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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="id1060611"></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="id1060652"></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="id1062804"></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="id1062903"></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="id1062944"></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="id1063272"></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> |
| <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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