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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;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">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;20.&#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 <span class="emphasis"><em>&#956;</em></span> and scale parameter <span class="emphasis"><em>&#963;</em></span>,
  60           it is defined by the probability density function:
  61         </p>
  62 <div class="blockquote"><blockquote class="blockquote"><p>
  63             <span class="inlinemediaobject"><img src="../../../../equations/laplace_pdf.svg"></span>
  64
  65           </p></blockquote></div>
  66 <p>
  67           The location and scale parameters are equivalent to the mean and standard
  68           deviation of the normal or Gaussian distribution.
  69         </p>
  70 <p>
  71           The following graph illustrates the effect of the parameters &#956; and &#963; on the
  72           PDF. Note that the domain of the random variable remains [-&#8734;,+&#8734;] irrespective
  73           of the value of the location parameter:
  74         </p>
  75 <div class="blockquote"><blockquote class="blockquote"><p>
  76             <span class="inlinemediaobject"><img src="../../../../graphs/laplace_pdf.svg" align="middle"></span>
  77
  78           </p></blockquote></div>
  79 <h5>
  80 <a name="math_toolkit.dist_ref.dists.laplace_dist.h0"></a>
  81           <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
  82           Functions</a>
  83         </h5>
  84 <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>
  85 </pre>
  86 <p>
  87           Constructs a laplace distribution with location <span class="emphasis"><em>location</em></span>
  88           and scale <span class="emphasis"><em>scale</em></span>.
  89         </p>
  90 <p>
  91           The location parameter is the same as the mean of the random variate.
  92         </p>
  93 <p>
  94           The scale parameter is proportional to the standard deviation of the random
  95           variate.
  96         </p>
  97 <p>
  98           Requires that the scale parameter is greater than zero, otherwise calls
  99           <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 100         </p>
 101 <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>
 102 </pre>
 103 <p>
 104           Returns the <span class="emphasis"><em>location</em></span> parameter of this distribution.
 105         </p>
 106 <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>
 107 </pre>
 108 <p>
 109           Returns the <span class="emphasis"><em>scale</em></span> parameter of this distribution.
 110         </p>
 111 <h5>
 112 <a name="math_toolkit.dist_ref.dists.laplace_dist.h1"></a>
 113           <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
 114           Accessors</a>
 115         </h5>
 116 <p>
 117           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 118           functions</a> that are generic to all distributions are supported:
 119           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 120           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 121           <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>,
 122           <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>,
 123           <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>,
 124           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 125           <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>,
 126           <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>.
 127         </p>
 128 <p>
 129           The domain of the random variable is [-&#8734;,+&#8734;].
 130         </p>
 131 <h5>
 132 <a name="math_toolkit.dist_ref.dists.laplace_dist.h2"></a>
 133           <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>
 134         </h5>
 135 <p>
 136           The laplace distribution is implemented in terms of the standard library
 137           log and exp functions and as such should have very small errors.
 138         </p>
 139 <h5>
 140 <a name="math_toolkit.dist_ref.dists.laplace_dist.h3"></a>
 141           <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>
 142         </h5>
 143 <p>
 144           In the following table &#956; is the location parameter of the distribution, &#963; is
 145           its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span>
 146           is the probability and its complement <span class="emphasis"><em>q = 1-p</em></span>.
 147         </p>
 148 <div class="informaltable"><table class="table">
 149 <colgroup>
 150 <col>
 151 <col>
 152 </colgroup>
 153 <thead><tr>
 154 <th>
 155                   <p>
 156                     Function
 157                   </p>
 158                 </th>
 159 <th>
 160                   <p>
 161                     Implementation Notes
 162                   </p>
 163                 </th>
 164 </tr></thead>
 165 <tbody>
 166 <tr>
 167 <td>
 168                   <p>
 169                     pdf
 170                   </p>
 171                 </td>
 172 <td>
 173                   <p>
 174                     Using the relation: pdf = e<sup>-abs(x-&#956;) / &#963;</sup> / (2 * &#963;)
 175                   </p>
 176                 </td>
 177 </tr>
 178 <tr>
 179 <td>
 180                   <p>
 181                     cdf
 182                   </p>
 183                 </td>
 184 <td>
 185                   <p>
 186                     Using the relations:
 187                   </p>
 188                   <p>
 189                     x &lt; &#956; : p = e<sup>(x-&#956;)/&#963; </sup> / &#963;
 190                   </p>
 191                   <p>
 192                     x &gt;= &#956; : p = 1 - e<sup>(&#956;-x)/&#963; </sup> / &#963;
 193
 194                   </p>
 195                 </td>
 196 </tr>
 197 <tr>
 198 <td>
 199                   <p>
 200                     cdf complement
 201                   </p>
 202                 </td>
 203 <td>
 204                   <p>
 205                     Using the relation:
 206                   </p>
 207                   <p>
 208                     -x &lt; &#956; : q = e<sup>(-x-&#956;)/&#963; </sup> / &#963;
 209                   </p>
 210                   <p>
 211                     -x &gt;= &#956; : q = 1 - e<sup>(&#956;+x)/&#963; </sup> / &#963;
 212
 213                   </p>
 214                 </td>
 215 </tr>
 216 <tr>
 217 <td>
 218                   <p>
 219                     quantile
 220                   </p>
 221                 </td>
 222 <td>
 223                   <p>
 224                     Using the relations:
 225                   </p>
 226                   <p>
 227                     p &lt; 0.5 : x = &#956; + &#963; * log(2*p)
 228                   </p>
 229                   <p>
 230                     p &gt;= 0.5 : x = &#956; - &#963; * log(2-2*p)
 231                   </p>
 232                 </td>
 233 </tr>
 234 <tr>
 235 <td>
 236                   <p>
 237                     quantile from the complement
 238                   </p>
 239                 </td>
 240 <td>
 241                   <p>
 242                     Using the relation:
 243                   </p>
 244                   <p>
 245                     q &gt; 0.5: x = &#956; + &#963;*log(2-2*q)
 246                   </p>
 247                   <p>
 248                     q &lt;=0.5: x = &#956; - &#963;*log( 2*q )
 249                   </p>
 250                 </td>
 251 </tr>
 252 <tr>
 253 <td>
 254                   <p>
 255                     mean
 256                   </p>
 257                 </td>
 258 <td>
 259                   <p>
 260                     &#956;
 261                   </p>
 262                 </td>
 263 </tr>
 264 <tr>
 265 <td>
 266                   <p>
 267                     variance
 268                   </p>
 269                 </td>
 270 <td>
 271                   <p>
 272                     2 * &#963;<sup>2</sup>
 273                   </p>
 274                 </td>
 275 </tr>
 276 <tr>
 277 <td>
 278                   <p>
 279                     mode
 280                   </p>
 281                 </td>
 282 <td>
 283                   <p>
 284                     &#956;
 285                   </p>
 286                 </td>
 287 </tr>
 288 <tr>
 289 <td>
 290                   <p>
 291                     skewness
 292                   </p>
 293                 </td>
 294 <td>
 295                   <p>
 296                     0
 297                   </p>
 298                 </td>
 299 </tr>
 300 <tr>
 301 <td>
 302                   <p>
 303                     kurtosis
 304                   </p>
 305                 </td>
 306 <td>
 307                   <p>
 308                     6
 309                   </p>
 310                 </td>
 311 </tr>
 312 <tr>
 313 <td>
 314                   <p>
 315                     kurtosis excess
 316                   </p>
 317                 </td>
 318 <td>
 319                   <p>
 320                     3
 321                   </p>
 322                 </td>
 323 </tr>
 324 </tbody>
 325 </table></div>
 326 <h5>
 327 <a name="math_toolkit.dist_ref.dists.laplace_dist.h4"></a>
 328           <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>
 329         </h5>
 330 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 331 <li class="listitem">
 332               <a href="http://mathworld.wolfram.com/LaplaceDistribution.html" target="_top">Weisstein,
 333               Eric W. "Laplace Distribution."</a> From MathWorld--A
 334               Wolfram Web Resource.
 335             </li>
 336 <li class="listitem">
 337               <a href="http://en.wikipedia.org/wiki/Laplace_distribution" target="_top">Laplace
 338               Distribution</a>
 339             </li>
 340 <li class="listitem">
 341               M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,
 342               1972, p. 930.
 343             </li>
 344 </ul></div>
 345 </div>
 346 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 347 <td align="left"></td>
 348 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 349       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 350       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 351       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 352       Daryle Walker and Xiaogang Zhang<p>
 353         Distributed under the Boost Software License, Version 1.0. (See accompanying
 354         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>)
 355       </p>
 356 </div></td>
 357 </tr></table>
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