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<div class="titlepage"><div><div><h5 class="title">
<a name="math_toolkit.dist.dist_ref.dists.weibull"></a><a class="link" href="weibull.html" title="Weibull Distribution"> Weibull
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">weibull</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">weibull_distribution</span><span class="special">;</span>
<span class="keyword">typedef</span> <span class="identifier">weibull_distribution</span><span class="special"><></span> <span class="identifier">weibull</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">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a><span class="special">></span>
<span class="keyword">class</span> <span class="identifier">weibull_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">// Construct:
</span> <span class="identifier">weibull_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">shape</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">// Accessors:
</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="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
<span class="special">}}</span> <span class="comment">// namespaces
</span></pre>
<p>
The <a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">Weibull
distribution</a> is a continuous distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
density function</a>:
</p>
<p>
f(x; α, β) = (α/β) * (x / β)<sup>α - 1</sup> * e<sup>-(x/β)<sup>α</sup></sup>
</p>
<p>
For shape parameter α > 0, and scale parameter β > 0, and x > 0.
</p>
<p>
The Weibull distribution is often used in the field of failure analysis;
in particular it can mimic distributions where the failure rate varies
over time. If the failure rate is:
</p>
<div class="itemizedlist"><ul type="disc">
<li>
constant over time, then α = 1, suggests that items are failing from
random events.
</li>
<li>
decreases over time, then α < 1, suggesting "infant mortality".
</li>
<li>
increases over time, then α > 1, suggesting "wear out"
- more likely to fail as time goes by.
</li>
</ul></div>
<p>
The following graph illustrates how the PDF varies with the shape parameter
α:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../../graphs/weibull_pdf1.png" align="middle"></span>
</p>
<p>
While this graph illustrates how the PDF varies with the scale parameter
β:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../../../graphs/weibull_pdf2.png" align="middle"></span>
</p>
<a name="math_toolkit.dist.dist_ref.dists.weibull.related_distributions"></a><h5>
<a name="id1065900"></a>
<a class="link" href="weibull.html#math_toolkit.dist.dist_ref.dists.weibull.related_distributions">Related
distributions</a>
</h5>
<p>
When α = 3, the <a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">Weibull
distribution</a> appears similar to the <a href="http://en.wikipedia.org/wiki/Normal_distribution" target="_top">normal
distribution</a>. When α = 1, the Weibull distribution reduces to the
<a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">exponential
distribution</a>. The relationship of the types of extreme value
distributions, of which the Weibull is but one, is discussed by <a href="http://www.worldscibooks.com/mathematics/p191.html" target="_top">Extreme Value
Distributions, Theory and Applications Samuel Kotz & Saralees Nadarajah</a>.
</p>
<a name="math_toolkit.dist.dist_ref.dists.weibull.member_functions"></a><h5>
<a name="id1065942"></a>
<a class="link" href="weibull.html#math_toolkit.dist.dist_ref.dists.weibull.member_functions">Member
Functions</a>
</h5>
<pre class="programlisting"><span class="identifier">weibull_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">shape</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>
</pre>
<p>
Constructs a <a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">Weibull
distribution</a> with shape <span class="emphasis"><em>shape</em></span> and scale
<span class="emphasis"><em>scale</em></span>.
</p>
<p>
Requires that the <span class="emphasis"><em>shape</em></span> and <span class="emphasis"><em>scale</em></span>
parameters are both 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">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
Returns the <span class="emphasis"><em>shape</em></span> 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 <span class="emphasis"><em>scale</em></span> parameter of this distribution.
</p>
<a name="math_toolkit.dist.dist_ref.dists.weibull.non_member_accessors"></a><h5>
<a name="id1066100"></a>
<a class="link" href="weibull.html#math_toolkit.dist.dist_ref.dists.weibull.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 variable is [0, ∞].
</p>
<a name="math_toolkit.dist.dist_ref.dists.weibull.accuracy"></a><h5>
<a name="id1066200"></a>
<a class="link" href="weibull.html#math_toolkit.dist.dist_ref.dists.weibull.accuracy">Accuracy</a>
</h5>
<p>
The Weibull distribution is implemented in terms of the standard library
<code class="computeroutput"><span class="identifier">log</span></code> and <code class="computeroutput"><span class="identifier">exp</span></code> functions plus <a class="link" href="../../../special/powers/expm1.html" title="expm1">expm1</a>
and <a class="link" href="../../../special/powers/log1p.html" title="log1p">log1p</a> and
as such should have very low error rates.
</p>
<a name="math_toolkit.dist.dist_ref.dists.weibull.implementation"></a><h5>
<a name="id1066241"></a>
<a class="link" href="weibull.html#math_toolkit.dist.dist_ref.dists.weibull.implementation">Implementation</a>
</h5>
<p>
In the following table α is the shape parameter of the distribution, β is
it's scale parameter, <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>.
</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 the relation: pdf = αβ<sup>-α </sup>x<sup>α - 1</sup> e<sup>-(x/beta)<sup>alpha</sup></sup>
</p>
</td>
</tr>
<tr>
<td>
<p>
cdf
</p>
</td>
<td>
<p>
Using the relation: p = -<a class="link" href="../../../special/powers/expm1.html" title="expm1">expm1</a>(-(x/β)<sup>α</sup>)
</p>
</td>
</tr>
<tr>
<td>
<p>
cdf complement
</p>
</td>
<td>
<p>
Using the relation: q = e<sup>-(x/β)<sup>α</sup></sup>
</p>
</td>
</tr>
<tr>
<td>
<p>
quantile
</p>
</td>
<td>
<p>
Using the relation: x = β * (-<a class="link" href="../../../special/powers/log1p.html" title="log1p">log1p</a>(-p))<sup>1/α</sup>
</p>
</td>
</tr>
<tr>
<td>
<p>
quantile from the complement
</p>
</td>
<td>
<p>
Using the relation: x = β * (-log(q))<sup>1/α</sup>
</p>
</td>
</tr>
<tr>
<td>
<p>
mean
</p>
</td>
<td>
<p>
β * Γ(1 + 1/α)
</p>
</td>
</tr>
<tr>
<td>
<p>
variance
</p>
</td>
<td>
<p>
β<sup>2</sup>(Γ(1 + 2/α) - Γ<sup>2</sup>(1 + 1/α))
</p>
</td>
</tr>
<tr>
<td>
<p>
mode
</p>
</td>
<td>
<p>
β((α - 1) / α)<sup>1/α</sup>
</p>
</td>
</tr>
<tr>
<td>
<p>
skewness
</p>
</td>
<td>
<p>
Refer to <a href="http://mathworld.wolfram.com/WeibullDistribution.html" target="_top">Weisstein,
Eric W. "Weibull Distribution." From MathWorld--A
Wolfram Web Resource.</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
kurtosis
</p>
</td>
<td>
<p>
Refer to <a href="http://mathworld.wolfram.com/WeibullDistribution.html" target="_top">Weisstein,
Eric W. "Weibull Distribution." From MathWorld--A
Wolfram Web Resource.</a>
</p>
</td>
</tr>
<tr>
<td>
<p>
kurtosis excess
</p>
</td>
<td>
<p>
Refer to <a href="http://mathworld.wolfram.com/WeibullDistribution.html" target="_top">Weisstein,
Eric W. "Weibull Distribution." From MathWorld--A
Wolfram Web Resource.</a>
</p>
</td>
</tr>
</tbody>
</table></div>
<a name="math_toolkit.dist.dist_ref.dists.weibull.references"></a><h5>
<a name="id1066568"></a>
<a class="link" href="weibull.html#math_toolkit.dist.dist_ref.dists.weibull.references">References</a>
</h5>
<div class="itemizedlist"><ul type="disc">
<li>
<a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">http://en.wikipedia.org/wiki/Weibull_distribution</a>
</li>
<li>
<a href="http://mathworld.wolfram.com/WeibullDistribution.html" target="_top">Weisstein,
Eric W. "Weibull Distribution." From MathWorld--A Wolfram
Web Resource.</a>
</li>
<li>
<a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm" target="_top">Weibull
in NIST Exploratory Data Analysis</a>
</li>
</ul></div>
</div>
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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>)
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