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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.laplace_dist"></a><a class="link" href="laplace_dist.html" title="Laplace Distribution">Laplace 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">laplace</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">laplace_distribution</span><span class="special">;</span>
  35
  36 <span class="keyword">typedef</span> <span class="identifier">laplace_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">laplace</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">laplace_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">laplace_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</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">location</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           Laplace distribution is the distribution of differences between two independent
  55           variates with identical exponential distributions (Abramowitz and Stegun
  56           1972, p. 930). It is also called the double exponential distribution.
  57         </p>
  58 <p>
  59           For location parameter &#956; &#160; and scale parameter &#963; &#160;, it is defined by the probability
  60           density function:
  61         </p>
  62 <p>
  63           <span class="inlinemediaobject"><img src="../../../../equations/laplace_pdf.svg"></span>
  64         </p>
  65 <p>
  66           The location and scale parameters are equivalent to the mean and standard
  67           deviation of the normal or Gaussian distribution.
  68         </p>
  69 <p>
  70           The following graph illustrates the effect of the parameters &#956; &#160; and &#963; &#160; on the
  71           PDF. Note that the domain of the random variable remains [-&#8734;,+&#8734;] irrespective
  72           of the value of the location parameter:
  73         </p>
  74 <p>
  75           <span class="inlinemediaobject"><img src="../../../../graphs/laplace_pdf.svg" align="middle"></span>
  76         </p>
  77 <h5>
  78 <a name="math_toolkit.dist_ref.dists.laplace_dist.h0"></a>
  79           <span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.member_functions"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.member_functions">Member
  80           Functions</a>
  81         </h5>
  82 <pre class="programlisting"><span class="identifier">laplace_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</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>
  83 </pre>
  84 <p>
  85           Constructs a laplace distribution with location <span class="emphasis"><em>location</em></span>
  86           and scale <span class="emphasis"><em>scale</em></span>.
  87         </p>
  88 <p>
  89           The location parameter is the same as the mean of the random variate.
  90         </p>
  91 <p>
  92           The scale parameter is proportional to the standard deviation of the random
  93           variate.
  94         </p>
  95 <p>
  96           Requires that the scale parameter is greater than zero, otherwise calls
  97           <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
  98         </p>
  99 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 100 </pre>
 101 <p>
 102           Returns the <span class="emphasis"><em>location</em></span> parameter of this distribution.
 103         </p>
 104 <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>
 105 </pre>
 106 <p>
 107           Returns the <span class="emphasis"><em>scale</em></span> parameter of this distribution.
 108         </p>
 109 <h5>
 110 <a name="math_toolkit.dist_ref.dists.laplace_dist.h1"></a>
 111           <span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.non_member_accessors"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.non_member_accessors">Non-member
 112           Accessors</a>
 113         </h5>
 114 <p>
 115           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 116           functions</a> that are generic to all distributions are supported:
 117           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 118           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 119           <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>,
 120           <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>,
 121           <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>,
 122           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 123           <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>,
 124           <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>.
 125         </p>
 126 <p>
 127           The domain of the random variable is [-&#8734;,+&#8734;].
 128         </p>
 129 <h5>
 130 <a name="math_toolkit.dist_ref.dists.laplace_dist.h2"></a>
 131           <span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.accuracy"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.accuracy">Accuracy</a>
 132         </h5>
 133 <p>
 134           The laplace distribution is implemented in terms of the standard library
 135           log and exp functions and as such should have very small errors.
 136         </p>
 137 <h5>
 138 <a name="math_toolkit.dist_ref.dists.laplace_dist.h3"></a>
 139           <span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.implementation"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.implementation">Implementation</a>
 140         </h5>
 141 <p>
 142           In the following table &#956; is the location parameter of the distribution, &#963; is
 143           its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span>
 144           is the probability and its complement <span class="emphasis"><em>q = 1-p</em></span>.
 145         </p>
 146 <div class="informaltable"><table class="table">
 147 <colgroup>
 148 <col>
 149 <col>
 150 </colgroup>
 151 <thead><tr>
 152 <th>
 153                   <p>
 154                     Function
 155                   </p>
 156                 </th>
 157 <th>
 158                   <p>
 159                     Implementation Notes
 160                   </p>
 161                 </th>
 162 </tr></thead>
 163 <tbody>
 164 <tr>
 165 <td>
 166                   <p>
 167                     pdf
 168                   </p>
 169                 </td>
 170 <td>
 171                   <p>
 172                     Using the relation: pdf = e<sup>-abs(x-&#956;) / &#963;</sup> / (2 * &#963;)
 173                   </p>
 174                 </td>
 175 </tr>
 176 <tr>
 177 <td>
 178                   <p>
 179                     cdf
 180                   </p>
 181                 </td>
 182 <td>
 183                   <p>
 184                     Using the relations:
 185                   </p>
 186                   <p>
 187                     x &lt; &#956; : p = e<sup>(x-&#956;)/&#963; </sup> / &#963;
 188                   </p>
 189                   <p>
 190                     x &gt;= &#956; : p = 1 - e<sup>(&#956;-x)/&#963; </sup> / &#963;
 191
 192                   </p>
 193                 </td>
 194 </tr>
 195 <tr>
 196 <td>
 197                   <p>
 198                     cdf complement
 199                   </p>
 200                 </td>
 201 <td>
 202                   <p>
 203                     Using the relation:
 204                   </p>
 205                   <p>
 206                     -x &lt; &#956; : q = e<sup>(-x-&#956;)/&#963; </sup> / &#963;
 207                   </p>
 208                   <p>
 209                     -x &gt;= &#956; : q = 1 - e<sup>(&#956;+x)/&#963; </sup> / &#963;
 210
 211                   </p>
 212                 </td>
 213 </tr>
 214 <tr>
 215 <td>
 216                   <p>
 217                     quantile
 218                   </p>
 219                 </td>
 220 <td>
 221                   <p>
 222                     Using the relations:
 223                   </p>
 224                   <p>
 225                     p &lt; 0.5 : x = &#956; + &#963; * log(2*p)
 226                   </p>
 227                   <p>
 228                     p &gt;= 0.5 : x = &#956; - &#963; * log(2-2*p)
 229                   </p>
 230                 </td>
 231 </tr>
 232 <tr>
 233 <td>
 234                   <p>
 235                     quantile from the complement
 236                   </p>
 237                 </td>
 238 <td>
 239                   <p>
 240                     Using the relation:
 241                   </p>
 242                   <p>
 243                     q &gt; 0.5: x = &#956; + &#963;*log(2-2*q)
 244                   </p>
 245                   <p>
 246                     q &lt;=0.5: x = &#956; - &#963;*log( 2*q )
 247                   </p>
 248                 </td>
 249 </tr>
 250 <tr>
 251 <td>
 252                   <p>
 253                     mean
 254                   </p>
 255                 </td>
 256 <td>
 257                   <p>
 258                     &#956;
 259                   </p>
 260                 </td>
 261 </tr>
 262 <tr>
 263 <td>
 264                   <p>
 265                     variance
 266                   </p>
 267                 </td>
 268 <td>
 269                   <p>
 270                     2 * &#963;<sup>2</sup>
 271                   </p>
 272                 </td>
 273 </tr>
 274 <tr>
 275 <td>
 276                   <p>
 277                     mode
 278                   </p>
 279                 </td>
 280 <td>
 281                   <p>
 282                     &#956;
 283                   </p>
 284                 </td>
 285 </tr>
 286 <tr>
 287 <td>
 288                   <p>
 289                     skewness
 290                   </p>
 291                 </td>
 292 <td>
 293                   <p>
 294                     0
 295                   </p>
 296                 </td>
 297 </tr>
 298 <tr>
 299 <td>
 300                   <p>
 301                     kurtosis
 302                   </p>
 303                 </td>
 304 <td>
 305                   <p>
 306                     6
 307                   </p>
 308                 </td>
 309 </tr>
 310 <tr>
 311 <td>
 312                   <p>
 313                     kurtosis excess
 314                   </p>
 315                 </td>
 316 <td>
 317                   <p>
 318                     3
 319                   </p>
 320                 </td>
 321 </tr>
 322 </tbody>
 323 </table></div>
 324 <h5>
 325 <a name="math_toolkit.dist_ref.dists.laplace_dist.h4"></a>
 326           <span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.references"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.references">References</a>
 327         </h5>
 328 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 329 <li class="listitem">
 330               <a href="http://mathworld.wolfram.com/LaplaceDistribution.html" target="_top">Weisstein,
 331               Eric W. "Laplace Distribution."</a> From MathWorld--A
 332               Wolfram Web Resource.
 333             </li>
 334 <li class="listitem">
 335               <a href="http://en.wikipedia.org/wiki/Laplace_distribution" target="_top">Laplace
 336               Distribution</a>
 337             </li>
 338 <li class="listitem">
 339               M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,
 340               1972, p. 930.
 341             </li>
 342 </ul></div>
 343 </div>
 344 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 345 <td align="left"></td>
 346 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 347       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 348       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 349       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 350         Distributed under the Boost Software License, Version 1.0. (See accompanying
 351         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>)
 352       </p>
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