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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.bivariate_statistics"></a><a class="link" href="bivariate_statistics.html" title="Bivariate Statistics">Bivariate Statistics</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.bivariate_statistics.h0"></a>
<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>
</h4>
<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>
<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>
<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">ExecutionPolicy</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">></span>
<span class="keyword">auto</span> <span class="identifier">covariance</span><span class="special">(</span><span class="identifier">ExecutionPolicy</span><span class="special">&&</span> <span class="identifier">exec</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>
<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">></span>
<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>
<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">ExecutionPolicy</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">></span>
<span class="keyword">auto</span> <span class="identifier">means_and_covariance</span><span class="special">(</span><span class="identifier">ExecutionPolicy</span><span class="special">&&</span> <span class="identifier">exec</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>
<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">></span>
<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>
<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">ExecutionPolicy</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">></span>
<span class="keyword">auto</span> <span class="identifier">correlation_coefficient</span><span class="special">(</span><span class="identifier">ExecutionPolicy</span><span class="special">&&</span> <span class="identifier">exec</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>
<span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">Container</span><span class="special">></span>
<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>
<span class="special">}}}</span>
</pre>
<h4>
<a name="math_toolkit.bivariate_statistics.h1"></a>
<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>
</h4>
<p>
This file provides functions for computing bivariate statistics. The functions
are C++11 compatible, but require C++17 to use execution policies. If an execution
policy is not passed to the function the default is std::execution::seq.
</p>
<h4>
<a name="math_toolkit.bivariate_statistics.h2"></a>
<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>
</h4>
<p>
Computes the population covariance of two datasets:
</p>
<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>
<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>
<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>
</pre>
<p>
The implementation follows <a href="https://doi.org/10.1109/CLUSTR.2009.5289161" target="_top">Bennet
et al</a>. The parallel implementation follows <a href="https://dl.acm.org/doi/10.1145/3221269.3223036" target="_top">Schubert
et al</a>. The data is not modified. Works with real-valued inputs and
does not work with complex-valued inputs.
</p>
<p>
<span class="emphasis"><em>Nota bene:</em></span> If the input is an integer type the output
will be a double precision type.
</p>
<p>
The algorithm used herein simultaneously generates the mean values of the input
data <span class="emphasis"><em>u</em></span> and <span class="emphasis"><em>v</em></span>. For certain applications,
it might be useful to get them in a single pass through the data. As such,
we provide <code class="computeroutput"><span class="identifier">means_and_covariance</span></code>:
</p>
<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>
<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>
<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>
</pre>
<h4>
<a name="math_toolkit.bivariate_statistics.h3"></a>
<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
Coefficient</a>
</h4>
<p>
Computes the <a href="https://en.wikipedia.org/wiki/Pearson_correlation_coefficient" target="_top">Pearson
correlation coefficient</a> of two datasets <span class="emphasis"><em>u</em></span> and
<span class="emphasis"><em>v</em></span>:
</p>
<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>
<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>
<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>
<span class="comment">// rho_uv = 1.</span>
</pre>
<p>
Works with real-valued inputs and does not work with complex-valued inputs.
</p>
<p>
<span class="emphasis"><em>Nota bene:</em></span> If the input is an integer type the output
will be a double precision type.
</p>
<p>
If one or both of the datasets is constant, the correlation coefficient is
an indeterminant form (0/0). In this case the returned value is a <code class="computeroutput"><span class="identifier">quiet_NaN</span><span class="special">()</span></code>.
</p>
<h4>
<a name="math_toolkit.bivariate_statistics.h4"></a>
<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>
</h4>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
Bennett, Janine, et al. <span class="emphasis"><em>Numerically stable, single-pass, parallel
statistics algorithms.</em></span> Cluster Computing and Workshops, 2009.
CLUSTER'09. IEEE International Conference on. IEEE, 2009.
</li>
<li class="listitem">
Schubert, Erich; Gertz, Michael <span class="emphasis"><em>Numerically stable parallel computation
of (co-)variance'</em></span> Proceedings of the 30th International Conference
on Scientific and Statistical Database Management, 2018.
</li>
</ul></div>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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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