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  25 <div class="section">
  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.skew_normal_dist"></a><a class="link" href="skew_normal_dist.html" title="Skew Normal Distribution">Skew
  28         Normal Distribution</a>
  29 </h4></div></div></div>
  30 <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">distributions</span><span class="special">/</span><span class="identifier">skew_normal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
  31 <pre class="programlisting"><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>
  32
  33 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
  34           <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
  35 <span class="keyword">class</span> <span class="identifier">skew_normal_distribution</span><span class="special">;</span>
  36
  37 <span class="keyword">typedef</span> <span class="identifier">skew_normal_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">normal</span><span class="special">;</span>
  38
  39 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  40 <span class="keyword">class</span> <span class="identifier">skew_normal_distribution</span>
  41 <span class="special">{</span>
  42 <span class="keyword">public</span><span class="special">:</span>
  43    <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
  44    <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
  45    <span class="comment">// Constructor:</span>
  46    <span class="identifier">skew_normal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">shape</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span>
  47    <span class="comment">// Accessors:</span>
  48    <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// mean if normal.</span>
  49    <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// width, standard deviation if normal.</span>
  50    <span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// The distribution is right skewed if shape &gt; 0 and is left skewed if shape &lt; 0.</span>
  51                           <span class="comment">// The distribution is normal if shape is zero.</span>
  52 <span class="special">};</span>
  53
  54 <span class="special">}}</span> <span class="comment">// namespaces</span>
  55 </pre>
  56 <p>
  57           The skew normal distribution is a variant of the most well known Gaussian
  58           statistical distribution.
  59         </p>
  60 <p>
  61           The skew normal distribution with shape zero resembles the <a href="http://en.wikipedia.org/wiki/Normal_distribution" target="_top">Normal
  62           Distribution</a>, hence the latter can be regarded as a special case
  63           of the more generic skew normal distribution.
  64         </p>
  65 <p>
  66           If the standard (mean = 0, scale = 1) normal distribution probability density
  67           function is
  68         </p>
  69 <p>
  70           &#160; &#160;  <span class="inlinemediaobject"><img src="../../../../equations/normal01_pdf.svg"></span>
  71         </p>
  72 <p>
  73           and the cumulative distribution function
  74         </p>
  75 <p>
  76           &#160; &#160;  <span class="inlinemediaobject"><img src="../../../../equations/normal01_cdf.svg"></span>
  77         </p>
  78 <p>
  79           then the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">PDF</a>
  80           of the <a href="http://en.wikipedia.org/wiki/Skew_normal_distribution" target="_top">skew
  81           normal distribution</a> with shape parameter &#945;, defined by O'Hagan and
  82           Leonhard (1976) is
  83         </p>
  84 <p>
  85           &#160; &#160;  <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf0.svg"></span>
  86         </p>
  87 <p>
  88           Given <a href="http://en.wikipedia.org/wiki/Location_parameter" target="_top">location</a>
  89           &#958;, <a href="http://en.wikipedia.org/wiki/Scale_parameter" target="_top">scale</a>
  90           &#969;, and <a href="http://en.wikipedia.org/wiki/Shape_parameter" target="_top">shape</a>
  91           &#945;, it can be <a href="http://en.wikipedia.org/wiki/Skew_normal_distribution" target="_top">transformed</a>,
  92           to the form:
  93         </p>
  94 <p>
  95           &#160; &#160;  <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf.svg"></span>
  96         </p>
  97 <p>
  98           and <a href="http://en.wikipedia.org/wiki/Cumulative_distribution_function" target="_top">CDF</a>:
  99         </p>
 100 <p>
 101           &#160; &#160;  <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_cdf.svg"></span>
 102         </p>
 103 <p>
 104           where <span class="emphasis"><em>T(h,a)</em></span> is Owen's T function, and <span class="emphasis"><em>&#934;(x)</em></span>
 105           is the normal distribution.
 106         </p>
 107 <p>
 108           The variation the PDF and CDF with its parameters is illustrated in the
 109           following graphs:
 110         </p>
 111 <p>
 112           <span class="inlinemediaobject"><img src="../../../../graphs/skew_normal_pdf.svg" align="middle"></span>
 113   <span class="inlinemediaobject"><img src="../../../../graphs/skew_normal_cdf.svg" align="middle"></span>
 114         </p>
 115 <h5>
 116 <a name="math_toolkit.dist_ref.dists.skew_normal_dist.h0"></a>
 117           <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.member_functions"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.member_functions">Member
 118           Functions</a>
 119         </h5>
 120 <pre class="programlisting"><span class="identifier">skew_normal_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">shape</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span>
 121 </pre>
 122 <p>
 123           Constructs a skew_normal distribution with location &#958;, scale &#969; and shape &#945;.
 124         </p>
 125 <p>
 126           Requires scale &gt; 0, otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 127           is called.
 128         </p>
 129 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 130 </pre>
 131 <p>
 132           returns the location &#958; of this distribution,
 133         </p>
 134 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 135 </pre>
 136 <p>
 137           returns the scale &#969; of this distribution,
 138         </p>
 139 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 140 </pre>
 141 <p>
 142           returns the shape &#945; of this distribution.
 143         </p>
 144 <p>
 145           (Location and scale function match other similar distributions, allowing
 146           the functions <code class="computeroutput"><span class="identifier">find_location</span></code>
 147           and <code class="computeroutput"><span class="identifier">find_scale</span></code> to be used
 148           generically).
 149         </p>
 150 <div class="note"><table border="0" summary="Note">
 151 <tr>
 152 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 153 <th align="left">Note</th>
 154 </tr>
 155 <tr><td align="left" valign="top">
 156 <p>
 157             While the shape parameter may be chosen arbitrarily (finite), the resulting
 158             <span class="bold"><strong>skewness</strong></span> of the distribution is in fact
 159             limited to about (-1, 1); strictly, the interval is (-0.9952717, 0.9952717).
 160           </p>
 161 <p>
 162             A parameter &#948; is related to the shape &#945; by &#948; = &#945; / (1 + &#945;&#178;), and used in the expression
 163             for skewness <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_skewness.svg"></span>
 164
 165           </p>
 166 </td></tr>
 167 </table></div>
 168 <h5>
 169 <a name="math_toolkit.dist_ref.dists.skew_normal_dist.h1"></a>
 170           <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.references"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.references">References</a>
 171         </h5>
 172 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 173 <li class="listitem">
 174               <a href="http://azzalini.stat.unipd.it/SN/" target="_top">Skew-Normal Probability
 175               Distribution</a> for many links and bibliography.
 176             </li>
 177 <li class="listitem">
 178               <a href="http://azzalini.stat.unipd.it/SN/Intro/intro.html" target="_top">A very
 179               brief introduction to the skew-normal distribution</a> by Adelchi
 180               Azzalini (2005-11-2).
 181             </li>
 182 <li class="listitem">
 183               See a <a href="http://www.tri.org.au/azzalini.html" target="_top">skew-normal
 184               function animation</a>.
 185             </li>
 186 </ul></div>
 187 <h5>
 188 <a name="math_toolkit.dist_ref.dists.skew_normal_dist.h2"></a>
 189           <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.non_member_accessors"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.non_member_accessors">Non-member
 190           Accessors</a>
 191         </h5>
 192 <p>
 193           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 194           functions</a> that are generic to all distributions are supported:
 195           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 196           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 197           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
 198           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
 199           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
 200           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 201           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
 202           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
 203         </p>
 204 <p>
 205           The domain of the random variable is <span class="emphasis"><em>-[max_value], +[min_value]</em></span>.
 206           Infinite values are not supported.
 207         </p>
 208 <p>
 209           There are no <a href="http://en.wikipedia.org/wiki/Closed-form_expression" target="_top">closed-form
 210           expression</a> known for the mode and median, but these are computed
 211           for the
 212         </p>
 213 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 214 <li class="listitem">
 215               mode - by finding the maximum of the PDF.
 216             </li>
 217 <li class="listitem">
 218               median - by computing <code class="computeroutput"><span class="identifier">quantile</span><span class="special">(</span><span class="number">1</span><span class="special">/</span><span class="number">2</span><span class="special">)</span></code>.
 219             </li>
 220 </ul></div>
 221 <p>
 222           The maximum of the PDF is sought through searching the root of f'(x)=0.
 223         </p>
 224 <p>
 225           Both involve iterative methods that will have lower accuracy than other
 226           estimates.
 227         </p>
 228 <h5>
 229 <a name="math_toolkit.dist_ref.dists.skew_normal_dist.h3"></a>
 230           <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.testing"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.testing">Testing</a>
 231         </h5>
 232 <p>
 233           <a href="http://www.r-project.org/" target="_top">The R Project for Statistical Computing</a>
 234           using library(sn) described at <a href="http://azzalini.stat.unipd.it/SN/" target="_top">Skew-Normal
 235           Probability Distribution</a>, and at <a href="http://cran.r-project.org/web/packages/sn/sn.pd" target="_top">R
 236           skew-normal(sn) package</a>.
 237         </p>
 238 <p>
 239           Package sn provides functions related to the skew-normal (SN) and the skew-t
 240           (ST) probability distributions, both for the univariate and for the the
 241           multivariate case, including regression models.
 242         </p>
 243 <p>
 244           <a href="http://www.wolfram.com/products/mathematica/index.html" target="_top">Wolfram
 245           Mathematica</a> was also used to generate some more accurate spot test
 246           data.
 247         </p>
 248 <h5>
 249 <a name="math_toolkit.dist_ref.dists.skew_normal_dist.h4"></a>
 250           <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.accuracy"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.accuracy">Accuracy</a>
 251         </h5>
 252 <p>
 253           The skew_normal distribution with shape = zero is implemented as a special
 254           case, equivalent to the normal distribution in terms of the <a class="link" href="../../sf_erf/error_function.html" title="Error Functions">error
 255           function</a>, and therefore should have excellent accuracy.
 256         </p>
 257 <p>
 258           The PDF and mean, variance, skewness and kurtosis are also accurately evaluated
 259           using <a href="http://en.wikipedia.org/wiki/Analytical_expression" target="_top">analytical
 260           expressions</a>. The CDF requires <a href="http://en.wikipedia.org/wiki/Owen%27s_T_function" target="_top">Owen's
 261           T function</a> that is evaluated using a Boost C++ <a class="link" href="../../owens_t.html" title="Owen's T function">Owens
 262           T</a> implementation of the algorithms of M. Patefield and D. Tandy,
 263           Journal of Statistical Software, 5(5), 1-25 (2000); the complicated accuracy
 264           of this function is discussed in detail at <a class="link" href="../../owens_t.html" title="Owen's T function">Owens
 265           T</a>.
 266         </p>
 267 <p>
 268           The median and mode are calculated by iterative root finding, and both
 269           will be less accurate.
 270         </p>
 271 <h5>
 272 <a name="math_toolkit.dist_ref.dists.skew_normal_dist.h5"></a>
 273           <span class="phrase"><a name="math_toolkit.dist_ref.dists.skew_normal_dist.implementation"></a></span><a class="link" href="skew_normal_dist.html#math_toolkit.dist_ref.dists.skew_normal_dist.implementation">Implementation</a>
 274         </h5>
 275 <p>
 276           In the following table, &#958; is the location of the distribution, and &#969; is its
 277           scale, and &#945; is its shape.
 278         </p>
 279 <div class="informaltable"><table class="table">
 280 <colgroup>
 281 <col>
 282 <col>
 283 </colgroup>
 284 <thead><tr>
 285 <th>
 286                   <p>
 287                     Function
 288                   </p>
 289                 </th>
 290 <th>
 291                   <p>
 292                     Implementation Notes
 293                   </p>
 294                 </th>
 295 </tr></thead>
 296 <tbody>
 297 <tr>
 298 <td>
 299                   <p>
 300                     pdf
 301                   </p>
 302                 </td>
 303 <td>
 304                   <p>
 305                     Using: <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf.svg"></span>
 306                   </p>
 307                 </td>
 308 </tr>
 309 <tr>
 310 <td>
 311                   <p>
 312                     cdf
 313                   </p>
 314                 </td>
 315 <td>
 316                   <p>
 317                     Using: <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_cdf.svg"></span><br> where <span class="emphasis"><em>T(h,a)</em></span>
 318                     is Owen's T function, and <span class="emphasis"><em>&#934;(x)</em></span> is the normal
 319                     distribution.
 320                   </p>
 321                 </td>
 322 </tr>
 323 <tr>
 324 <td>
 325                   <p>
 326                     cdf complement
 327                   </p>
 328                 </td>
 329 <td>
 330                   <p>
 331                     Using: complement of normal distribution + 2 * Owens_t
 332                   </p>
 333                 </td>
 334 </tr>
 335 <tr>
 336 <td>
 337                   <p>
 338                     quantile
 339                   </p>
 340                 </td>
 341 <td>
 342                   <p>
 343                     Maximum of the pdf is sought through searching the root of f'(x)=0
 344                   </p>
 345                 </td>
 346 </tr>
 347 <tr>
 348 <td>
 349                   <p>
 350                     quantile from the complement
 351                   </p>
 352                 </td>
 353 <td>
 354                   <p>
 355                     -quantile(SN(-location &#958;, scale &#969;, -shape&#945;), p)
 356                   </p>
 357                 </td>
 358 </tr>
 359 <tr>
 360 <td>
 361                   <p>
 362                     location
 363                   </p>
 364                 </td>
 365 <td>
 366                   <p>
 367                     location &#958;
 368                   </p>
 369                 </td>
 370 </tr>
 371 <tr>
 372 <td>
 373                   <p>
 374                     scale
 375                   </p>
 376                 </td>
 377 <td>
 378                   <p>
 379                     scale &#969;
 380                   </p>
 381                 </td>
 382 </tr>
 383 <tr>
 384 <td>
 385                   <p>
 386                     shape
 387                   </p>
 388                 </td>
 389 <td>
 390                   <p>
 391                     shape &#945;
 392                   </p>
 393                 </td>
 394 </tr>
 395 <tr>
 396 <td>
 397                   <p>
 398                     median
 399                   </p>
 400                 </td>
 401 <td>
 402                   <p>
 403                     quantile(1/2)
 404                   </p>
 405                 </td>
 406 </tr>
 407 <tr>
 408 <td>
 409                   <p>
 410                     mean
 411                   </p>
 412                 </td>
 413 <td>
 414                   <p>
 415                     <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_mean.svg"></span>
 416                   </p>
 417                 </td>
 418 </tr>
 419 <tr>
 420 <td>
 421                   <p>
 422                     mode
 423                   </p>
 424                 </td>
 425 <td>
 426                   <p>
 427                     Maximum of the pdf is sought through searching the root of f'(x)=0
 428                   </p>
 429                 </td>
 430 </tr>
 431 <tr>
 432 <td>
 433                   <p>
 434                     variance
 435                   </p>
 436                 </td>
 437 <td>
 438                   <p>
 439                     <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_variance.svg"></span>
 440                   </p>
 441                 </td>
 442 </tr>
 443 <tr>
 444 <td>
 445                   <p>
 446                     skewness
 447                   </p>
 448                 </td>
 449 <td>
 450                   <p>
 451                     <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_skewness.svg"></span>
 452                   </p>
 453                 </td>
 454 </tr>
 455 <tr>
 456 <td>
 457                   <p>
 458                     kurtosis
 459                   </p>
 460                 </td>
 461 <td>
 462                   <p>
 463                     kurtosis excess-3
 464                   </p>
 465                 </td>
 466 </tr>
 467 <tr>
 468 <td>
 469                   <p>
 470                     kurtosis excess
 471                   </p>
 472                 </td>
 473 <td>
 474                   <p>
 475                     <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_kurt_ex.svg"></span>
 476                   </p>
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 484 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 485       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 486       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 487       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 488         Distributed under the Boost Software License, Version 1.0. (See accompanying
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