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  27 <a name="math_toolkit.bivariate_statistics"></a><a class="link" href="bivariate_statistics.html" title="Bivariate Statistics">Bivariate Statistics</a>
  28 </h2></div></div></div>
  29 <h4>
  30 <a name="math_toolkit.bivariate_statistics.h0"></a>
  31       <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>
  32     </h4>
  33 <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">bivariate_statistics</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</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="identifier">statistics</span> <span class="special">{</span>
  36
  37     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</span>
  38     <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">&amp;</span> <span class="identifier">u</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v</span><span class="special">);</span>
  39
  40     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</span>
  41     <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">&amp;</span> <span class="identifier">u</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v</span><span class="special">);</span>
  42
  43     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Container</span><span class="special">&gt;</span>
  44     <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">&amp;</span> <span class="identifier">u</span><span class="special">,</span> <span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">v</span><span class="special">);</span>
  45
  46 <span class="special">}}}</span>
  47 </pre>
  48 <h4>
  49 <a name="math_toolkit.bivariate_statistics.h1"></a>
  50       <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>
  51     </h4>
  52 <p>
  53       This file provides functions for computing bivariate statistics.
  54     </p>
  55 <h4>
  56 <a name="math_toolkit.bivariate_statistics.h2"></a>
  57       <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>
  58     </h4>
  59 <p>
  60       Computes the population covariance of two datasets:
  61     </p>
  62 <pre class="programlisting"><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">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>
  63 <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">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>
  64 <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>
  65 </pre>
  66 <p>
  67       The implementation follows <a href="https://doi.org/10.1109/CLUSTR.2009.5289161" target="_top">Bennet
  68       et al</a>. The data is not modified. Requires a random-access container.
  69       Works with real-valued inputs and does not work with complex-valued inputs.
  70     </p>
  71 <p>
  72       The algorithm used herein simultaneously generates the mean values of the input
  73       data <span class="emphasis"><em>u</em></span> and <span class="emphasis"><em>v</em></span>. For certain applications,
  74       it might be useful to get them in a single pass through the data. As such,
  75       we provide <code class="computeroutput"><span class="identifier">means_and_covariance</span></code>:
  76     </p>
  77 <pre class="programlisting"><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">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>
  78 <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">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>
  79 <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>
  80 </pre>
  81 <h4>
  82 <a name="math_toolkit.bivariate_statistics.h3"></a>
  83       <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
  84       Coefficient</a>
  85     </h4>
  86 <p>
  87       Computes the <a href="https://en.wikipedia.org/wiki/Pearson_correlation_coefficient" target="_top">Pearson
  88       correlation coefficient</a> of two datasets <span class="emphasis"><em>u</em></span> and
  89       <span class="emphasis"><em>v</em></span>:
  90     </p>
  91 <pre class="programlisting"><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">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>
  92 <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">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>
  93 <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>
  94 <span class="comment">// rho_uv = 1.</span>
  95 </pre>
  96 <p>
  97       The data must be random access and cannot be complex.
  98     </p>
  99 <p>
 100       If one or both of the datasets is constant, the correlation coefficient is
 101       an indeterminant form (0/0) and definitions must be introduced to assign it
 102       a value. We use the following: If both datasets are constant, then the correlation
 103       coefficient is 1. If one dataset is constant, and the other is not, then the
 104       correlation coefficient is zero.
 105     </p>
 106 <h4>
 107 <a name="math_toolkit.bivariate_statistics.h4"></a>
 108       <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>
 109     </h4>
 110 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
 111           Bennett, Janine, et al. <span class="emphasis"><em>Numerically stable, single-pass, parallel
 112           statistics algorithms.</em></span> Cluster Computing and Workshops, 2009.
 113           CLUSTER'09. IEEE International Conference on. IEEE, 2009.
 114         </li></ul></div>
 115 </div>
 116 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 117 <td align="left"></td>
 118 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 119       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 120       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 121       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 122       Daryle Walker and Xiaogang Zhang<p>
 123         Distributed under the Boost Software License, Version 1.0. (See accompanying
 124         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>)
 125       </p>
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