1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 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 2.12.0"> 8<link rel="up" href="../statistics.html" title="Chapter 6. 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"><</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">></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"><</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">></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">&</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"><</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><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"><</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><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"><</span><span class="identifier">vector</span><span class="special">></span> 77<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">random</span><span class="special">></span> 78<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> 79<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">statistics</span><span class="special">/</span><span class="identifier">anderson_darling</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></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"><</span><span class="keyword">double</span><span class="special">></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"><</span><span class="keyword">double</span><span class="special">></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">&</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"><<</span> <span class="string">"A^2/n = "</span> <span class="special"><<</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"><<</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"><<</span> <span class="string">"A^2/n = "</span> <span class="special"><<</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"><<</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">>/</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"><</span><span class="keyword">float</span><span class="special">></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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">>/</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"><</span><span class="keyword">double</span><span class="special">></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 © 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 Råde, 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> 211
