variance <span class="emphasis"><em>variance</em></span>.
|
||
</p>
|
||
<pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
|
||
<span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">,</span> <span class="comment">// Expected value of mean.</span>
|
||
<span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">);</span> <span class="comment">// Expected value of variance.</span>
|
||
</pre>
|
||
<p>
|
||
Returns the unique value of β that corresponds to a beta distribution with
|
||
mean <span class="emphasis"><em>mean</em></span> and variance <span class="emphasis"><em>variance</em></span>.
|
||
</p>
|
||
<pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_alpha</span><span class="special">(</span>
|
||
<span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> <span class="comment">// from beta.</span>
|
||
<span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">// x.</span>
|
||
<span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// probability cdf</span>
|
||
</pre>
|
||
<p>
|
||
Returns the value of α that gives: <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">beta_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>(</span><span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">beta</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">probability</span></code>.
|
||
</p>
|
||
<pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_beta</span><span class="special">(</span>
|
||
<span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="comment">// alpha.</span>
|
||
<span class="identifier">RealType</span> <span class="identifier">x</span><span class="special">,</span> <span class="comment">// probability x.</span>
|
||
<span class="identifier">RealType</span> <span class="identifier">probability</span><span class="special">);</span> <span class="comment">// probability cdf.</span>
|
||
</pre>
|
||
<p>
|
||
Returns the value of β that gives: <code class="computeroutput"><span class="identifier">cdf</span><span class="special">(</span><span class="identifier">beta_distribution</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>(</span><span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">beta</span><span class="special">),</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">==</span> <span class="identifier">probability</span></code>.
|
||
</p>
|
||
<h5>
|
||
<a name="math_toolkit.dist_ref.dists.beta_dist.h4"></a>
|
||
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.non_member_accessor_functions"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.non_member_accessor_functions">Non-member
|
||
Accessor Functions</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 formulae for calculating these are shown in the table below, and at
|
||
<a href="http://mathworld.wolfram.com/BetaDistribution.html" target="_top">Wolfram
|
||
Mathworld</a>.
|
||
</p>
|
||
<h5>
|
||
<a name="math_toolkit.dist_ref.dists.beta_dist.h5"></a>
|
||
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.applications"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.applications">Applications</a>
|
||
</h5>
|
||
<p>
|
||
The beta distribution can be used to model events constrained to take place
|
||
within an interval defined by a minimum and maximum value: so it is used
|
||
in project management systems.
|
||
</p>
|
||
<p>
|
||
It is also widely used in <a href="http://en.wikipedia.org/wiki/Bayesian_inference" target="_top">Bayesian
|
||
statistical inference</a>.
|
||
</p>
|
||
<h5>
|
||
<a name="math_toolkit.dist_ref.dists.beta_dist.h6"></a>
|
||
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.related_distributions"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.related_distributions">Related
|
||
distributions</a>
|
||
</h5>
|
||
<p>
|
||
The beta distribution with both α and β = 1 follows a <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform
|
||
distribution</a>.
|
||
</p>
|
||
<p>
|
||
The <a href="http://en.wikipedia.org/wiki/Triangular_distribution" target="_top">triangular</a>
|
||
is used when less precise information is available.
|
||
</p>
|
||
<p>
|
||
The <a href="http://en.wikipedia.org/wiki/Binomial_distribution" target="_top">binomial
|
||
distribution</a> is closely related when α and β are integers.
|
||
</p>
|
||
<p>
|
||
With integer values of α and β the distribution B(i, j) is that of the j-th
|
||
highest of a sample of i + j + 1 independent random variables uniformly
|
||
distributed between 0 and 1. The cumulative probability from 0 to x is
|
||
thus the probability that the j-th highest value is less than x. Or it
|
||
is the probability that at least i of the random variables are less than
|
||
x, a probability given by summing over the <a class="link" href="binomial_dist.html" title="Binomial Distribution">Binomial
|
||
Distribution</a> with its p parameter set to x.
|
||
</p>
|
||
<h5>
|
||
<a name="math_toolkit.dist_ref.dists.beta_dist.h7"></a>
|
||
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.accuracy"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.accuracy">Accuracy</a>
|
||
</h5>
|
||
<p>
|
||
This distribution is implemented using the <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta
|
||
functions</a> <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta</a>
|
||
and <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">incomplete beta
|
||
functions</a> <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>
|
||
and <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>;
|
||
please refer to these functions for information on accuracy.
|
||
</p>
|
||
<h5>
|
||
<a name="math_toolkit.dist_ref.dists.beta_dist.h8"></a>
|
||
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.implementation"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.implementation">Implementation</a>
|
||
</h5>
|
||
<p>
|
||
In the following table <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>
|
||
are the parameters α and β, <span class="emphasis"><em>x</em></span> is the random variable,
|
||
<span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
|
||
</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>
|
||
<span class="serif_italic">f(x;α,β) = x<sup>α - 1</sup> (1 - x)<sup>β -1</sup> / B(α, β)</span>
|
||
</p>
|
||
<p>
|
||
Implemented using <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>(a,
|
||
b, x).
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
cdf
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
Using the incomplete beta function <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(a,
|
||
b, x)
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
cdf complement
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(a,
|
||
b, x)
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
quantile
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
Using the inverse incomplete beta function <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>(a,
|
||
b, p)
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
quantile from the complement
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>(a,
|
||
b, q)
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
mean
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<code class="computeroutput"><span class="identifier">a</span><span class="special">/(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">)</span></code>
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
variance
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<code class="computeroutput"><span class="identifier">a</span> <span class="special">*</span>
|
||
<span class="identifier">b</span> <span class="special">/</span>
|
||
<span class="special">(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">)^</span><span class="number">2</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span>
|
||
<span class="identifier">b</span> <span class="special">+</span>
|
||
<span class="number">1</span><span class="special">)</span></code>
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
mode
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<code class="computeroutput"><span class="special">(</span><span class="identifier">a</span><span class="special">-</span><span class="number">1</span><span class="special">)</span> <span class="special">/</span>
|
||
<span class="special">(</span><span class="identifier">a</span>
|
||
<span class="special">+</span> <span class="identifier">b</span>
|
||
<span class="special">-</span> <span class="number">2</span><span class="special">)</span></code>
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
skewness
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<code class="computeroutput"><span class="number">2</span> <span class="special">(</span><span class="identifier">b</span><span class="special">-</span><span class="identifier">a</span><span class="special">)</span>
|
||
<span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">+</span><span class="number">1</span><span class="special">)/(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">+</span><span class="number">2</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">a</span>
|
||
<span class="special">*</span> <span class="identifier">b</span><span class="special">)</span></code>
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
kurtosis excess
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<div class="blockquote"><blockquote class="blockquote"><p>
|
||
<span class="inlinemediaobject"><img src="../../../../equations/beta_dist_kurtosis.svg"></span>
|
||
|
||
</p></blockquote></div>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
kurtosis
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<code class="computeroutput"><span class="identifier">kurtosis</span> <span class="special">+</span>
|
||
<span class="number">3</span></code>
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
parameter estimation
|
||
</p>
|
||
</td>
|
||
<td>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
alpha (from mean and variance)
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<code class="computeroutput"><span class="identifier">mean</span> <span class="special">*</span>
|
||
<span class="special">((</span> <span class="special">(</span><span class="identifier">mean</span> <span class="special">*</span>
|
||
<span class="special">(</span><span class="number">1</span>
|
||
<span class="special">-</span> <span class="identifier">mean</span><span class="special">))</span> <span class="special">/</span>
|
||
<span class="identifier">variance</span><span class="special">)-</span>
|
||
<span class="number">1</span><span class="special">)</span></code>
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
beta (from mean and variance)
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<code class="computeroutput"><span class="special">(</span><span class="number">1</span>
|
||
<span class="special">-</span> <span class="identifier">mean</span><span class="special">)</span> <span class="special">*</span>
|
||
<span class="special">(((</span><span class="identifier">mean</span>
|
||
<span class="special">*</span> <span class="special">(</span><span class="number">1</span> <span class="special">-</span> <span class="identifier">mean</span><span class="special">))</span>
|
||
<span class="special">/</span><span class="identifier">variance</span><span class="special">)-</span><span class="number">1</span><span class="special">)</span></code>
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
The member functions <code class="computeroutput"><span class="identifier">find_alpha</span></code>
|
||
and <code class="computeroutput"><span class="identifier">find_beta</span></code>
|
||
</p>
|
||
<p>
|
||
from cdf and probability x
|
||
</p>
|
||
<p>
|
||
and <span class="bold"><strong>either</strong></span> <code class="computeroutput"><span class="identifier">alpha</span></code>
|
||
or <code class="computeroutput"><span class="identifier">beta</span></code>
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
Implemented in terms of the inverse incomplete beta functions
|
||
</p>
|
||
<p>
|
||
<a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inva</a>,
|
||
and <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_invb</a>
|
||
respectively.
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
<code class="computeroutput"><span class="identifier">find_alpha</span></code>
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<code class="computeroutput"><span class="identifier">ibeta_inva</span><span class="special">(</span><span class="identifier">beta</span><span class="special">,</span>
|
||
<span class="identifier">x</span><span class="special">,</span>
|
||
<span class="identifier">probability</span><span class="special">)</span></code>
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td>
|
||
<p>
|
||
<code class="computeroutput"><span class="identifier">find_beta</span></code>
|
||
</p>
|
||
</td>
|
||
<td>
|
||
<p>
|
||
<code class="computeroutput"><span class="identifier">ibeta_invb</span><span class="special">(</span><span class="identifier">alpha</span><span class="special">,</span>
|
||
<span class="identifier">x</span><span class="special">,</span>
|
||
<span class="identifier">probability</span><span class="special">)</span></code>
|
||
</p>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table></div>
|
||
<h5>
|
||
<a name="math_toolkit.dist_ref.dists.beta_dist.h9"></a>
|
||
<span class="phrase"><a name="math_toolkit.dist_ref.dists.beta_dist.references"></a></span><a class="link" href="beta_dist.html#math_toolkit.dist_ref.dists.beta_dist.references">References</a>
|
||
</h5>
|
||
<p>
|
||
<a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">Wikipedia Beta
|
||
distribution</a>
|
||
</p>
|
||
<p>
|
||
<a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm" target="_top">NIST
|
||
Exploratory Data Analysis</a>
|
||
</p>
|
||
<p>
|
||
<a href="http://mathworld.wolfram.com/BetaDistribution.html" target="_top">Wolfram
|
||
MathWorld</a>
|
||
</p>
|
||
</div>
|
||
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
||
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
||
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
||
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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>
|
||
<hr>
|
||
<div class="spirit-nav">
|
||
<a accesskey="p" href="bernoulli_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="binomial_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
|
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|
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</body>
|
||
</html>
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