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  27 <a name="math_toolkit.dist_ref.dists.empirical_cdf"></a><a class="link" href="empirical_cdf.html" title="Empirical Cumulative Distribution Function">Empirical
  28         Cumulative Distribution Function</a>
  29 </h4></div></div></div>
  30 <h6>
  31 <a name="math_toolkit.dist_ref.dists.empirical_cdf.h0"></a>
  32           <span class="phrase"><a name="math_toolkit.dist_ref.dists.empirical_cdf.synopsis"></a></span><a class="link" href="empirical_cdf.html#math_toolkit.dist_ref.dists.empirical_cdf.synopsis">Synopsis</a>
  33         </h6>
  34 <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">empirical_cumulative_distribution_function</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  35
  36 <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>
  37
  38 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">&gt;</span>
  39 <span class="keyword">class</span> <span class="identifier">empirical_cumulative_distribution_function</span>
  40 <span class="special">{</span>
  41 <span class="keyword">public</span><span class="special">:</span>
  42     <span class="keyword">using</span> <span class="identifier">Real</span> <span class="special">=</span> <span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">;</span>
  43     <span class="identifier">empirical_cumulative_distribution_function</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="special">&amp;&amp;</span> <span class="identifier">v</span><span class="special">,</span> <span class="keyword">bool</span> <span class="identifier">sorted</span> <span class="special">=</span> <span class="keyword">false</span><span class="special">);</span>
  44
  45     <span class="keyword">auto</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">Real</span> <span class="identifier">t</span><span class="special">)</span> <span class="keyword">const</span><span class="special">;</span>
  46
  47     <span class="identifier">RandomAccessContainer</span><span class="special">&amp;&amp;</span> <span class="identifier">return_data</span><span class="special">();</span>
  48 <span class="special">};</span>
  49
  50 <span class="special">}}</span>
  51 </pre>
  52 <h6>
  53 <a name="math_toolkit.dist_ref.dists.empirical_cdf.h1"></a>
  54           <span class="phrase"><a name="math_toolkit.dist_ref.dists.empirical_cdf.empirical_cumulative_distributio"></a></span><a class="link" href="empirical_cdf.html#math_toolkit.dist_ref.dists.empirical_cdf.empirical_cumulative_distributio">Empirical
  55           Cumulative Distribution Function</a>
  56         </h6>
  57 <p>
  58           The empirical cumulative distribution function is a step function constructed
  59           from observed data which converges to the true cumulative distribution
  60           function in the limit of infinite data. This function is a basic building
  61           block of hypothesis testing workflows that attempt to answer the question
  62           "does my data come from a given distribution?" These tests require
  63           computing quadratures over some function of the empirical CDF and the supposed
  64           CDF to create a distance measurement, and hence it is occasionally useful
  65           to construct a continuous callable from the data.
  66         </p>
  67 <p>
  68           An example usage is demonstrated below:
  69         </p>
  70 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">vector</span><span class="special">&gt;</span>
  71 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">random</span><span class="special">&gt;</span>
  72 <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">empirical_cumulative_distribution_function</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  73 <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">empirical_cumulative_distribution_function</span><span class="special">;</span>
  74 <span class="identifier">std</span><span class="special">::</span><span class="identifier">random_device</span> <span class="identifier">rd</span><span class="special">;</span>
  75 <span class="identifier">std</span><span class="special">::</span><span class="identifier">mt19937</span> <span class="identifier">gen</span><span class="special">{</span><span class="identifier">rd</span><span class="special">()};</span>
  76 <span class="identifier">std</span><span class="special">::</span><span class="identifier">normal_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">dis</span><span class="special">(</span><span class="number">0</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span>
  77 <span class="identifier">size_t</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">128</span><span class="special">;</span>
  78 <span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">v</span><span class="special">(</span><span class="identifier">n</span><span class="special">);</span>
  79 <span class="keyword">for</span> <span class="special">(</span><span class="identifier">size_t</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special">&lt;</span> <span class="identifier">n</span><span class="special">;</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span> <span class="special">{</span>
  80   <span class="identifier">v</span><span class="special">[</span><span class="identifier">i</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">dis</span><span class="special">(</span><span class="identifier">gen</span><span class="special">);</span>
  81 <span class="special">}</span>
  82
  83 <span class="keyword">auto</span> <span class="identifier">ecdf</span> <span class="special">=</span> <span class="identifier">empirical_cumulative_distribution_function</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">v</span><span class="special">));</span>
  84 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"ecdf(0.0) = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">ecdf</span><span class="special">(</span><span class="number">0.0</span><span class="special">)</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  85 <span class="comment">// should print approximately 0.5 . . .</span>
  86 </pre>
  87 <p>
  88           The empirical distribution function requires sorted data. By default, the
  89           constructor sorts it for you at O(Nlog(N)) cost.
  90         </p>
  91 <p>
  92           If your data is already sorted, you can specify this and the constructor
  93           simply moves your data into the class:
  94         </p>
  95 <pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">sort</span><span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span>
  96 <span class="keyword">auto</span> <span class="identifier">ecdf</span> <span class="special">=</span> <span class="identifier">empirical_cumulative_distribution_function</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">move</span><span class="special">(</span><span class="identifier">v</span><span class="special">),</span> <span class="comment">/* already sorted = */</span> <span class="keyword">true</span><span class="special">);</span>
  97 </pre>
  98 <p>
  99           If you want your data back after being done with the object, use
 100         </p>
 101 <pre class="programlisting"><span class="identifier">v</span> <span class="special">=</span> <span class="identifier">ecdf</span><span class="special">.</span><span class="identifier">return_data</span><span class="special">();</span>
 102 </pre>
 103 <p>
 104           This operation invalidates <code class="computeroutput"><span class="identifier">ecdf</span></code>;
 105           it can no longer be used.
 106         </p>
 107 <p>
 108           The call operator complexity is O(log(N)), as it requires a call to <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">upper_bound</span></code>.
 109         </p>
 110 <p>
 111           Works with both integer and floating point types. If the input data consists
 112           of integers, the output of the call operator is a double. Requires C++17.
 113         </p>
 114 <p>
 115           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../../graphs/empiricial_cumulative_distribution_gauss.svg"></object></span>
 116         </p>
 117 <p>
 118           <span class="inlinemediaobject"><object type="image/svg+xml" data="../../../../graphs/empiricial_cumulative_distribution_uniform.svg"></object></span>
 119         </p>
 120 <h6>
 121 <a name="math_toolkit.dist_ref.dists.empirical_cdf.h2"></a>
 122           <span class="phrase"><a name="math_toolkit.dist_ref.dists.empirical_cdf.performance"></a></span><a class="link" href="empirical_cdf.html#math_toolkit.dist_ref.dists.empirical_cdf.performance">Performance</a>
 123         </h6>
 124 <pre class="programlisting"><span class="special">------------------------------------------------------</span>
 125 <span class="identifier">Benchmark</span>                                         <span class="identifier">Time</span>
 126 <span class="special">------------------------------------------------------</span>
 127 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span>                <span class="number">4.52</span> <span class="identifier">ns</span>
 128 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16</span>               <span class="number">5.20</span> <span class="identifier">ns</span>
 129 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32</span>               <span class="number">5.22</span> <span class="identifier">ns</span>
 130 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span>               <span class="number">7.37</span> <span class="identifier">ns</span>
 131 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">128</span>              <span class="number">7.16</span> <span class="identifier">ns</span>
 132 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span>              <span class="number">8.97</span> <span class="identifier">ns</span>
 133 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span>              <span class="number">8.44</span> <span class="identifier">ns</span>
 134 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span>             <span class="number">9.07</span> <span class="identifier">ns</span>
 135 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span>             <span class="number">11.4</span> <span class="identifier">ns</span>
 136 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span>             <span class="number">12.6</span> <span class="identifier">ns</span>
 137 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span>             <span class="number">11.4</span> <span class="identifier">ns</span>
 138 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span>            <span class="number">16.0</span> <span class="identifier">ns</span>
 139 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span>            <span class="number">17.0</span> <span class="identifier">ns</span>
 140 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span>            <span class="number">19.5</span> <span class="identifier">ns</span>
 141 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span>           <span class="number">15.8</span> <span class="identifier">ns</span>
 142 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span>           <span class="number">17.9</span> <span class="identifier">ns</span>
 143 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span>           <span class="number">26.7</span> <span class="identifier">ns</span>
 144 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span>          <span class="number">29.5</span> <span class="identifier">ns</span>
 145 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2097152</span>          <span class="number">31.8</span> <span class="identifier">ns</span>
 146 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4194304</span>          <span class="number">32.8</span> <span class="identifier">ns</span>
 147 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8388608</span>          <span class="number">35.4</span> <span class="identifier">ns</span>
 148 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16777216</span>         <span class="number">30.4</span> <span class="identifier">ns</span>
 149 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span>             <span class="number">1.27</span> <span class="identifier">lgN</span>
 150 <span class="identifier">ECDFConstructorSorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_RMS</span>                <span class="number">20</span> <span class="special">%</span>
 151 <span class="identifier">ECDFConstructorUnsorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span>               <span class="number">155</span> <span class="identifier">ns</span>
 152 <span class="identifier">ECDFConstructorUnsorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span>             <span class="number">2095</span> <span class="identifier">ns</span>
 153 <span class="identifier">ECDFConstructorUnsorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span>           <span class="number">22212</span> <span class="identifier">ns</span>
 154 <span class="identifier">ECDFConstructorUnsorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span>         <span class="number">220821</span> <span class="identifier">ns</span>
 155 <span class="identifier">ECDFConstructorUnsorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span>       <span class="number">1996380</span> <span class="identifier">ns</span>
 156 <span class="identifier">ECDFConstructorUnsorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span>     <span class="number">18916039</span> <span class="identifier">ns</span>
 157 <span class="identifier">ECDFConstructorUnsorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2097152</span>   <span class="number">194250013</span> <span class="identifier">ns</span>
 158 <span class="identifier">ECDFConstructorUnsorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16777216</span> <span class="number">2281469424</span> <span class="identifier">ns</span>
 159 <span class="identifier">ECDFConstructorUnsorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span>           <span class="number">5.65</span> <span class="identifier">NlgN</span>
 160 <span class="identifier">ECDFConstructorUnsorted</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_RMS</span>               <span class="number">6</span> <span class="special">%</span>
 161 <span class="identifier">Shuffle</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span>                              <span class="number">82.4</span> <span class="identifier">ns</span>
 162 <span class="identifier">Shuffle</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span>                              <span class="number">731</span> <span class="identifier">ns</span>
 163 <span class="identifier">Shuffle</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span>                            <span class="number">5876</span> <span class="identifier">ns</span>
 164 <span class="identifier">Shuffle</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span>                          <span class="number">46864</span> <span class="identifier">ns</span>
 165 <span class="identifier">Shuffle</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span>                        <span class="number">385265</span> <span class="identifier">ns</span>
 166 <span class="identifier">Shuffle</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span>                      <span class="number">4663866</span> <span class="identifier">ns</span>
 167 <span class="identifier">Shuffle</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2097152</span>                    <span class="number">54686332</span> <span class="identifier">ns</span>
 168 <span class="identifier">Shuffle</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16777216</span>                  <span class="number">875309099</span> <span class="identifier">ns</span>
 169 <span class="identifier">Shuffle</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span>                           <span class="number">2.16</span> <span class="identifier">NlgN</span>
 170 <span class="identifier">Shuffle</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_RMS</span>                              <span class="number">12</span> <span class="special">%</span>
 171 <span class="identifier">ECDFEvaluation</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span>                       <span class="number">48.6</span> <span class="identifier">ns</span>
 172 <span class="identifier">ECDFEvaluation</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span>                      <span class="number">61.3</span> <span class="identifier">ns</span>
 173 <span class="identifier">ECDFEvaluation</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span>                     <span class="number">85.1</span> <span class="identifier">ns</span>
 174 <span class="identifier">ECDFEvaluation</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span>                     <span class="number">105</span> <span class="identifier">ns</span>
 175 <span class="identifier">ECDFEvaluation</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span>                    <span class="number">131</span> <span class="identifier">ns</span>
 176 <span class="identifier">ECDFEvaluation</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span>                   <span class="number">196</span> <span class="identifier">ns</span>
 177 <span class="identifier">ECDFEvaluation</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2097152</span>                  <span class="number">391</span> <span class="identifier">ns</span>
 178 <span class="identifier">ECDFEvaluation</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16777216</span>                 <span class="number">715</span> <span class="identifier">ns</span>
 179 <span class="identifier">ECDFEvaluation</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span>                   <span class="number">18.19</span> <span class="identifier">lgN</span>
 180 <span class="identifier">ECDFEvaluation</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_RMS</span>                       <span class="number">60</span> <span class="special">%</span>
 181 </pre>
 182 <p>
 183           The call to the unsorted constructor is in fact a little faster than indicated,
 184           as the data must be shuffled after being sorted in the benchmark. This
 185           is itself a fairly expensive operation.
 186         </p>
 187 </div>
 188 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 189 <td align="left"></td>
 190 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 191       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 192       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 193       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 194       Daryle Walker and Xiaogang Zhang<p>
 195         Distributed under the Boost Software License, Version 1.0. (See accompanying
 196         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>)
 197       </p>
 198 </div></td>
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