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  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;20.&#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;20.&#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 <div class="blockquote"><blockquote class="blockquote"><p>
  70             <span class="inlinemediaobject"><img src="../../../../equations/normal01_pdf.svg"></span>
  71
  72           </p></blockquote></div>
  73 <p>
  74           and the cumulative distribution function
  75         </p>
  76 <div class="blockquote"><blockquote class="blockquote"><p>
  77             <span class="inlinemediaobject"><img src="../../../../equations/normal01_cdf.svg"></span>
  78
  79           </p></blockquote></div>
  80 <p>
  81           then the <a href="http://en.wikipedia.org/wiki/Probability_density_function" target="_top">PDF</a>
  82           of the <a href="http://en.wikipedia.org/wiki/Skew_normal_distribution" target="_top">skew
  83           normal distribution</a> with shape parameter &#945;, defined by O'Hagan and
  84           Leonhard (1976) is
  85         </p>
  86 <div class="blockquote"><blockquote class="blockquote"><p>
  87             <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf0.svg"></span>
  88
  89           </p></blockquote></div>
  90 <p>
  91           Given <a href="http://en.wikipedia.org/wiki/Location_parameter" target="_top">location</a>
  92           &#958;, <a href="http://en.wikipedia.org/wiki/Scale_parameter" target="_top">scale</a>
  93           &#969;, and <a href="http://en.wikipedia.org/wiki/Shape_parameter" target="_top">shape</a>
  94           &#945;, it can be <a href="http://en.wikipedia.org/wiki/Skew_normal_distribution" target="_top">transformed</a>,
  95           to the form:
  96         </p>
  97 <div class="blockquote"><blockquote class="blockquote"><p>
  98             <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf.svg"></span>
  99
 100           </p></blockquote></div>
 101 <p>
 102           and <a href="http://en.wikipedia.org/wiki/Cumulative_distribution_function" target="_top">CDF</a>:
 103         </p>
 104 <div class="blockquote"><blockquote class="blockquote"><p>
 105             <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_cdf.svg"></span>
 106
 107           </p></blockquote></div>
 108 <p>
 109           where <span class="emphasis"><em>T(h,a)</em></span> is Owen's T function, and <span class="emphasis"><em>&#934;(x)</em></span>
 110           is the normal distribution.
 111         </p>
 112 <p>
 113           The variation the PDF and CDF with its parameters is illustrated in the
 114           following graphs:
 115         </p>
 116 <div class="blockquote"><blockquote class="blockquote"><p>
 117             <span class="inlinemediaobject"><img src="../../../../graphs/skew_normal_pdf.svg" align="middle"></span>
 118
 119           </p></blockquote></div>
 120 <div class="blockquote"><blockquote class="blockquote"><p>
 121             <span class="inlinemediaobject"><img src="../../../../graphs/skew_normal_cdf.svg" align="middle"></span>
 122
 123           </p></blockquote></div>
 124 <h5>
 125 <a name="math_toolkit.dist_ref.dists.skew_normal_dist.h0"></a>
 126           <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
 127           Functions</a>
 128         </h5>
 129 <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>
 130 </pre>
 131 <p>
 132           Constructs a skew_normal distribution with location &#958;, scale &#969; and shape &#945;.
 133         </p>
 134 <p>
 135           Requires scale &gt; 0, otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
 136           is called.
 137         </p>
 138 <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>
 139 </pre>
 140 <p>
 141           returns the location &#958; of this distribution,
 142         </p>
 143 <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>
 144 </pre>
 145 <p>
 146           returns the scale &#969; of this distribution,
 147         </p>
 148 <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>
 149 </pre>
 150 <p>
 151           returns the shape &#945; of this distribution.
 152         </p>
 153 <p>
 154           (Location and scale function match other similar distributions, allowing
 155           the functions <code class="computeroutput"><span class="identifier">find_location</span></code>
 156           and <code class="computeroutput"><span class="identifier">find_scale</span></code> to be used
 157           generically).
 158         </p>
 159 <div class="note"><table border="0" summary="Note">
 160 <tr>
 161 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
 162 <th align="left">Note</th>
 163 </tr>
 164 <tr><td align="left" valign="top">
 165 <p>
 166             While the shape parameter may be chosen arbitrarily (finite), the resulting
 167             <span class="bold"><strong>skewness</strong></span> of the distribution is in fact
 168             limited to about (-1, 1); strictly, the interval is (-0.9952717, 0.9952717).
 169           </p>
 170 <p>
 171             A parameter &#948; is related to the shape &#945; by &#948; = &#945; / (1 + &#945;&#178;), and used in the expression
 172             for skewness
 173           </p>
 174 <div class="blockquote"><blockquote class="blockquote"><p>
 175               <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_skewness.svg"></span>
 176
 177             </p></blockquote></div>
 178 </td></tr>
 179 </table></div>
 180 <h5>
 181 <a name="math_toolkit.dist_ref.dists.skew_normal_dist.h1"></a>
 182           <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>
 183         </h5>
 184 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 185 <li class="listitem">
 186               <a href="http://azzalini.stat.unipd.it/SN/" target="_top">Skew-Normal Probability
 187               Distribution</a> for many links and bibliography.
 188             </li>
 189 <li class="listitem">
 190               <a href="http://azzalini.stat.unipd.it/SN/Intro/intro.html" target="_top">A very
 191               brief introduction to the skew-normal distribution</a> by Adelchi
 192               Azzalini (2005-11-2).
 193             </li>
 194 <li class="listitem">
 195               See a <a href="http://www.tri.org.au/azzalini.html" target="_top">skew-normal
 196               function animation</a>.
 197             </li>
 198 </ul></div>
 199 <h5>
 200 <a name="math_toolkit.dist_ref.dists.skew_normal_dist.h2"></a>
 201           <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
 202           Accessors</a>
 203         </h5>
 204 <p>
 205           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 206           functions</a> that are generic to all distributions are supported:
 207           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 208           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 209           <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>,
 210           <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>,
 211           <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>,
 212           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 213           <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>,
 214           <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>.
 215         </p>
 216 <p>
 217           The domain of the random variable is <span class="emphasis"><em>-[max_value], +[min_value]</em></span>.
 218           Infinite values are not supported.
 219         </p>
 220 <p>
 221           There are no <a href="http://en.wikipedia.org/wiki/Closed-form_expression" target="_top">closed-form
 222           expression</a> known for the mode and median, but these are computed
 223           for the
 224         </p>
 225 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 226 <li class="listitem">
 227               mode - by finding the maximum of the PDF.
 228             </li>
 229 <li class="listitem">
 230               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>.
 231             </li>
 232 </ul></div>
 233 <p>
 234           The maximum of the PDF is sought through searching the root of f'(x)=0.
 235         </p>
 236 <p>
 237           Both involve iterative methods that will have lower accuracy than other
 238           estimates.
 239         </p>
 240 <h5>
 241 <a name="math_toolkit.dist_ref.dists.skew_normal_dist.h3"></a>
 242           <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>
 243         </h5>
 244 <p>
 245           <a href="http://www.r-project.org/" target="_top">The R Project for Statistical Computing</a>
 246           using library(sn) described at <a href="http://azzalini.stat.unipd.it/SN/" target="_top">Skew-Normal
 247           Probability Distribution</a>, and at <a href="http://cran.r-project.org/web/packages/sn/sn.pd" target="_top">R
 248           skew-normal(sn) package</a>.
 249         </p>
 250 <p>
 251           Package sn provides functions related to the skew-normal (SN) and the skew-t
 252           (ST) probability distributions, both for the univariate and for the the
 253           multivariate case, including regression models.
 254         </p>
 255 <p>
 256           <a href="http://www.wolfram.com/products/mathematica/index.html" target="_top">Wolfram
 257           Mathematica</a> was also used to generate some more accurate spot test
 258           data.
 259         </p>
 260 <h5>
 261 <a name="math_toolkit.dist_ref.dists.skew_normal_dist.h4"></a>
 262           <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>
 263         </h5>
 264 <p>
 265           The skew_normal distribution with shape = zero is implemented as a special
 266           case, equivalent to the normal distribution in terms of the <a class="link" href="../../sf_erf/error_function.html" title="Error Function erf and complement erfc">error
 267           function</a>, and therefore should have excellent accuracy.
 268         </p>
 269 <p>
 270           The PDF and mean, variance, skewness and kurtosis are also accurately evaluated
 271           using <a href="http://en.wikipedia.org/wiki/Analytical_expression" target="_top">analytical
 272           expressions</a>. The CDF requires <a href="http://en.wikipedia.org/wiki/Owen%27s_T_function" target="_top">Owen's
 273           T function</a> that is evaluated using a Boost C++ <a class="link" href="../../owens_t.html" title="Owen's T function">Owens
 274           T</a> implementation of the algorithms of M. Patefield and D. Tandy,
 275           Journal of Statistical Software, 5(5), 1-25 (2000); the complicated accuracy
 276           of this function is discussed in detail at <a class="link" href="../../owens_t.html" title="Owen's T function">Owens
 277           T</a>.
 278         </p>
 279 <p>
 280           The median and mode are calculated by iterative root finding, and both
 281           will be less accurate.
 282         </p>
 283 <h5>
 284 <a name="math_toolkit.dist_ref.dists.skew_normal_dist.h5"></a>
 285           <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>
 286         </h5>
 287 <p>
 288           In the following table, &#958; is the location of the distribution, and &#969; is its
 289           scale, and &#945; is its shape.
 290         </p>
 291 <div class="informaltable"><table class="table">
 292 <colgroup>
 293 <col>
 294 <col>
 295 </colgroup>
 296 <thead><tr>
 297 <th>
 298                   <p>
 299                     Function
 300                   </p>
 301                 </th>
 302 <th>
 303                   <p>
 304                     Implementation Notes
 305                   </p>
 306                 </th>
 307 </tr></thead>
 308 <tbody>
 309 <tr>
 310 <td>
 311                   <p>
 312                     pdf
 313                   </p>
 314                 </td>
 315 <td>
 316                   <p>
 317                     Using:
 318                   </p>
 319                   <div class="blockquote"><blockquote class="blockquote"><p>
 320                       <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_pdf.svg"></span>
 321
 322                     </p></blockquote></div>
 323                 </td>
 324 </tr>
 325 <tr>
 326 <td>
 327                   <p>
 328                     cdf
 329                   </p>
 330                 </td>
 331 <td>
 332                   <p>
 333                     Using:
 334                   </p>
 335                   <div class="blockquote"><blockquote class="blockquote"><p>
 336                       <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_cdf.svg"></span>
 337
 338                     </p></blockquote></div>
 339                   <p>
 340                     <br> where <span class="emphasis"><em>T(h,a)</em></span> is Owen's T function,
 341                     and <span class="emphasis"><em>&#934;(x)</em></span> is the normal distribution.
 342                   </p>
 343                 </td>
 344 </tr>
 345 <tr>
 346 <td>
 347                   <p>
 348                     cdf complement
 349                   </p>
 350                 </td>
 351 <td>
 352                   <p>
 353                     Using: complement of normal distribution + 2 * Owens_t
 354                   </p>
 355                 </td>
 356 </tr>
 357 <tr>
 358 <td>
 359                   <p>
 360                     quantile
 361                   </p>
 362                 </td>
 363 <td>
 364                   <p>
 365                     Maximum of the pdf is sought through searching the root of f'(x)=0
 366                   </p>
 367                 </td>
 368 </tr>
 369 <tr>
 370 <td>
 371                   <p>
 372                     quantile from the complement
 373                   </p>
 374                 </td>
 375 <td>
 376                   <p>
 377                     -quantile(SN(-location &#958;, scale &#969;, -shape&#945;), p)
 378                   </p>
 379                 </td>
 380 </tr>
 381 <tr>
 382 <td>
 383                   <p>
 384                     location
 385                   </p>
 386                 </td>
 387 <td>
 388                   <p>
 389                     location &#958;
 390                   </p>
 391                 </td>
 392 </tr>
 393 <tr>
 394 <td>
 395                   <p>
 396                     scale
 397                   </p>
 398                 </td>
 399 <td>
 400                   <p>
 401                     scale &#969;
 402                   </p>
 403                 </td>
 404 </tr>
 405 <tr>
 406 <td>
 407                   <p>
 408                     shape
 409                   </p>
 410                 </td>
 411 <td>
 412                   <p>
 413                     shape &#945;
 414                   </p>
 415                 </td>
 416 </tr>
 417 <tr>
 418 <td>
 419                   <p>
 420                     median
 421                   </p>
 422                 </td>
 423 <td>
 424                   <p>
 425                     quantile(1/2)
 426                   </p>
 427                 </td>
 428 </tr>
 429 <tr>
 430 <td>
 431                   <p>
 432                     mean
 433                   </p>
 434                 </td>
 435 <td>
 436                   <div class="blockquote"><blockquote class="blockquote"><p>
 437                       <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_mean.svg"></span>
 438
 439                     </p></blockquote></div>
 440                 </td>
 441 </tr>
 442 <tr>
 443 <td>
 444                   <p>
 445                     mode
 446                   </p>
 447                 </td>
 448 <td>
 449                   <p>
 450                     Maximum of the pdf is sought through searching the root of f'(x)=0
 451                   </p>
 452                 </td>
 453 </tr>
 454 <tr>
 455 <td>
 456                   <p>
 457                     variance
 458                   </p>
 459                 </td>
 460 <td>
 461                   <div class="blockquote"><blockquote class="blockquote"><p>
 462                       <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_variance.svg"></span>
 463
 464                     </p></blockquote></div>
 465                 </td>
 466 </tr>
 467 <tr>
 468 <td>
 469                   <p>
 470                     skewness
 471                   </p>
 472                 </td>
 473 <td>
 474                   <div class="blockquote"><blockquote class="blockquote"><p>
 475                       <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_skewness.svg"></span>
 476
 477                     </p></blockquote></div>
 478                 </td>
 479 </tr>
 480 <tr>
 481 <td>
 482                   <p>
 483                     kurtosis
 484                   </p>
 485                 </td>
 486 <td>
 487                   <p>
 488                     kurtosis excess-3
 489                   </p>
 490                 </td>
 491 </tr>
 492 <tr>
 493 <td>
 494                   <p>
 495                     kurtosis excess
 496                   </p>
 497                 </td>
 498 <td>
 499                   <div class="blockquote"><blockquote class="blockquote"><p>
 500                       <span class="inlinemediaobject"><img src="../../../../equations/skew_normal_kurt_ex.svg"></span>
 501
 502                     </p></blockquote></div>
 503                 </td>
 504 </tr>
 505 </tbody>
 506 </table></div>
 507 </div>
 508 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 509 <td align="left"></td>
 510 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 511       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 512       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 513       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 514       Daryle Walker and Xiaogang Zhang<p>
 515         Distributed under the Boost Software License, Version 1.0. (See accompanying
 516         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>)
 517       </p>
 518 </div></td>
 519 </tr></table>
 520 <hr>
 521 <div class="spirit-nav">
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