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<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.ljung_box"></a><a class="link" href="ljung_box.html" title="The Ljung-Box Test">The Ljung-Box Test</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.ljung_box.h0"></a>
      <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>
    </h4>
<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">ljung_box</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>

<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>

<span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RandomAccessIterator</span><span class="special">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">&gt;</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>


<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>
<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">&amp;</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>

<span class="special">}</span>
</pre>
<h4>
<a name="math_toolkit.ljung_box.h1"></a>
      <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>
    </h4>
<p>
      The Ljung-Box test is used to test if residuals from a fitted model have unwanted
      autocorrelation. If autocorrelation exists in the residuals, then presumably
      a model with more parameters can be fitted to the original data and explain
      more of the structure it contains.
    </p>
<p>
      The test statistic is
    </p>
<p>
      <span class="inlinemediaobject"><object type="image/svg+xml" data="../../graphs/ljung_box_definition.svg" width="479" height="55"></object></span>
    </p>
<p>
      where <span class="emphasis"><em>n</em></span> is the length of <span class="emphasis"><em>v</em></span> and ℓ
      is the number of lags.
    </p>
<p>
      The variance of the statistic slightly exceeds the variance of the chi squared
      distribution, but nonetheless it still is a fairly good test with reasonable
      computational cost.
    </p>
<p>
      An example use is given below:
    </p>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">vector</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">random</span><span class="special">&gt;</span>
<span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">iostream</span><span class="special">&gt;</span>
<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">ljung_box</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
<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>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">random_device</span> <span class="identifier">rd</span><span class="special">;</span>
<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>
<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>
<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>
<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="comment">// Possible output: Q = 5.94734, p = 0.819668</span>
</pre>
<p>
      Now if the result is clearly autocorrelated:
    </p>
<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">&lt;</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>
<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="comment">// Possible output: Q = 81665.1, p = 0</span>
</pre>
<p>
      By default, the number of lags is taken to be the logarithm of the number of
      samples, so that the default complexity is [bigO](<span class="emphasis"><em>n</em></span> ln
      <span class="emphasis"><em>n</em></span>). If you want to calculate a given number of lags, use
      the second argument:
    </p>
<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>
<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>
</pre>
<p>
      Finally, it is sometimes relevant to specify how many degrees of freedom were
      used in creating the model from which the residuals were computed. This does
      not affect the test statistic <span class="emphasis"><em>Q</em></span>, but only the <span class="emphasis"><em>p</em></span>-value.
      If you need to specify the number of degrees of freedom, use
    </p>
<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>
<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>
</pre>
<p>
      For example, if you fit your data with an ARIMA(<span class="emphasis"><em>p</em></span>, <span class="emphasis"><em>q</em></span>)
      model, then <code class="computeroutput"><span class="identifier">fit_dof</span> <span class="special">=</span>
      <span class="identifier">p</span> <span class="special">+</span> <span class="identifier">q</span></code>.
    </p>
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