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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.rayleigh"></a><a class="link" href="rayleigh.html" title="Rayleigh Distribution">Rayleigh 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">rayleigh</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">rayleigh_distribution</span><span class="special">;</span>
  35
  36 <span class="keyword">typedef</span> <span class="identifier">rayleigh_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">rayleigh</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">rayleigh_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">rayleigh_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">sigma</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">sigma</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  48 <span class="special">};</span>
  49
  50 <span class="special">}}</span> <span class="comment">// namespaces</span>
  51 </pre>
  52 <p>
  53           The <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh
  54           distribution</a> is a continuous distribution with the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">probability
  55           density function</a>:
  56         </p>
  57 <p>
  58           f(x; sigma) = x * exp(-x<sup>2</sup>/2 &#963;<sup>2</sup>) / &#963;<sup>2</sup>
  59         </p>
  60 <p>
  61           For sigma parameter &#963; &#160; &gt; 0, and x &gt; 0.
  62         </p>
  63 <p>
  64           The Rayleigh distribution is often used where two orthogonal components
  65           have an absolute value, for example, wind velocity and direction may be
  66           combined to yield a wind speed, or real and imaginary components may have
  67           absolute values that are Rayleigh distributed.
  68         </p>
  69 <p>
  70           The following graph illustrates how the Probability density Function(pdf)
  71           varies with the shape parameter &#963;:
  72         </p>
  73 <p>
  74           <span class="inlinemediaobject"><img src="../../../../graphs/rayleigh_pdf.svg" align="middle"></span>
  75         </p>
  76 <p>
  77           and the Cumulative Distribution Function (cdf)
  78         </p>
  79 <p>
  80           <span class="inlinemediaobject"><img src="../../../../graphs/rayleigh_cdf.svg" align="middle"></span>
  81         </p>
  82 <h5>
  83 <a name="math_toolkit.dist_ref.dists.rayleigh.h0"></a>
  84           <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.related_distributions"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.related_distributions">Related
  85           distributions</a>
  86         </h5>
  87 <p>
  88           The absolute value of two independent normal distributions X and Y, &#8730; (X<sup>2</sup> +
  89           Y<sup>2</sup>) is a Rayleigh distribution.
  90         </p>
  91 <p>
  92           The <a href="http://en.wikipedia.org/wiki/Chi_distribution" target="_top">Chi</a>,
  93           <a href="http://en.wikipedia.org/wiki/Rice_distribution" target="_top">Rice</a>
  94           and <a href="http://en.wikipedia.org/wiki/Weibull_distribution" target="_top">Weibull</a>
  95           distributions are generalizations of the <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh
  96           distribution</a>.
  97         </p>
  98 <h5>
  99 <a name="math_toolkit.dist_ref.dists.rayleigh.h1"></a>
 100           <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.member_functions"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.member_functions">Member
 101           Functions</a>
 102         </h5>
 103 <pre class="programlisting"><span class="identifier">rayleigh_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">sigma</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
 104 </pre>
 105 <p>
 106           Constructs a <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">Rayleigh
 107           distribution</a> with &#963; <span class="emphasis"><em>sigma</em></span>.
 108         </p>
 109 <p>
 110           Requires that the &#963; parameter is greater than zero, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 111         </p>
 112 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">sigma</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 113 </pre>
 114 <p>
 115           Returns the <span class="emphasis"><em>sigma</em></span> parameter of this distribution.
 116         </p>
 117 <h5>
 118 <a name="math_toolkit.dist_ref.dists.rayleigh.h2"></a>
 119           <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.non_member_accessors"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.non_member_accessors">Non-member
 120           Accessors</a>
 121         </h5>
 122 <p>
 123           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 124           functions</a> that are generic to all distributions are supported:
 125           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 126           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 127           <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>,
 128           <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>,
 129           <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>,
 130           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 131           <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>,
 132           <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>.
 133         </p>
 134 <p>
 135           The domain of the random variable is [0, max_value].
 136         </p>
 137 <h5>
 138 <a name="math_toolkit.dist_ref.dists.rayleigh.h3"></a>
 139           <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.accuracy"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.accuracy">Accuracy</a>
 140         </h5>
 141 <p>
 142           The Rayleigh distribution is implemented in terms of the standard library
 143           <code class="computeroutput"><span class="identifier">sqrt</span></code> and <code class="computeroutput"><span class="identifier">exp</span></code> and as such should have very low
 144           error rates. Some constants such as skewness and kurtosis were calculated
 145           using NTL RR type with 150-bit accuracy, about 50 decimal digits.
 146         </p>
 147 <h5>
 148 <a name="math_toolkit.dist_ref.dists.rayleigh.h4"></a>
 149           <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.implementation"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.implementation">Implementation</a>
 150         </h5>
 151 <p>
 152           In the following table &#963; &#160; is the sigma parameter of the distribution, <span class="emphasis"><em>x</em></span>
 153           is the random variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q
 154           = 1-p</em></span>.
 155         </p>
 156 <div class="informaltable"><table class="table">
 157 <colgroup>
 158 <col>
 159 <col>
 160 </colgroup>
 161 <thead><tr>
 162 <th>
 163                   <p>
 164                     Function
 165                   </p>
 166                 </th>
 167 <th>
 168                   <p>
 169                     Implementation Notes
 170                   </p>
 171                 </th>
 172 </tr></thead>
 173 <tbody>
 174 <tr>
 175 <td>
 176                   <p>
 177                     pdf
 178                   </p>
 179                 </td>
 180 <td>
 181                   <p>
 182                     Using the relation: pdf = x * exp(-x<sup>2</sup>)/2 &#963;<sup>2</sup>
 183                   </p>
 184                 </td>
 185 </tr>
 186 <tr>
 187 <td>
 188                   <p>
 189                     cdf
 190                   </p>
 191                 </td>
 192 <td>
 193                   <p>
 194                     Using the relation: p = 1 - exp(-x<sup>2</sup>/2) &#963;<sup>2</sup> &#160; = -<a class="link" href="../../powers/expm1.html" title="expm1">expm1</a>(-x<sup>2</sup>/2)
 195                     &#963;<sup>2</sup>
 196                   </p>
 197                 </td>
 198 </tr>
 199 <tr>
 200 <td>
 201                   <p>
 202                     cdf complement
 203                   </p>
 204                 </td>
 205 <td>
 206                   <p>
 207                     Using the relation: q = exp(-x<sup>2</sup>/ 2) * &#963;<sup>2</sup>
 208                   </p>
 209                 </td>
 210 </tr>
 211 <tr>
 212 <td>
 213                   <p>
 214                     quantile
 215                   </p>
 216                 </td>
 217 <td>
 218                   <p>
 219                     Using the relation: x = sqrt(-2 * &#963; <sup>2</sup>) * log(1 - p)) = sqrt(-2
 220                     * &#963; <sup>2</sup>) * <a class="link" href="../../powers/log1p.html" title="log1p">log1p</a>(-p))
 221                   </p>
 222                 </td>
 223 </tr>
 224 <tr>
 225 <td>
 226                   <p>
 227                     quantile from the complement
 228                   </p>
 229                 </td>
 230 <td>
 231                   <p>
 232                     Using the relation: x = sqrt(-2 * &#963; <sup>2</sup>) * log(q))
 233                   </p>
 234                 </td>
 235 </tr>
 236 <tr>
 237 <td>
 238                   <p>
 239                     mean
 240                   </p>
 241                 </td>
 242 <td>
 243                   <p>
 244                     &#963; * sqrt(&#960;/2)
 245                   </p>
 246                 </td>
 247 </tr>
 248 <tr>
 249 <td>
 250                   <p>
 251                     variance
 252                   </p>
 253                 </td>
 254 <td>
 255                   <p>
 256                     &#963;<sup>2</sup> * (4 - &#960;/2)
 257                   </p>
 258                 </td>
 259 </tr>
 260 <tr>
 261 <td>
 262                   <p>
 263                     mode
 264                   </p>
 265                 </td>
 266 <td>
 267                   <p>
 268                     &#963;
 269                   </p>
 270                 </td>
 271 </tr>
 272 <tr>
 273 <td>
 274                   <p>
 275                     skewness
 276                   </p>
 277                 </td>
 278 <td>
 279                   <p>
 280                     Constant from <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
 281                     Eric W. "Weibull Distribution." From MathWorld--A Wolfram
 282                     Web Resource.</a>
 283                   </p>
 284                 </td>
 285 </tr>
 286 <tr>
 287 <td>
 288                   <p>
 289                     kurtosis
 290                   </p>
 291                 </td>
 292 <td>
 293                   <p>
 294                     Constant from <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
 295                     Eric W. "Weibull Distribution." From MathWorld--A Wolfram
 296                     Web Resource.</a>
 297                   </p>
 298                 </td>
 299 </tr>
 300 <tr>
 301 <td>
 302                   <p>
 303                     kurtosis excess
 304                   </p>
 305                 </td>
 306 <td>
 307                   <p>
 308                     Constant from <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
 309                     Eric W. "Weibull Distribution." From MathWorld--A Wolfram
 310                     Web Resource.</a>
 311                   </p>
 312                 </td>
 313 </tr>
 314 </tbody>
 315 </table></div>
 316 <h5>
 317 <a name="math_toolkit.dist_ref.dists.rayleigh.h5"></a>
 318           <span class="phrase"><a name="math_toolkit.dist_ref.dists.rayleigh.references"></a></span><a class="link" href="rayleigh.html#math_toolkit.dist_ref.dists.rayleigh.references">References</a>
 319         </h5>
 320 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 321 <li class="listitem">
 322               <a href="http://en.wikipedia.org/wiki/Rayleigh_distribution" target="_top">http://en.wikipedia.org/wiki/Rayleigh_distribution</a>
 323             </li>
 324 <li class="listitem">
 325               <a href="http://mathworld.wolfram.com/RayleighDistribution.html" target="_top">Weisstein,
 326               Eric W. "Rayleigh Distribution." From MathWorld--A Wolfram
 327               Web Resource.</a>
 328             </li>
 329 </ul></div>
 330 </div>
 331 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 332 <td align="left"></td>
 333 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 334       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 335       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 336       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 337         Distributed under the Boost Software License, Version 1.0. (See accompanying
 338         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>)
 339       </p>
 340 </div></td>
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