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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.bivariate_statistics"></a><a class="link" href="bivariate_statistics.html" title="Bivariate Statistics">Bivariate Statistics</a>
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</h2></div></div></div>
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<h4>
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<a name="math_toolkit.bivariate_statistics.h0"></a>
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<span class="phrase"><a name="math_toolkit.bivariate_statistics.synopsis"></a></span><a class="link" href="bivariate_statistics.html#math_toolkit.bivariate_statistics.synopsis">Synopsis</a>
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</h4>
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<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">bivariate_statistics</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<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="identifier">statistics</span> <span class="special">{</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">covariance</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">u</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">v</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">means_and_covariance</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">u</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">v</span><span class="special">);</span>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">></span>
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<span class="keyword">auto</span> <span class="identifier">correlation_coefficient</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">u</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&</span> <span class="identifier">v</span><span class="special">);</span>
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<span class="special">}}}</span>
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</pre>
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<h4>
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<a name="math_toolkit.bivariate_statistics.h1"></a>
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<span class="phrase"><a name="math_toolkit.bivariate_statistics.description"></a></span><a class="link" href="bivariate_statistics.html#math_toolkit.bivariate_statistics.description">Description</a>
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</h4>
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<p>
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This file provides functions for computing bivariate statistics.
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</p>
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<h4>
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<a name="math_toolkit.bivariate_statistics.h2"></a>
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<span class="phrase"><a name="math_toolkit.bivariate_statistics.covariance"></a></span><a class="link" href="bivariate_statistics.html#math_toolkit.bivariate_statistics.covariance">Covariance</a>
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</h4>
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<p>
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Computes the population covariance of two datasets:
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</p>
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<pre class="programlisting"><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">u</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">,</span><span class="number">4</span><span class="special">,</span><span class="number">5</span><span class="special">};</span>
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<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">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">,</span><span class="number">4</span><span class="special">,</span><span class="number">5</span><span class="special">};</span>
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<span class="keyword">double</span> <span class="identifier">cov_uv</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">covariance</span><span class="special">(</span><span class="identifier">u</span><span class="special">,</span> <span class="identifier">v</span><span class="special">);</span>
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</pre>
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<p>
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The implementation follows <a href="https://doi.org/10.1109/CLUSTR.2009.5289161" target="_top">Bennet
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et al</a>. The data is not modified. Requires a random-access container.
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Works with real-valued inputs and does not work with complex-valued inputs.
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</p>
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<p>
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The algorithm used herein simultaneously generates the mean values of the input
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data <span class="emphasis"><em>u</em></span> and <span class="emphasis"><em>v</em></span>. For certain applications,
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it might be useful to get them in a single pass through the data. As such,
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we provide <code class="computeroutput"><span class="identifier">means_and_covariance</span></code>:
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</p>
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<pre class="programlisting"><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">u</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">,</span><span class="number">4</span><span class="special">,</span><span class="number">5</span><span class="special">};</span>
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<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">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">,</span><span class="number">4</span><span class="special">,</span><span class="number">5</span><span class="special">};</span>
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<span class="keyword">auto</span> <span class="special">[</span><span class="identifier">mu_u</span><span class="special">,</span> <span class="identifier">mu_v</span><span class="special">,</span> <span class="identifier">cov_uv</span><span class="special">]</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">means_and_covariance</span><span class="special">(</span><span class="identifier">u</span><span class="special">,</span> <span class="identifier">v</span><span class="special">);</span>
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</pre>
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<h4>
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<a name="math_toolkit.bivariate_statistics.h3"></a>
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<span class="phrase"><a name="math_toolkit.bivariate_statistics.correlation_coefficient"></a></span><a class="link" href="bivariate_statistics.html#math_toolkit.bivariate_statistics.correlation_coefficient">Correlation
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Coefficient</a>
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</h4>
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<p>
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Computes the <a href="https://en.wikipedia.org/wiki/Pearson_correlation_coefficient" target="_top">Pearson
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correlation coefficient</a> of two datasets <span class="emphasis"><em>u</em></span> and
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<span class="emphasis"><em>v</em></span>:
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</p>
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<pre class="programlisting"><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">u</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">,</span><span class="number">4</span><span class="special">,</span><span class="number">5</span><span class="special">};</span>
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<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">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">,</span><span class="number">4</span><span class="special">,</span><span class="number">5</span><span class="special">};</span>
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<span class="keyword">double</span> <span class="identifier">rho_uv</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">correlation_coefficient</span><span class="special">(</span><span class="identifier">u</span><span class="special">,</span> <span class="identifier">v</span><span class="special">);</span>
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<span class="comment">// rho_uv = 1.</span>
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</pre>
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<p>
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The data must be random access and cannot be complex.
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</p>
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<p>
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If one or both of the datasets is constant, the correlation coefficient is
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an indeterminant form (0/0) and definitions must be introduced to assign it
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a value. We use the following: If both datasets are constant, then the correlation
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coefficient is 1. If one dataset is constant, and the other is not, then the
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correlation coefficient is zero.
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</p>
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<h4>
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<a name="math_toolkit.bivariate_statistics.h4"></a>
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<span class="phrase"><a name="math_toolkit.bivariate_statistics.references"></a></span><a class="link" href="bivariate_statistics.html#math_toolkit.bivariate_statistics.references">References</a>
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</h4>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
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Bennett, Janine, et al. <span class="emphasis"><em>Numerically stable, single-pass, parallel
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statistics algorithms.</em></span> Cluster Computing and Workshops, 2009.
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CLUSTER'09. IEEE International Conference on. IEEE, 2009.
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</li></ul></div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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>)
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