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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.weibull_dist"></a><a class="link" href="weibull_dist.html" title="Weibull Distribution">Weibull Distribution</a>
  28 </h4></div></div></div>
  29 <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">weibull</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
  30 <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>
  31
  32 <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>
  33           <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>
  34 <span class="keyword">class</span> <span class="identifier">weibull_distribution</span><span class="special">;</span>
  35
  36 <span class="keyword">typedef</span> <span class="identifier">weibull_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">weibull</span><span class="special">;</span>
  37
  38 <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>
  39 <span class="keyword">class</span> <span class="identifier">weibull_distribution</span>
  40 <span class="special">{</span>
  41 <span class="keyword">public</span><span class="special">:</span>
  42    <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
  43    <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
  44    <span class="comment">// Construct:</span>
  45    <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>
  46    <span class="comment">// Accessors:</span>
  47    <span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  48    <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  49 <span class="special">};</span>
  50
  51 <span class="special">}}</span> <span class="comment">// namespaces</span>
  52 </pre>
  53 <p>
  54           The <a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">Weibull
  55           distribution</a> is a continuous distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
  56           density function</a>:
  57         </p>
  58 <p>
  59           f(x; &#945;, &#946;) = (&#945;/&#946;) * (x / &#946;)<sup>&#945; - 1</sup> * e<sup>-(x/&#946;)<sup>&#945;</sup></sup>
  60         </p>
  61 <p>
  62           For shape parameter &#945; &#160; &gt; 0, and scale parameter &#946; &#160; &gt; 0, and x &gt; 0.
  63         </p>
  64 <p>
  65           The Weibull distribution is often used in the field of failure analysis;
  66           in particular it can mimic distributions where the failure rate varies
  67           over time. If the failure rate is:
  68         </p>
  69 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
  70 <li class="listitem">
  71               constant over time, then &#945; &#160; = 1, suggests that items are failing from
  72               random events.
  73             </li>
  74 <li class="listitem">
  75               decreases over time, then &#945; &#160; &lt; 1, suggesting "infant mortality".
  76             </li>
  77 <li class="listitem">
  78               increases over time, then &#945; &#160; &gt; 1, suggesting "wear out" -
  79               more likely to fail as time goes by.
  80             </li>
  81 </ul></div>
  82 <p>
  83           The following graph illustrates how the PDF varies with the shape parameter
  84           &#945;:
  85         </p>
  86 <p>
  87           <span class="inlinemediaobject"><img src="../../../../graphs/weibull_pdf1.svg" align="middle"></span>
  88         </p>
  89 <p>
  90           While this graph illustrates how the PDF varies with the scale parameter
  91           &#946;:
  92         </p>
  93 <p>
  94           <span class="inlinemediaobject"><img src="../../../../graphs/weibull_pdf2.svg" align="middle"></span>
  95         </p>
  96 <h5>
  97 <a name="math_toolkit.dist_ref.dists.weibull_dist.h0"></a>
  98           <span class="phrase"><a name="math_toolkit.dist_ref.dists.weibull_dist.related_distributions"></a></span><a class="link" href="weibull_dist.html#math_toolkit.dist_ref.dists.weibull_dist.related_distributions">Related
  99           distributions</a>
 100         </h5>
 101 <p>
 102           When &#945; &#160; = 3, the <a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">Weibull
 103           distribution</a> appears similar to the <a href="http://en.wikipedia.org/wiki/Normal_distribution" target="_top">normal
 104           distribution</a>. When &#945; &#160; = 1, the Weibull distribution reduces to the
 105           <a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">exponential
 106           distribution</a>. The relationship of the types of extreme value distributions,
 107           of which the Weibull is but one, is discussed by <a href="http://www.worldscibooks.com/mathematics/p191.html" target="_top">Extreme
 108           Value Distributions, Theory and Applications Samuel Kotz &amp; Saralees
 109           Nadarajah</a>.
 110         </p>
 111 <h5>
 112 <a name="math_toolkit.dist_ref.dists.weibull_dist.h1"></a>
 113           <span class="phrase"><a name="math_toolkit.dist_ref.dists.weibull_dist.member_functions"></a></span><a class="link" href="weibull_dist.html#math_toolkit.dist_ref.dists.weibull_dist.member_functions">Member
 114           Functions</a>
 115         </h5>
 116 <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>
 117 </pre>
 118 <p>
 119           Constructs a <a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">Weibull
 120           distribution</a> with shape <span class="emphasis"><em>shape</em></span> and scale <span class="emphasis"><em>scale</em></span>.
 121         </p>
 122 <p>
 123           Requires that the <span class="emphasis"><em>shape</em></span> and <span class="emphasis"><em>scale</em></span>
 124           parameters are both greater than zero, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 125         </p>
 126 <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>
 127 </pre>
 128 <p>
 129           Returns the <span class="emphasis"><em>shape</em></span> parameter of this distribution.
 130         </p>
 131 <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>
 132 </pre>
 133 <p>
 134           Returns the <span class="emphasis"><em>scale</em></span> parameter of this distribution.
 135         </p>
 136 <h5>
 137 <a name="math_toolkit.dist_ref.dists.weibull_dist.h2"></a>
 138           <span class="phrase"><a name="math_toolkit.dist_ref.dists.weibull_dist.non_member_accessors"></a></span><a class="link" href="weibull_dist.html#math_toolkit.dist_ref.dists.weibull_dist.non_member_accessors">Non-member
 139           Accessors</a>
 140         </h5>
 141 <p>
 142           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 143           functions</a> that are generic to all distributions are supported:
 144           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 145           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 146           <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>,
 147           <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>,
 148           <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>,
 149           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 150           <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>,
 151           <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>.
 152         </p>
 153 <p>
 154           The domain of the random variable is [0, &#8734;].
 155         </p>
 156 <h5>
 157 <a name="math_toolkit.dist_ref.dists.weibull_dist.h3"></a>
 158           <span class="phrase"><a name="math_toolkit.dist_ref.dists.weibull_dist.accuracy"></a></span><a class="link" href="weibull_dist.html#math_toolkit.dist_ref.dists.weibull_dist.accuracy">Accuracy</a>
 159         </h5>
 160 <p>
 161           The Weibull distribution is implemented in terms of the standard library
 162           <code class="computeroutput"><span class="identifier">log</span></code> and <code class="computeroutput"><span class="identifier">exp</span></code>
 163           functions plus <a class="link" href="../../powers/expm1.html" title="expm1">expm1</a> and
 164           <a class="link" href="../../powers/log1p.html" title="log1p">log1p</a> and as such should
 165           have very low error rates.
 166         </p>
 167 <h5>
 168 <a name="math_toolkit.dist_ref.dists.weibull_dist.h4"></a>
 169           <span class="phrase"><a name="math_toolkit.dist_ref.dists.weibull_dist.implementation"></a></span><a class="link" href="weibull_dist.html#math_toolkit.dist_ref.dists.weibull_dist.implementation">Implementation</a>
 170         </h5>
 171 <p>
 172           In the following table &#945; &#160; is the shape parameter of the distribution, &#946; &#160; is its
 173           scale parameter, <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span>
 174           is the probability and <span class="emphasis"><em>q = 1-p</em></span>.
 175         </p>
 176 <div class="informaltable"><table class="table">
 177 <colgroup>
 178 <col>
 179 <col>
 180 </colgroup>
 181 <thead><tr>
 182 <th>
 183                   <p>
 184                     Function
 185                   </p>
 186                 </th>
 187 <th>
 188                   <p>
 189                     Implementation Notes
 190                   </p>
 191                 </th>
 192 </tr></thead>
 193 <tbody>
 194 <tr>
 195 <td>
 196                   <p>
 197                     pdf
 198                   </p>
 199                 </td>
 200 <td>
 201                   <p>
 202                     Using the relation: pdf = &#945;&#946;<sup>-&#945; </sup>x<sup>&#945; - 1</sup> e<sup>-(x/beta)<sup>alpha</sup></sup>
 203                   </p>
 204                 </td>
 205 </tr>
 206 <tr>
 207 <td>
 208                   <p>
 209                     cdf
 210                   </p>
 211                 </td>
 212 <td>
 213                   <p>
 214                     Using the relation: p = -<a class="link" href="../../powers/expm1.html" title="expm1">expm1</a>(-(x/&#946;)<sup>&#945;</sup>)
 215                   </p>
 216                 </td>
 217 </tr>
 218 <tr>
 219 <td>
 220                   <p>
 221                     cdf complement
 222                   </p>
 223                 </td>
 224 <td>
 225                   <p>
 226                     Using the relation: q = e<sup>-(x/&#946;)<sup>&#945;</sup></sup>
 227                   </p>
 228                 </td>
 229 </tr>
 230 <tr>
 231 <td>
 232                   <p>
 233                     quantile
 234                   </p>
 235                 </td>
 236 <td>
 237                   <p>
 238                     Using the relation: x = &#946; * (-<a class="link" href="../../powers/log1p.html" title="log1p">log1p</a>(-p))<sup>1/&#945;</sup>
 239                   </p>
 240                 </td>
 241 </tr>
 242 <tr>
 243 <td>
 244                   <p>
 245                     quantile from the complement
 246                   </p>
 247                 </td>
 248 <td>
 249                   <p>
 250                     Using the relation: x = &#946; * (-log(q))<sup>1/&#945;</sup>
 251                   </p>
 252                 </td>
 253 </tr>
 254 <tr>
 255 <td>
 256                   <p>
 257                     mean
 258                   </p>
 259                 </td>
 260 <td>
 261                   <p>
 262                     &#946; * &#915;(1 + 1/&#945;)
 263                   </p>
 264                 </td>
 265 </tr>
 266 <tr>
 267 <td>
 268                   <p>
 269                     variance
 270                   </p>
 271                 </td>
 272 <td>
 273                   <p>
 274                     &#946;<sup>2</sup>(&#915;(1 + 2/&#945;) - &#915;<sup>2</sup>(1 + 1/&#945;))
 275                   </p>
 276                 </td>
 277 </tr>
 278 <tr>
 279 <td>
 280                   <p>
 281                     mode
 282                   </p>
 283                 </td>
 284 <td>
 285                   <p>
 286                     &#946;((&#945; - 1) / &#945;)<sup>1/&#945;</sup>
 287                   </p>
 288                 </td>
 289 </tr>
 290 <tr>
 291 <td>
 292                   <p>
 293                     skewness
 294                   </p>
 295                 </td>
 296 <td>
 297                   <p>
 298                     Refer to <a href="http://mathworld.wolfram.com/WeibullDistribution.html" target="_top">Weisstein,
 299                     Eric W. "Weibull Distribution." From MathWorld--A Wolfram
 300                     Web Resource.</a>
 301                   </p>
 302                 </td>
 303 </tr>
 304 <tr>
 305 <td>
 306                   <p>
 307                     kurtosis
 308                   </p>
 309                 </td>
 310 <td>
 311                   <p>
 312                     Refer to <a href="http://mathworld.wolfram.com/WeibullDistribution.html" target="_top">Weisstein,
 313                     Eric W. "Weibull Distribution." From MathWorld--A Wolfram
 314                     Web Resource.</a>
 315                   </p>
 316                 </td>
 317 </tr>
 318 <tr>
 319 <td>
 320                   <p>
 321                     kurtosis excess
 322                   </p>
 323                 </td>
 324 <td>
 325                   <p>
 326                     Refer to <a href="http://mathworld.wolfram.com/WeibullDistribution.html" target="_top">Weisstein,
 327                     Eric W. "Weibull Distribution." From MathWorld--A Wolfram
 328                     Web Resource.</a>
 329                   </p>
 330                 </td>
 331 </tr>
 332 </tbody>
 333 </table></div>
 334 <h5>
 335 <a name="math_toolkit.dist_ref.dists.weibull_dist.h5"></a>
 336           <span class="phrase"><a name="math_toolkit.dist_ref.dists.weibull_dist.references"></a></span><a class="link" href="weibull_dist.html#math_toolkit.dist_ref.dists.weibull_dist.references">References</a>
 337         </h5>
 338 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 339 <li class="listitem">
 340               <a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">http://en.wikipedia.org/wiki/Weibull_distribution</a>
 341             </li>
 342 <li class="listitem">
 343               <a href="http://mathworld.wolfram.com/WeibullDistribution.html" target="_top">Weisstein,
 344               Eric W. "Weibull Distribution." From MathWorld--A Wolfram
 345               Web Resource.</a>
 346             </li>
 347 <li class="listitem">
 348               <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm" target="_top">Weibull
 349               in NIST Exploratory Data Analysis</a>
 350             </li>
 351 </ul></div>
 352 </div>
 353 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 354 <td align="left"></td>
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 356       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
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