libs/math/doc/html/math_toolkit/anderson_darling.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   4 <title>The Anderson-Darling Test</title>
   5 <link rel="stylesheet" href="../math.css" type="text/css">
   6 <meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
   7 <link rel="home" href="../index.html" title="Math Toolkit 2.11.0">
   8 <link rel="up" href="../statistics.html" title="Chapter&#160;6.&#160;Statistics">
   9 <link rel="prev" href="signal_statistics.html" title="Signal Statistics">
  10 <link rel="next" href="t_test.html" title="t-tests">
  11 </head>
  12 <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
  13 <table cellpadding="2" width="100%"><tr>
  14 <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td>
  15 <td align="center"><a href="../../../../../index.html">Home</a></td>
  16 <td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td>
  17 <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
  18 <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
  19 <td align="center"><a href="../../../../../more/index.htm">More</a></td>
  20 </tr></table>
  21 <hr>
  22 <div class="spirit-nav">
  23 <a accesskey="p" href="signal_statistics.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../statistics.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="t_test.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h2 class="title" style="clear: both">
  27 <a name="math_toolkit.anderson_darling"></a><a class="link" href="anderson_darling.html" title="The Anderson-Darling Test">The Anderson-Darling Test</a>
  28 </h2></div></div></div>
  29 <h4>
  30 <a name="math_toolkit.anderson_darling.h0"></a>
  31       <span class="phrase"><a name="math_toolkit.anderson_darling.synopsis"></a></span><a class="link" href="anderson_darling.html#math_toolkit.anderson_darling.synopsis">Synopsis</a>
  32     </h4>
  33 <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">statistics</span><span class="special">/</span><span class="identifier">anderson_darling</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  34
  35 <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> <span class="keyword">namespace</span> <span class="special">{</span> <span class="identifier">statistics</span> <span class="special">{</span>
  36
  37 <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>
  38 <span class="keyword">auto</span> <span class="identifier">anderson_darling_normality_statistic</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v</span><span class="special">,</span>
  39                                           <span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">mu</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">(),</span>
  40                                           <span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">sd</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">());</span>
  41
  42 <span class="special">}}}</span>
  43 </pre>
  44 <h4>
  45 <a name="math_toolkit.anderson_darling.h1"></a>
  46       <span class="phrase"><a name="math_toolkit.anderson_darling.background"></a></span><a class="link" href="anderson_darling.html#math_toolkit.anderson_darling.background">Background</a>
  47     </h4>
  48 <p>
  49       The Anderson-Darling test for normality asks if a given sequence of numbers
  50       are drawn from a normal distribution by computing an integral over the empirical
  51       cumulative distribution function. The test statistic <span class="emphasis"><em>A</em></span><sup>2</sup> is
  52       given by
  53     </p>
  54 <p>
  55       <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/anderson_darling_definition.svg" width="346" height="56"></object></span>
  56     </p>
  57 <p>
  58       where <span class="emphasis"><em>F</em></span><sub><span class="emphasis"><em>n</em></span></sub> is the empirical cumulative
  59       distribution and <span class="emphasis"><em>F</em></span> is the CDF of the normal distribution.
  60     </p>
  61 <p>
  62       The value returned by the routine is <span class="emphasis"><em>A</em></span><sup>2</sup>.
  63     </p>
  64 <p>
  65       If <span class="emphasis"><em>A</em></span><sup>2</sup>/n converges to zero as <span class="emphasis"><em>n</em></span> goes
  66       to infinity, then the hypothesis that the data is normally distributed is supported
  67       by the test.
  68     </p>
  69 <p>
  70       If <span class="emphasis"><em>A</em></span><sup>2</sup>/n converges to a finite positive value as <span class="emphasis"><em>n</em></span>
  71       goes to infinity, then the hypothesis is not supported by the test.
  72     </p>
  73 <p>
  74       An example usage is demonstrated below:
  75     </p>
  76 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">vector</span><span class="special">&gt;</span>
  77 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">random</span><span class="special">&gt;</span>
  78 <span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
  79 <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">statistics</span><span class="special">/</span><span class="identifier">anderson_darling</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
  80 <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">statistics</span><span class="special">::</span><span class="identifier">anderson_darling_normality_statistic</span><span class="special">;</span>
  81 <span class="identifier">std</span><span class="special">::</span><span class="identifier">random_device</span> <span class="identifier">rd</span><span class="special">;</span>
  82 <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>
  83 <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="number">8192</span><span class="special">);</span>
  84 <span class="keyword">for</span> <span class="special">(</span><span class="keyword">auto</span> <span class="special">&amp;</span> <span class="identifier">x</span> <span class="special">:</span> <span class="identifier">v</span><span class="special">)</span> <span class="special">{</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">dis</span><span class="special">(</span><span class="identifier">rd</span><span class="special">);</span> <span class="special">}</span>
  85 <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>
  86 <span class="keyword">double</span> <span class="identifier">presumed_mean</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span>
  87 <span class="keyword">double</span> <span class="identifier">presumed_standard_deviation</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span>
  88 <span class="keyword">double</span> <span class="identifier">Asq</span> <span class="special">=</span> <span class="identifier">anderson_darling_normality_statistic</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">presumed_mean</span><span class="special">,</span> <span class="identifier">presumed_standard_deviation</span><span class="special">);</span>
  89 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"A^2/n = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">Asq</span><span class="special">/</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  90 <span class="number">5.39e-05</span> <span class="comment">// should be small . . .</span>
  91 <span class="comment">// Now use an incorrect hypothesis:</span>
  92 <span class="identifier">presumed_mean</span> <span class="special">=</span> <span class="number">4</span><span class="special">;</span>
  93 <span class="identifier">Asq</span> <span class="special">=</span> <span class="identifier">anderson_darling_normality_statistic</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">presumed_mean</span><span class="special">,</span> <span class="identifier">presumed_standard_deviation</span><span class="special">);</span>
  94 <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special">&lt;&lt;</span> <span class="string">"A^2/n = "</span> <span class="special">&lt;&lt;</span> <span class="identifier">Asq</span><span class="special">/</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">size</span><span class="special">()</span> <span class="special">&lt;&lt;</span> <span class="string">"\n"</span><span class="special">;</span>
  95 <span class="number">7.41</span> <span class="comment">// should be somewhat large . . .</span>
  96 </pre>
  97 <p>
  98       The Anderson-Darling normality requires sorted data. If the data are not sorted
  99       an exception is thrown.
 100     </p>
 101 <p>
 102       If you simply wish to know whether or not data is normally distributed, and
 103       not whether it is normally distributed with a presumed mean and variance, then
 104       you can call the function without the final two arguments, and the mean and
 105       variance will be estimated from the data themselves:
 106     </p>
 107 <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">Asq</span> <span class="special">=</span> <span class="identifier">anderson_darling_normality_statistic</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
 108 </pre>
 109 <p>
 110       The following graph demonstrates the convergence of the test statistic. Each
 111       data point represents a vector of length <span class="emphasis"><em>n</em></span> which is filled
 112       with normally distributed data. The test statistic is computed over this vector,
 113       divided by <span class="emphasis"><em>n</em></span>, and passed to the natural logarithm. This
 114       exhibits the (admittedly slow) convergence of the integral to zero when the
 115       hypothesis is true.
 116     </p>
 117 <p>
 118       <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/anderson_darling_simulation.svg"></object></span>
 119     </p>
 120 <h4>
 121 <a name="math_toolkit.anderson_darling.h2"></a>
 122       <span class="phrase"><a name="math_toolkit.anderson_darling.performance"></a></span><a class="link" href="anderson_darling.html#math_toolkit.anderson_darling.performance">Performance</a>
 123     </h4>
 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">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8</span>                <span class="number">224</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">136.509</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 128 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16</span>               <span class="number">435</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">140.254</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 129 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">32</span>               <span class="number">898</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">135.995</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 130 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">64</span>              <span class="number">1773</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">137.675</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 131 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">128</span>             <span class="number">3455</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">141.338</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 132 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">256</span>             <span class="number">7001</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.488</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 133 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">512</span>            <span class="number">13996</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.551</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 134 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">1024</span>           <span class="number">28129</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">138.868</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 135 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">2048</span>           <span class="number">55723</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">140.206</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 136 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">4096</span>          <span class="number">112008</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.501</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 137 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8192</span>          <span class="number">224643</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.11</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 138 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16384</span>         <span class="number">450320</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">138.791</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 139 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">32768</span>         <span class="number">896409</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.45</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 140 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">65536</span>        <span class="number">1797800</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">139.058</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 141 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">131072</span>       <span class="number">3604995</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">138.698</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 142 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">262144</span>       <span class="number">7235625</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">138.207</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 143 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">524288</span>      <span class="number">14502815</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">137.904</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 144 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">1048576</span>     <span class="number">29058087</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">137.659</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 145 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">2097152</span>     <span class="number">58470439</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">136.824</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 146 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">4194304</span>    <span class="number">117476365</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">136.201</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 147 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">8388608</span>    <span class="number">239887895</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">133.397</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 148 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;/</span><span class="number">16777216</span>   <span class="number">488787211</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">130.94</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 149 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">float</span><span class="special">&gt;</span><span class="identifier">_BigO</span>           <span class="number">28.96</span> <span class="identifier">N</span>         <span class="number">28.96</span> <span class="identifier">N</span>
 150 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8</span>               <span class="number">470</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">129.733</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 151 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16</span>              <span class="number">911</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">133.989</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 152 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32</span>             <span class="number">1773</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">137.723</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 153 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">64</span>             <span class="number">3368</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">144.966</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 154 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">128</span>            <span class="number">6627</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">147.357</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 155 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">256</span>           <span class="number">12458</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">156.777</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 156 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">512</span>           <span class="number">23060</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">169.395</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 157 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1024</span>          <span class="number">44529</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">175.45</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 158 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2048</span>          <span class="number">88735</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">176.087</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 159 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4096</span>         <span class="number">175583</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">177.978</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 160 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8192</span>         <span class="number">348042</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">179.577</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 161 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16384</span>        <span class="number">701439</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">178.206</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 162 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">32768</span>       <span class="number">1394597</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">179.262</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 163 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">65536</span>       <span class="number">2777943</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">179.994</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 164 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">131072</span>      <span class="number">5571455</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">179.487</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 165 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">262144</span>     <span class="number">11161456</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">179.193</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 166 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">524288</span>     <span class="number">22048950</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.417</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 167 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">1048576</span>    <span class="number">44094409</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.429</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 168 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">2097152</span>    <span class="number">88300185</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.199</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 169 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">4194304</span>   <span class="number">176140378</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.678</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 170 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">8388608</span>   <span class="number">352102955</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.769</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 171 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;/</span><span class="number">16777216</span>  <span class="number">706160246</span> <span class="identifier">ns</span>    <span class="identifier">bytes_per_second</span><span class="special">=</span><span class="number">181.267</span><span class="identifier">M</span><span class="special">/</span><span class="identifier">s</span>
 172 <span class="identifier">AndersonDarlingNormalityTest</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span><span class="identifier">_BigO</span>          <span class="number">42.06</span> <span class="identifier">N</span>
 173 </pre>
 174 <h4>
 175 <a name="math_toolkit.anderson_darling.h3"></a>
 176       <span class="phrase"><a name="math_toolkit.anderson_darling.caveats"></a></span><a class="link" href="anderson_darling.html#math_toolkit.anderson_darling.caveats">Caveats</a>
 177     </h4>
 178 <p>
 179       Some authors, including <a href="https://www.itl.nist.gov/div898/handbook/eda/section3/eda35e.htm" target="_top">NIST</a>,
 180       give the following definition of the Anderson-Darling test statistic:
 181     </p>
 182 <p>
 183       <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/alternative_anderson_darling_definition.svg" width="461" height="54"></object></span>
 184     </p>
 185 <p>
 186       This is an approximation to the quadrature sum we use as our definition. Boost.Math
 187       <span class="emphasis"><em>does not compute this quantity</em></span>. (However, with a sufficiently
 188       large amount of data the two definitions seem to agree to two digits, so the
 189       importance of making a clear distinction between the two is unclear.) Our computation
 190       of the Anderson-Darling test statistic agrees with Mathematica.
 191     </p>
 192 </div>
 193 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 194 <td align="left"></td>
 195 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 196       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 197       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 198       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 199       Daryle Walker and Xiaogang Zhang<p>
 200         Distributed under the Boost Software License, Version 1.0. (See accompanying
 201         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>)
 202       </p>
 203 </div></td>
 204 </tr></table>
 205 <hr>
 206 <div class="spirit-nav">
 207 <a accesskey="p" href="signal_statistics.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../statistics.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="t_test.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
 208 </div>
 209 </body>
 210 </html>