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  26 <div class="titlepage"><div><div><h2 class="title" style="clear: both">
  27 <a name="math_toolkit.univariate_statistics"></a><a class="link" href="univariate_statistics.html" title="Univariate Statistics">Univariate Statistics</a>
  28 </h2></div></div></div>
  29 <h4>
  30 <a name="math_toolkit.univariate_statistics.h0"></a>
  31       <span class="phrase"><a name="math_toolkit.univariate_statistics.synopsis"></a></span><a class="link" href="univariate_statistics.html#math_toolkit.univariate_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">univariate_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">mean</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">c</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">ForwardIterator</span><span class="special">&gt;</span>
  41     <span class="keyword">auto</span> <span class="identifier">mean</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</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">variance</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">c</span><span class="special">);</span>
  45
  46     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</span>
  47     <span class="keyword">auto</span> <span class="identifier">variance</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span>
  48
  49     <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>
  50     <span class="keyword">auto</span> <span class="identifier">sample_variance</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">c</span><span class="special">);</span>
  51
  52     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</span>
  53     <span class="keyword">auto</span> <span class="identifier">sample_variance</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span>
  54
  55     <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>
  56     <span class="keyword">auto</span> <span class="identifier">mean_and_sample_variance</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">c</span><span class="special">);</span>
  57
  58     <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>
  59     <span class="keyword">auto</span> <span class="identifier">skewness</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">c</span><span class="special">);</span>
  60
  61     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</span>
  62     <span class="keyword">auto</span> <span class="identifier">skewness</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span>
  63
  64     <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>
  65     <span class="keyword">auto</span> <span class="identifier">kurtosis</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">c</span><span class="special">);</span>
  66
  67     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</span>
  68     <span class="keyword">auto</span> <span class="identifier">kurtosis</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span>
  69
  70     <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>
  71     <span class="keyword">auto</span> <span class="identifier">excess_kurtosis</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">c</span><span class="special">);</span>
  72
  73     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</span>
  74     <span class="keyword">auto</span> <span class="identifier">excess_kurtosis</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span>
  75
  76     <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>
  77     <span class="keyword">auto</span> <span class="identifier">first_four_moments</span><span class="special">(</span><span class="identifier">Container</span> <span class="keyword">const</span> <span class="special">&amp;</span> <span class="identifier">c</span><span class="special">);</span>
  78
  79     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</span>
  80     <span class="keyword">auto</span> <span class="identifier">first_four_moments</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span>
  81
  82     <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>
  83     <span class="keyword">auto</span> <span class="identifier">median</span><span class="special">(</span><span class="identifier">Container</span> <span class="special">&amp;</span> <span class="identifier">c</span><span class="special">);</span>
  84
  85     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</span>
  86     <span class="keyword">auto</span> <span class="identifier">median</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span>
  87
  88     <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>
  89     <span class="keyword">auto</span> <span class="identifier">median_absolute_deviation</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">iterator_traits</span><span class="special">&lt;</span><span class="identifier">RandomAccessIterator</span><span class="special">&gt;::</span><span class="identifier">value_type</span> <span class="identifier">center</span><span class="special">=</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">());</span>
  90
  91     <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>
  92     <span class="keyword">auto</span> <span class="identifier">median_absolute_deviation</span><span class="special">(</span><span class="identifier">RandomAccessContainer</span> <span class="identifier">v</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">RandomAccessContainer</span><span class="special">::</span><span class="identifier">value_type</span> <span class="identifier">center</span><span class="special">=</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">Real</span><span class="special">&gt;::</span><span class="identifier">quiet_NaN</span><span class="special">());</span>
  93
  94     <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>
  95     <span class="keyword">auto</span> <span class="identifier">gini_coefficient</span><span class="special">(</span><span class="identifier">Container</span> <span class="special">&amp;</span> <span class="identifier">c</span><span class="special">);</span>
  96
  97     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</span>
  98     <span class="keyword">auto</span> <span class="identifier">gini_coefficient</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span>
  99
 100     <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>
 101     <span class="keyword">auto</span> <span class="identifier">sample_gini_coefficient</span><span class="special">(</span><span class="identifier">Container</span> <span class="special">&amp;</span> <span class="identifier">c</span><span class="special">);</span>
 102
 103     <span class="keyword">template</span><span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">ForwardIterator</span><span class="special">&gt;</span>
 104     <span class="keyword">auto</span> <span class="identifier">sample_gini_coefficient</span><span class="special">(</span><span class="identifier">ForwardIterator</span> <span class="identifier">first</span><span class="special">,</span> <span class="identifier">ForwardIterator</span> <span class="identifier">last</span><span class="special">);</span>
 105
 106 <span class="special">}}}</span>
 107 </pre>
 108 <h4>
 109 <a name="math_toolkit.univariate_statistics.h1"></a>
 110       <span class="phrase"><a name="math_toolkit.univariate_statistics.description"></a></span><a class="link" href="univariate_statistics.html#math_toolkit.univariate_statistics.description">Description</a>
 111     </h4>
 112 <p>
 113       The file <code class="computeroutput"><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">univariate_statistics</span><span class="special">.</span><span class="identifier">hpp</span></code> is a
 114       set of facilities for computing scalar values from vectors.
 115     </p>
 116 <p>
 117       Many of these functionals have trivial naive implementations, but experienced
 118       programmers will recognize that even trivial algorithms are easy to screw up,
 119       and that numerical instabilities often lurk in corner cases. We have attempted
 120       to do our "due diligence" to root out these problems-scouring the
 121       literature for numerically stable algorithms for even the simplest of functionals.
 122     </p>
 123 <p>
 124       <span class="emphasis"><em>Nota bene</em></span>: Some similar functionality is provided in
 125       <a href="https://www.boost.org/doc/libs/1_68_0/doc/html/accumulators/user_s_guide.html" target="_top">Boost
 126       Accumulators Framework</a>. These accumulators should be used in real-time
 127       applications; <code class="computeroutput"><span class="identifier">univariate_statistics</span><span class="special">.</span><span class="identifier">hpp</span></code> should
 128       be used when CPU vectorization is needed. As a reminder, remember that to actually
 129       <span class="emphasis"><em>get</em></span> vectorization, compile with <code class="computeroutput"><span class="special">-</span><span class="identifier">march</span><span class="special">=</span><span class="identifier">native</span>
 130       <span class="special">-</span><span class="identifier">O3</span></code>
 131       flags.
 132     </p>
 133 <p>
 134       We now describe each functional in detail. Our examples use <code class="computeroutput"><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></code>
 135       to hold the data, but this not required. In general, you can store your data
 136       in an Eigen array, and Armadillo vector, <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">array</span></code>,
 137       and for many of the routines, a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">forward_list</span></code>.
 138       These routines are usable in float, double, long double, and Boost.Multiprecision
 139       precision, as well as their complex extensions whenever the computation is
 140       well-defined. For certain operations (total variation, for example) integer
 141       inputs are supported.
 142     </p>
 143 <h4>
 144 <a name="math_toolkit.univariate_statistics.h2"></a>
 145       <span class="phrase"><a name="math_toolkit.univariate_statistics.mean"></a></span><a class="link" href="univariate_statistics.html#math_toolkit.univariate_statistics.mean">Mean</a>
 146     </h4>
 147 <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">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>
 148 <span class="keyword">double</span> <span class="identifier">mu</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">mean</span><span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">cbegin</span><span class="special">(),</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">cend</span><span class="special">());</span>
 149 <span class="comment">// Alternative syntax if you want to use entire container:</span>
 150 <span class="identifier">mu</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">mean</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
 151 </pre>
 152 <p>
 153       The implementation follows <a href="https://doi.org/10.1137/1.9780898718027" target="_top">Higham
 154       1.6a</a>. The data is not modified and must be forward iterable. Works
 155       with real and integer data. If the input is an integer type, the output is
 156       a double precision float.
 157     </p>
 158 <h4>
 159 <a name="math_toolkit.univariate_statistics.h3"></a>
 160       <span class="phrase"><a name="math_toolkit.univariate_statistics.variance"></a></span><a class="link" href="univariate_statistics.html#math_toolkit.univariate_statistics.variance">Variance</a>
 161     </h4>
 162 <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">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>
 163 <span class="identifier">Real</span> <span class="identifier">sigma_sq</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">variance</span><span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">cbegin</span><span class="special">(),</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">cend</span><span class="special">());</span>
 164 </pre>
 165 <p>
 166       If you don't need to calculate on a subset of the input, then the range call
 167       is more terse:
 168     </p>
 169 <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">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>
 170 <span class="identifier">Real</span> <span class="identifier">sigma_sq</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">variance</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
 171 </pre>
 172 <p>
 173       The implementation follows <a href="https://doi.org/10.1137/1.9780898718027" target="_top">Higham
 174       1.6b</a>. The input data must be forward iterable and the range <code class="computeroutput"><span class="special">[</span><span class="identifier">first</span><span class="special">,</span>
 175       <span class="identifier">last</span><span class="special">)</span></code>
 176       must contain at least two elements. It is <span class="emphasis"><em>not</em></span> in general
 177       sensible to pass complex numbers to this routine. If integers are passed as
 178       input, then the output is a double precision float.
 179     </p>
 180 <p>
 181       <code class="computeroutput"><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">variance</span></code>
 182       returns the population variance. If you want a sample variance, use
 183     </p>
 184 <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">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>
 185 <span class="identifier">Real</span> <span class="identifier">sn_sq</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">sample_variance</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
 186 </pre>
 187 <h4>
 188 <a name="math_toolkit.univariate_statistics.h4"></a>
 189       <span class="phrase"><a name="math_toolkit.univariate_statistics.skewness"></a></span><a class="link" href="univariate_statistics.html#math_toolkit.univariate_statistics.skewness">Skewness</a>
 190     </h4>
 191 <p>
 192       Computes the skewness of a dataset:
 193     </p>
 194 <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">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>
 195 <span class="keyword">double</span> <span class="identifier">skewness</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">skewness</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
 196 <span class="comment">// skewness = 0.</span>
 197 </pre>
 198 <p>
 199       The input vector is not modified, works with integral and real data. If the
 200       input data is integral, the output is a double precision float.
 201     </p>
 202 <p>
 203       For a dataset consisting of a single constant value, we take the skewness to
 204       be zero by definition.
 205     </p>
 206 <p>
 207       The implementation follows <a href="https://prod.sandia.gov/techlib-noauth/access-control.cgi/2008/086212.pdf" target="_top">Pebay</a>.
 208     </p>
 209 <h4>
 210 <a name="math_toolkit.univariate_statistics.h5"></a>
 211       <span class="phrase"><a name="math_toolkit.univariate_statistics.kurtosis"></a></span><a class="link" href="univariate_statistics.html#math_toolkit.univariate_statistics.kurtosis">Kurtosis</a>
 212     </h4>
 213 <p>
 214       Computes the kurtosis of a dataset:
 215     </p>
 216 <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">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>
 217 <span class="keyword">double</span> <span class="identifier">kurtosis</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">kurtosis</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
 218 <span class="comment">// kurtosis = 17/10</span>
 219 </pre>
 220 <p>
 221       The implementation follows <a href="https://prod.sandia.gov/techlib-noauth/access-control.cgi/2008/086212.pdf" target="_top">Pebay</a>.
 222       The input data must be forward iterable and must consist of real or integral
 223       values. If the input data is integral, the output is a double precision float.
 224       Note that this is <span class="emphasis"><em>not</em></span> the excess kurtosis. If you require
 225       the excess kurtosis, use <code class="computeroutput"><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">excess_kurtosis</span></code>. This function simply subtracts
 226       3 from the kurtosis, but it makes eminently clear our definition of kurtosis.
 227     </p>
 228 <h4>
 229 <a name="math_toolkit.univariate_statistics.h6"></a>
 230       <span class="phrase"><a name="math_toolkit.univariate_statistics.first_four_moments"></a></span><a class="link" href="univariate_statistics.html#math_toolkit.univariate_statistics.first_four_moments">First
 231       four moments</a>
 232     </h4>
 233 <p>
 234       Simultaneously computes the first four <a href="https://en.wikipedia.org/wiki/Central_moment" target="_top">central
 235       moments</a> in a single pass through the data:
 236     </p>
 237 <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">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>
 238 <span class="keyword">auto</span> <span class="special">[</span><span class="identifier">M1</span><span class="special">,</span> <span class="identifier">M2</span><span class="special">,</span> <span class="identifier">M3</span><span class="special">,</span> <span class="identifier">M4</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">first_four_moments</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
 239 </pre>
 240 <h4>
 241 <a name="math_toolkit.univariate_statistics.h7"></a>
 242       <span class="phrase"><a name="math_toolkit.univariate_statistics.median"></a></span><a class="link" href="univariate_statistics.html#math_toolkit.univariate_statistics.median">Median</a>
 243     </h4>
 244 <p>
 245       Computes the median of a dataset:
 246     </p>
 247 <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">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>
 248 <span class="keyword">double</span> <span class="identifier">m</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">median</span><span class="special">(</span><span class="identifier">v</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">v</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span>
 249 </pre>
 250 <p>
 251       <span class="emphasis"><em>Nota bene: The input vector is modified.</em></span> The calculation
 252       of the median is a thin wrapper around the C++11 <a href="https://en.cppreference.com/w/cpp/algorithm/nth_element" target="_top"><code class="computeroutput"><span class="identifier">nth_element</span></code></a>. Therefore, all requirements
 253       of <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">nth_element</span></code> are inherited by the median calculation.
 254       In particular, the container must allow random access.
 255     </p>
 256 <h4>
 257 <a name="math_toolkit.univariate_statistics.h8"></a>
 258       <span class="phrase"><a name="math_toolkit.univariate_statistics.median_absolute_deviation"></a></span><a class="link" href="univariate_statistics.html#math_toolkit.univariate_statistics.median_absolute_deviation">Median
 259       Absolute Deviation</a>
 260     </h4>
 261 <p>
 262       Computes the <a href="https://en.wikipedia.org/wiki/Median_absolute_deviation" target="_top">median
 263       absolute deviation</a> of a dataset:
 264     </p>
 265 <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">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>
 266 <span class="keyword">double</span> <span class="identifier">mad</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">median_absolute_deviation</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
 267 </pre>
 268 <p>
 269       By default, the deviation from the median is used. If you have some prior that
 270       the median is zero, or wish to compute the median absolute deviation from the
 271       mean, use the following:
 272     </p>
 273 <pre class="programlisting"><span class="comment">// prior is that center is zero:</span>
 274 <span class="keyword">double</span> <span class="identifier">center</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span>
 275 <span class="keyword">double</span> <span class="identifier">mad</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">median_absolute_deviation</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">center</span><span class="special">);</span>
 276
 277 <span class="comment">// compute median absolute deviation from the mean:</span>
 278 <span class="keyword">double</span> <span class="identifier">mu</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">mean</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
 279 <span class="keyword">double</span> <span class="identifier">mad</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">median_absolute_deviation</span><span class="special">(</span><span class="identifier">v</span><span class="special">,</span> <span class="identifier">mu</span><span class="special">);</span>
 280 </pre>
 281 <p>
 282       <span class="emphasis"><em>Nota bene:</em></span> The input vector is modified. Again the vector
 283       is passed into a call to <a href="https://en.cppreference.com/w/cpp/algorithm/nth_element" target="_top"><code class="computeroutput"><span class="identifier">nth_element</span></code></a>.
 284     </p>
 285 <h4>
 286 <a name="math_toolkit.univariate_statistics.h9"></a>
 287       <span class="phrase"><a name="math_toolkit.univariate_statistics.gini_coefficient"></a></span><a class="link" href="univariate_statistics.html#math_toolkit.univariate_statistics.gini_coefficient">Gini
 288       Coefficient</a>
 289     </h4>
 290 <p>
 291       Compute the Gini coefficient of a dataset:
 292     </p>
 293 <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">v</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">,</span><span class="number">0</span><span class="special">};</span>
 294 <span class="keyword">double</span> <span class="identifier">gini</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">gini_coefficient</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
 295 <span class="comment">// gini = 3/4</span>
 296 <span class="keyword">double</span> <span class="identifier">s_gini</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">sample_gini_coefficient</span><span class="special">(</span><span class="identifier">v</span><span class="special">);</span>
 297 <span class="comment">// s_gini = 1.</span>
 298 <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">w</span><span class="special">{</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">};</span>
 299 <span class="identifier">gini</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">gini_coefficient</span><span class="special">(</span><span class="identifier">w</span><span class="special">.</span><span class="identifier">begin</span><span class="special">(),</span> <span class="identifier">w</span><span class="special">.</span><span class="identifier">end</span><span class="special">());</span>
 300 <span class="comment">// gini = 0, as all elements are now equal.</span>
 301 </pre>
 302 <p>
 303       <span class="emphasis"><em>Nota bene</em></span>: The input data is altered: in particular, it
 304       is sorted. Makes a call to <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">sort</span></code>, and
 305       as such requires random access iterators.
 306     </p>
 307 <p>
 308       The sample Gini coefficient lies in the range [0,1], whereas the population
 309       Gini coefficient is in the range [0, 1 - 1/ <span class="emphasis"><em>n</em></span>].
 310     </p>
 311 <p>
 312       <span class="emphasis"><em>Nota bene:</em></span> There is essentially no reason to pass negative
 313       values to the Gini coefficient function. However, a use case (measuring wealth
 314       inequality when some people have negative wealth) exists, so we do not throw
 315       an exception when negative values are encountered. You should have <span class="emphasis"><em>very</em></span>
 316       good cause to pass negative values to the Gini coefficient calculator. Another
 317       use case is found in signal processing, but the sorting is by magnitude and
 318       hence has a different implementation. See <code class="computeroutput"><span class="identifier">absolute_gini_coefficient</span></code>
 319       for details.
 320     </p>
 321 <h4>
 322 <a name="math_toolkit.univariate_statistics.h10"></a>
 323       <span class="phrase"><a name="math_toolkit.univariate_statistics.references"></a></span><a class="link" href="univariate_statistics.html#math_toolkit.univariate_statistics.references">References</a>
 324     </h4>
 325 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 326 <li class="listitem">
 327           Higham, Nicholas J. <span class="emphasis"><em>Accuracy and stability of numerical algorithms.</em></span>
 328           Vol. 80. Siam, 2002.
 329         </li>
 330 <li class="listitem">
 331           Philippe P. P&#233;bay: <span class="quote">&#8220;<span class="quote">Formulas for Robust, One-Pass Parallel Computation
 332           of Covariances and Arbitrary-Order Statistical Moments.</span>&#8221;</span> Technical
 333           Report SAND2008-6212, Sandia National Laboratories, September 2008.
 334         </li>
 335 </ul></div>
 336 </div>
 337 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 338 <td align="left"></td>
 339 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 340       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 341       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 342       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 343       Daryle Walker and Xiaogang Zhang<p>
 344         Distributed under the Boost Software License, Version 1.0. (See accompanying
 345         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>)
 346       </p>
 347 </div></td>
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