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<div class="titlepage"><div><div><h4 class="title">
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<a name="math_toolkit.dist_ref.dists.laplace_dist"></a><a class="link" href="laplace_dist.html" title="Laplace Distribution">Laplace Distribution</a>
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</h4></div></div></div>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</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">></span></pre>
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<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>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
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<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 15. 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<></a> <span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">laplace_distribution</span><span class="special">;</span>
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<span class="keyword">typedef</span> <span class="identifier">laplace_distribution</span><span class="special"><></span> <span class="identifier">laplace</span><span class="special">;</span>
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<span class="keyword">template</span> <span class="special"><</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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
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<span class="keyword">class</span> <span class="identifier">laplace_distribution</span>
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<span class="special">{</span>
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<span class="keyword">public</span><span class="special">:</span>
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<span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
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<span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span>
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<span class="comment">// Construct:</span>
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<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>
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<span class="comment">// Accessors:</span>
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<span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
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<span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
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<span class="special">};</span>
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<span class="special">}}</span> <span class="comment">// namespaces</span>
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</pre>
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<p>
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Laplace distribution is the distribution of differences between two independent
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variates with identical exponential distributions (Abramowitz and Stegun
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1972, p. 930). It is also called the double exponential distribution.
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</p>
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<p>
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For location parameter μ   and scale parameter σ  , it is defined by the probability
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density function:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../../equations/laplace_pdf.svg"></span>
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</p>
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<p>
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The location and scale parameters are equivalent to the mean and standard
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deviation of the normal or Gaussian distribution.
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</p>
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<p>
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The following graph illustrates the effect of the parameters μ   and σ   on the
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PDF. Note that the domain of the random variable remains [-∞,+∞] irrespective
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of the value of the location parameter:
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</p>
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<p>
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<span class="inlinemediaobject"><img src="../../../../graphs/laplace_pdf.svg" align="middle"></span>
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.laplace_dist.h0"></a>
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<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
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Functions</a>
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</h5>
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<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>
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</pre>
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<p>
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Constructs a laplace distribution with location <span class="emphasis"><em>location</em></span>
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and scale <span class="emphasis"><em>scale</em></span>.
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</p>
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<p>
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The location parameter is the same as the mean of the random variate.
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</p>
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<p>
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The scale parameter is proportional to the standard deviation of the random
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variate.
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</p>
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<p>
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Requires that the scale parameter is greater than zero, otherwise calls
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<a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
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</p>
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<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>
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</pre>
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<p>
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Returns the <span class="emphasis"><em>location</em></span> parameter of this distribution.
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</p>
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<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>
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</pre>
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<p>
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Returns the <span class="emphasis"><em>scale</em></span> parameter of this distribution.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.laplace_dist.h1"></a>
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<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
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Accessors</a>
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</h5>
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<p>
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All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
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functions</a> that are generic to all distributions are supported:
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
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<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>,
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<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>,
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<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>,
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<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
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<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>,
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<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>.
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</p>
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<p>
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The domain of the random variable is [-∞,+∞].
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.laplace_dist.h2"></a>
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<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>
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</h5>
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<p>
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The laplace distribution is implemented in terms of the standard library
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log and exp functions and as such should have very small errors.
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</p>
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<h5>
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<a name="math_toolkit.dist_ref.dists.laplace_dist.h3"></a>
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<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>
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</h5>
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<p>
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In the following table μ is the location parameter of the distribution, σ is
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its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span>
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is the probability and its complement <span class="emphasis"><em>q = 1-p</em></span>.
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</p>
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<div class="informaltable"><table class="table">
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<colgroup>
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<col>
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<col>
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</colgroup>
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<thead><tr>
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<th>
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<p>
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Function
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</p>
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</th>
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<th>
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<p>
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Implementation Notes
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</p>
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</th>
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</tr></thead>
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<tbody>
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<tr>
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<td>
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<p>
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pdf
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</p>
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</td>
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<td>
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<p>
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Using the relation: pdf = e<sup>-abs(x-μ) / σ</sup> / (2 * σ)
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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cdf
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</p>
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</td>
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<td>
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<p>
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Using the relations:
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</p>
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<p>
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x < μ : p = e<sup>(x-μ)/σ </sup> / σ
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</p>
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<p>
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x >= μ : p = 1 - e<sup>(μ-x)/σ </sup> / σ
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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cdf complement
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</p>
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</td>
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<td>
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<p>
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Using the relation:
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</p>
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<p>
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-x < μ : q = e<sup>(-x-μ)/σ </sup> / σ
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</p>
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<p>
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-x >= μ : q = 1 - e<sup>(μ+x)/σ </sup> / σ
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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quantile
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</p>
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</td>
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<td>
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<p>
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Using the relations:
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</p>
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<p>
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p < 0.5 : x = μ + σ * log(2*p)
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</p>
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<p>
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p >= 0.5 : x = μ - σ * log(2-2*p)
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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quantile from the complement
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</p>
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</td>
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<td>
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<p>
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Using the relation:
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</p>
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<p>
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q > 0.5: x = μ + σ*log(2-2*q)
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</p>
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<p>
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q <=0.5: x = μ - σ*log( 2*q )
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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mean
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</p>
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</td>
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<td>
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<p>
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μ
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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variance
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</p>
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</td>
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<td>
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<p>
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2 * σ<sup>2</sup>
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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mode
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</p>
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</td>
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<td>
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<p>
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μ
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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skewness
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</p>
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</td>
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<td>
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<p>
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0
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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kurtosis
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</p>
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</td>
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<td>
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<p>
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6
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</p>
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</td>
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</tr>
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<tr>
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<td>
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<p>
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kurtosis excess
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</p>
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</td>
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<td>
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<p>
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3
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</p>
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</td>
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</tr>
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</tbody>
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</table></div>
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<h5>
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<a name="math_toolkit.dist_ref.dists.laplace_dist.h4"></a>
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<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>
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</h5>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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<a href="http://mathworld.wolfram.com/LaplaceDistribution.html" target="_top">Weisstein,
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Eric W. "Laplace Distribution."</a> From MathWorld--A
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Wolfram Web Resource.
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</li>
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<li class="listitem">
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<a href="http://en.wikipedia.org/wiki/Laplace_distribution" target="_top">Laplace
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Distribution</a>
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</li>
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<li class="listitem">
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M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,
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1972, p. 930.
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</li>
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</ul></div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
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Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
|
|
Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
|
|
Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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>)
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</p>
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</div></td>
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</tr></table>
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<hr>
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