<title>Beta Distribution</title>
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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>
<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="Chapter 
19. 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>
+ <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 
20. 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>
<span class="keyword">class</span> <span class="identifier">beta_distribution</span><span class="special">;</span>
<span class="comment">// typedef beta_distribution<double> beta;</span>
<span class="comment">// Note that this is deliberately NOT provided,</span>
<span class="comment">// to avoid a clash with the function name beta.</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">class</span> <a class="link" href="../../../policy.html" title="Chapter 19. Policies: Controlling Precision, Error Handling etc">Policy</a><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">class</span> <a class="link" href="../../../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
<span class="keyword">class</span> <span class="identifier">beta_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
density function PDF</a> for the <a href="http://en.wikipedia.org/wiki/Beta_distribution" target="_top">beta
distribution</a> defined on the interval [0,1] is given by:
</p>
+<div class="blockquote"><blockquote class="blockquote"><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></blockquote></div>
<p>
- f(x;α,β) = x<sup>α - 1</sup> (1 - x)<sup>β -1</sup> / B(α, β)
- </p>
-<p>
- where B(α, β) is the <a href="http://en.wikipedia.org/wiki/Beta_function" target="_top">beta
+ where <span class="serif_italic">B(α, β)</span> is the <a href="http://en.wikipedia.org/wiki/Beta_function" target="_top">beta
function</a>, implemented in this library as <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta</a>.
Division by the beta function ensures that the pdf is normalized to the
range zero to unity.
</p>
<p>
The following graph illustrates examples of the pdf for various values
- of the shape parameters. Note the α = β = 2 (blue line) is dome-shaped, and
- might be approximated by a symmetrical triangular distribution.
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../../graphs/beta_pdf.svg" align="middle"></span>
+ of the shape parameters. Note the <span class="emphasis"><em>α = β = 2</em></span> (blue line)
+ is dome-shaped, and might be approximated by a symmetrical triangular distribution.
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../graphs/beta_pdf.svg" align="middle"></span>
+
+ </p></blockquote></div>
<p>
- If α = β = 1, then it is a __space <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform
+ If α = β = 1, then it is a  
+<a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform
distribution</a>, equal to unity in the entire interval x = 0 to 1.
- If α __space and β __space are < 1, then the pdf is U-shaped. If α != β, then
- the shape is asymmetric and could be approximated by a triangle whose apex
- is away from the centre (where x = half).
+ If α and β are < 1, then the pdf is U-shaped. If α != β, then the shape is
+ asymmetric and could be approximated by a triangle whose apex is away from
+ the centre (where x = half).
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.beta_dist.h0"></a>
Two pairs of parameter estimators are provided.
</p>
<p>
- One estimates either α __space or β __space from presumed-known mean and variance.
+ One estimates either α or β
+from presumed-known mean and variance.
</p>
<p>
- The other pair estimates either α __space or β __space from the cdf and x.
+ The other pair estimates either α or β from the cdf and x.
</p>
<p>
- It is also possible to estimate α __space and β __space 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
+ 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>
<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
+ 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">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
+ 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">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>.
+ 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">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>.
+ 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>
distributions</a>
</h5>
<p>
- The beta distribution with both α
__space and β = 1 follows a <a href="http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29" target="_top">uniform
+ 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>
</p>
<p>
The <a href="http://en.wikipedia.org/wiki/Binomial_distribution" target="_top">binomial
- distribution</a> is closely related when α __space and β __space are integers.
+ distribution</a> is closely related when α and β are integers.
</p>
<p>
- With integer values of α __space and β __space 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
+ 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>
</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,
+ 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">
</td>
<td>
<p>
- f(x;α,β) = x<sup>α - 1</sup> (1 - x)<sup>β -1</sup> / B(α, β)
+ <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,
</p>
</td>
<td>
- <p>
- <span class="inlinemediaobject"><img src="../../../../equations/beta_dist_kurtosis.svg"></span>
- </p>
+ <div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../../equations/beta_dist_kurtosis.svg"></span>
+
+ </p></blockquote></div>
</td>
</tr>
<tr>
<tr>
<td>
<p>
- alpha
- </p>
- <p>
- from mean and variance
+ alpha (from mean and variance)
</p>
</td>
<td>
<tr>
<td>
<p>
- beta
- </p>
- <p>
- from mean and variance
+ beta (from mean and variance)
</p>
</td>
<td>
projects / platform / upstream / boost.git / blobdiff