1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>The Ljung-Box 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="runs_test.html" title="Runs tests"> 10<link rel="next" href="linear_regression.html" title="Linear Regression"> 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="runs_test.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="linear_regression.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.ljung_box"></a><a class="link" href="ljung_box.html" title="The Ljung-Box Test">The Ljung-Box Test</a> 28</h2></div></div></div> 29<h4> 30<a name="math_toolkit.ljung_box.h0"></a> 31 <span class="phrase"><a name="math_toolkit.ljung_box.synopsis"></a></span><a class="link" href="ljung_box.html#math_toolkit.ljung_box.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">ljung_box</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="identifier">math</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">RandomAccessIterator</span><span class="special">></span> 38<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">></span> <span class="identifier">ljung_box</span><span class="special">(</span><span class="identifier">RandomAccessIterator</span> <span class="identifier">begin</span><span class="special">,</span> <span class="identifier">RandomAccessIterator</span> <span class="identifier">end</span><span class="special">,</span> <span class="identifier">int64_t</span> <span class="identifier">lags</span> <span class="special">=</span> <span class="special">-</span><span class="number">1</span><span class="special">,</span> <span class="identifier">int64_t</span> <span class="identifier">fit_dof</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span> 39 40 41<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">RandomAccessContainer</span><span class="special">></span> 42<span class="keyword">auto</span> <span class="identifier">ljung_box</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> <span class="identifier">int64_t</span> <span class="identifier">lags</span> <span class="special">=</span> <span class="special">-</span><span class="number">1</span><span class="special">,</span> <span class="identifier">int64_t</span> <span class="identifier">fit_dof</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span> 43 44<span class="special">}</span> 45</pre> 46<h4> 47<a name="math_toolkit.ljung_box.h1"></a> 48 <span class="phrase"><a name="math_toolkit.ljung_box.background"></a></span><a class="link" href="ljung_box.html#math_toolkit.ljung_box.background">Background</a> 49 </h4> 50<p> 51 The Ljung-Box test is used to test if residuals from a fitted model have unwanted 52 autocorrelation. If autocorrelation exists in the residuals, then presumably 53 a model with more parameters can be fitted to the original data and explain 54 more of the structure it contains. 55 </p> 56<p> 57 The test statistic is 58 </p> 59<p> 60 <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/ljung_box_definition.svg" width="479" height="55"></object></span> 61 </p> 62<p> 63 where <span class="emphasis"><em>n</em></span> is the length of <span class="emphasis"><em>v</em></span> and ℓ 64 is the number of lags. 65 </p> 66<p> 67 The variance of the statistic slightly exceeds the variance of the chi squared 68 distribution, but nonetheless it still is a fairly good test with reasonable 69 computational cost. 70 </p> 71<p> 72 An example use is given below: 73 </p> 74<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">vector</span><span class="special">></span> 75<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">random</span><span class="special">></span> 76<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> 77<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">ljung_box</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 78<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">ljung_box</span><span class="special">;</span> 79<span class="identifier">std</span><span class="special">::</span><span class="identifier">random_device</span> <span class="identifier">rd</span><span class="special">;</span> 80<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> 81<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> 82<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> 83<span class="keyword">auto</span> <span class="special">[</span><span class="identifier">Q</span><span class="special">,</span> <span class="identifier">p</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">ljung_box</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span> 84<span class="comment">// Possible output: Q = 5.94734, p = 0.819668</span> 85</pre> 86<p> 87 Now if the result is clearly autocorrelated: 88 </p> 89<pre class="programlisting"><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"><</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="identifier">i</span><span class="special">)</span> <span class="special">{</span> <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">i</span><span class="special">;</span> <span class="special">}</span> 90<span class="keyword">auto</span> <span class="special">[</span><span class="identifier">Q</span><span class="special">,</span> <span class="identifier">p</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">ljung_box</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span> 91<span class="comment">// Possible output: Q = 81665.1, p = 0</span> 92</pre> 93<p> 94 By default, the number of lags is taken to be the logarithm of the number of 95 samples, so that the default complexity is [bigO](<span class="emphasis"><em>n</em></span> ln 96 <span class="emphasis"><em>n</em></span>). If you want to calculate a given number of lags, use 97 the second argument: 98 </p> 99<pre class="programlisting"><span class="identifier">int64_t</span> <span class="identifier">lags</span> <span class="special">=</span> <span class="number">10</span><span class="special">;</span> 100<span class="keyword">auto</span> <span class="special">[</span><span class="identifier">Q</span><span class="special">,</span> <span class="identifier">p</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">ljung_box</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span><span class="number">10</span><span class="special">);</span> 101</pre> 102<p> 103 Finally, it is sometimes relevant to specify how many degrees of freedom were 104 used in creating the model from which the residuals were computed. This does 105 not affect the test statistic <span class="emphasis"><em>Q</em></span>, but only the <span class="emphasis"><em>p</em></span>-value. 106 If you need to specify the number of degrees of freedom, use 107 </p> 108<pre class="programlisting"><span class="identifier">int64_t</span> <span class="identifier">fit_dof</span> <span class="special">=</span> <span class="number">2</span><span class="special">;</span> 109<span class="keyword">auto</span> <span class="special">[</span><span class="identifier">Q</span><span class="special">,</span> <span class="identifier">p</span><span class="special">]</span> <span class="special">=</span> <span class="identifier">ljung_box</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="special">-</span><span class="number">1</span><span class="special">,</span> <span class="identifier">fit_dof</span><span class="special">);</span> 110</pre> 111<p> 112 For example, if you fit your data with an ARIMA(<span class="emphasis"><em>p</em></span>, <span class="emphasis"><em>q</em></span>) 113 model, then <code class="computeroutput"><span class="identifier">fit_dof</span> <span class="special">=</span> 114 <span class="identifier">p</span> <span class="special">+</span> <span class="identifier">q</span></code>. 115 </p> 116</div> 117<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 118<td align="left"></td> 119<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 120 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 121 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 122 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 123 Daryle Walker and Xiaogang Zhang<p> 124 Distributed under the Boost Software License, Version 1.0. (See accompanying 125 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>) 126 </p> 127</div></td> 128</tr></table> 129<hr> 130<div class="spirit-nav"> 131<a accesskey="p" href="runs_test.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="linear_regression.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 132</div> 133</body> 134</html> 135
