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