ceph/src/boost/libs/math/doc/html/math_toolkit/dist_ref/dists/nc_f_dist.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   4 <title>Noncentral F Distribution</title>
   5 <link rel="stylesheet" href="../../../math.css" type="text/css">
   6 <meta name="generator" content="DocBook XSL Stylesheets V1.77.1">
   7 <link rel="home" href="../../../index.html" title="Math Toolkit 2.5.1">
   8 <link rel="up" href="../dists.html" title="Distributions">
   9 <link rel="prev" href="nc_chi_squared_dist.html" title="Noncentral Chi-Squared Distribution">
  10 <link rel="next" href="nc_t_dist.html" title="Noncentral T Distribution">
  11 </head>
  12 <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
  13 <table cellpadding="2" width="100%"><tr>
  14 <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td>
  15 <td align="center"><a href="../../../../../../../index.html">Home</a></td>
  16 <td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td>
  17 <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
  18 <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
  19 <td align="center"><a href="../../../../../../../more/index.htm">More</a></td>
  20 </tr></table>
  21 <hr>
  22 <div class="spirit-nav">
  23 <a accesskey="p" href="nc_chi_squared_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="nc_t_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.nc_f_dist"></a><a class="link" href="nc_f_dist.html" title="Noncentral F Distribution">Noncentral F
  28         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">non_central_f</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">non_central_f_distribution</span><span class="special">;</span>
  36
  37 <span class="keyword">typedef</span> <span class="identifier">non_central_f_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">non_central_f</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">non_central_f_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
  46    <span class="comment">// Constructor:</span>
  47    <span class="identifier">non_central_f_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">v1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">v2</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span>
  48
  49    <span class="comment">// Accessor to degrees_of_freedom parameters v1 &amp; v2:</span>
  50    <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom1</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  51    <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom2</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  52
  53    <span class="comment">// Accessor to non-centrality parameter lambda:</span>
  54    <span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  55 <span class="special">};</span>
  56
  57 <span class="special">}}</span> <span class="comment">// namespaces</span>
  58 </pre>
  59 <p>
  60           The noncentral F distribution is a generalization of the <a class="link" href="f_dist.html" title="F Distribution">Fisher
  61           F Distribution</a>. It is defined as the ratio
  62         </p>
  63 <pre class="programlisting"><span class="identifier">F</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">X</span><span class="special">/</span><span class="identifier">v1</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="identifier">Y</span><span class="special">/</span><span class="identifier">v2</span><span class="special">)</span>
  64 </pre>
  65 <p>
  66           where X is a noncentral &#967;<sup>2</sup>
  67 random variable with <span class="emphasis"><em>v1</em></span> degrees
  68           of freedom and non-centrality parameter &#955;, and Y is a central &#967;<sup>2</sup> random variable
  69           with <span class="emphasis"><em>v2</em></span> degrees of freedom.
  70         </p>
  71 <p>
  72           This gives the following PDF:
  73         </p>
  74 <p>
  75           <span class="inlinemediaobject"><img src="../../../../equations/nc_f_ref1.svg"></span>
  76         </p>
  77 <p>
  78           where L<sub>a</sub><sup>b</sup>(c) is a generalised Laguerre polynomial and B(a,b) is the <a class="link" href="../../sf_beta/beta_function.html" title="Beta">beta</a> function, or
  79         </p>
  80 <p>
  81           <span class="inlinemediaobject"><img src="../../../../equations/nc_f_ref2.svg"></span>
  82         </p>
  83 <p>
  84           The following graph illustrates how the distribution changes for different
  85           values of &#955;:
  86         </p>
  87 <p>
  88           <span class="inlinemediaobject"><img src="../../../../graphs/nc_f_pdf.svg" align="middle"></span>
  89         </p>
  90 <h5>
  91 <a name="math_toolkit.dist_ref.dists.nc_f_dist.h0"></a>
  92           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_f_dist.member_functions"></a></span><a class="link" href="nc_f_dist.html#math_toolkit.dist_ref.dists.nc_f_dist.member_functions">Member
  93           Functions</a>
  94         </h5>
  95 <pre class="programlisting"><span class="identifier">non_central_f_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">v1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">v2</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span>
  96 </pre>
  97 <p>
  98           Constructs a non-central beta distribution with parameters <span class="emphasis"><em>v1</em></span>
  99           and <span class="emphasis"><em>v2</em></span> and non-centrality parameter <span class="emphasis"><em>lambda</em></span>.
 100         </p>
 101 <p>
 102           Requires v1 &gt; 0, v2 &gt; 0 and lambda &gt;= 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 103         </p>
 104 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom1</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 105 </pre>
 106 <p>
 107           Returns the parameter <span class="emphasis"><em>v1</em></span> from which this object was
 108           constructed.
 109         </p>
 110 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom2</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 111 </pre>
 112 <p>
 113           Returns the parameter <span class="emphasis"><em>v2</em></span> from which this object was
 114           constructed.
 115         </p>
 116 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 117 </pre>
 118 <p>
 119           Returns the non-centrality parameter <span class="emphasis"><em>lambda</em></span> from which
 120           this object was constructed.
 121         </p>
 122 <h5>
 123 <a name="math_toolkit.dist_ref.dists.nc_f_dist.h1"></a>
 124           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_f_dist.non_member_accessors"></a></span><a class="link" href="nc_f_dist.html#math_toolkit.dist_ref.dists.nc_f_dist.non_member_accessors">Non-member
 125           Accessors</a>
 126         </h5>
 127 <p>
 128           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 129           functions</a> that are generic to all distributions are supported:
 130           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 131           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 132           <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>,
 133           <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>,
 134           <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>,
 135           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 136           <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>,
 137           <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>.
 138         </p>
 139 <p>
 140           The domain of the random variable is [0, +&#8734;].
 141         </p>
 142 <h5>
 143 <a name="math_toolkit.dist_ref.dists.nc_f_dist.h2"></a>
 144           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_f_dist.accuracy"></a></span><a class="link" href="nc_f_dist.html#math_toolkit.dist_ref.dists.nc_f_dist.accuracy">Accuracy</a>
 145         </h5>
 146 <p>
 147           This distribution is implemented in terms of the <a class="link" href="nc_beta_dist.html" title="Noncentral Beta Distribution">Noncentral
 148           Beta Distribution</a>: refer to that distribution for accuracy data.
 149         </p>
 150 <h5>
 151 <a name="math_toolkit.dist_ref.dists.nc_f_dist.h3"></a>
 152           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_f_dist.tests"></a></span><a class="link" href="nc_f_dist.html#math_toolkit.dist_ref.dists.nc_f_dist.tests">Tests</a>
 153         </h5>
 154 <p>
 155           Since this distribution is implemented by adapting another distribution,
 156           the tests consist of basic sanity checks computed by the <a href="http://www.r-project.org/" target="_top">R-2.5.1
 157           Math library statistical package</a> and its pbeta and dbeta functions.
 158         </p>
 159 <h5>
 160 <a name="math_toolkit.dist_ref.dists.nc_f_dist.h4"></a>
 161           <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_f_dist.implementation"></a></span><a class="link" href="nc_f_dist.html#math_toolkit.dist_ref.dists.nc_f_dist.implementation">Implementation</a>
 162         </h5>
 163 <p>
 164           In the following table <span class="emphasis"><em>v1</em></span> and <span class="emphasis"><em>v2</em></span>
 165           are the first and second degrees of freedom parameters of the distribution,
 166           &#955;
 167 is the non-centrality parameter, <span class="emphasis"><em>x</em></span> is the random variate,
 168           <span class="emphasis"><em>p</em></span> is the probability, and <span class="emphasis"><em>q = 1-p</em></span>.
 169         </p>
 170 <div class="informaltable"><table class="table">
 171 <colgroup>
 172 <col>
 173 <col>
 174 </colgroup>
 175 <thead><tr>
 176 <th>
 177                   <p>
 178                     Function
 179                   </p>
 180                 </th>
 181 <th>
 182                   <p>
 183                     Implementation Notes
 184                   </p>
 185                 </th>
 186 </tr></thead>
 187 <tbody>
 188 <tr>
 189 <td>
 190                   <p>
 191                     pdf
 192                   </p>
 193                 </td>
 194 <td>
 195                   <p>
 196                     Implemented in terms of the non-central beta PDF using the relation:
 197                   </p>
 198                   <p>
 199                     f(x;v1,v2;&#955;) = (v1/v2) / ((1+y)*(1+y)) * g(y/(1+y);v1/2,v2/2;&#955;)
 200                   </p>
 201                   <p>
 202                     where g(x; a, b; &#955;) is the non central beta PDF, and:
 203                   </p>
 204                   <p>
 205                     y = x * v1 / v2
 206                   </p>
 207                 </td>
 208 </tr>
 209 <tr>
 210 <td>
 211                   <p>
 212                     cdf
 213                   </p>
 214                 </td>
 215 <td>
 216                   <p>
 217                     Using the relation:
 218                   </p>
 219                   <p>
 220                     p = B<sub>y</sub>(v1/2, v2/2; &#955;)
 221                   </p>
 222                   <p>
 223                     where B<sub>x</sub>(a, b; &#955;) is the noncentral beta distribution CDF and
 224                   </p>
 225                   <p>
 226                     y = x * v1 / v2
 227                   </p>
 228                 </td>
 229 </tr>
 230 <tr>
 231 <td>
 232                   <p>
 233                     cdf complement
 234                   </p>
 235                 </td>
 236 <td>
 237                   <p>
 238                     Using the relation:
 239                   </p>
 240                   <p>
 241                     q = 1 - B<sub>y</sub>(v1/2, v2/2; &#955;)
 242                   </p>
 243                   <p>
 244                     where 1 - B<sub>x</sub>(a, b; &#955;) is the complement of the noncentral beta
 245                     distribution CDF and
 246                   </p>
 247                   <p>
 248                     y = x * v1 / v2
 249                   </p>
 250                 </td>
 251 </tr>
 252 <tr>
 253 <td>
 254                   <p>
 255                     quantile
 256                   </p>
 257                 </td>
 258 <td>
 259                   <p>
 260                     Using the relation:
 261                   </p>
 262                   <p>
 263                     x = (bx / (1-bx)) * (v1 / v2)
 264                   </p>
 265                   <p>
 266                     where
 267                   </p>
 268                   <p>
 269                     bx = Q<sub>p</sub><sup>-1</sup>(v1/2, v2/2; &#955;)
 270                   </p>
 271                   <p>
 272                     and
 273                   </p>
 274                   <p>
 275                     Q<sub>p</sub><sup>-1</sup>(v1/2, v2/2; &#955;)
 276                   </p>
 277                   <p>
 278                     is the noncentral beta quantile.
 279                   </p>
 280                 </td>
 281 </tr>
 282 <tr>
 283 <td>
 284                   <p>
 285                     quantile
 286                   </p>
 287                   <p>
 288                     from the complement
 289                   </p>
 290                 </td>
 291 <td>
 292                   <p>
 293                     Using the relation:
 294                   </p>
 295                   <p>
 296                     x = (bx / (1-bx)) * (v1 / v2)
 297                   </p>
 298                   <p>
 299                     where
 300                   </p>
 301                   <p>
 302                     bx = QC<sub>q</sub><sup>-1</sup>(v1/2, v2/2; &#955;)
 303                   </p>
 304                   <p>
 305                     and
 306                   </p>
 307                   <p>
 308                     QC<sub>q</sub><sup>-1</sup>(v1/2, v2/2; &#955;)
 309                   </p>
 310                   <p>
 311                     is the noncentral beta quantile from the complement.
 312                   </p>
 313                 </td>
 314 </tr>
 315 <tr>
 316 <td>
 317                   <p>
 318                     mean
 319                   </p>
 320                 </td>
 321 <td>
 322                   <p>
 323                     v2 * (v1 + l) / (v1 * (v2 - 2))
 324                   </p>
 325                 </td>
 326 </tr>
 327 <tr>
 328 <td>
 329                   <p>
 330                     mode
 331                   </p>
 332                 </td>
 333 <td>
 334                   <p>
 335                     By numeric maximalisation of the PDF.
 336                   </p>
 337                 </td>
 338 </tr>
 339 <tr>
 340 <td>
 341                   <p>
 342                     variance
 343                   </p>
 344                 </td>
 345 <td>
 346                   <p>
 347                     Refer to, <a href="http://mathworld.wolfram.com/NoncentralF-Distribution.html" target="_top">Weisstein,
 348                     Eric W. "Noncentral F-Distribution." From MathWorld--A
 349                     Wolfram Web Resource.</a>
 350                   </p>
 351                 </td>
 352 </tr>
 353 <tr>
 354 <td>
 355                   <p>
 356                     skewness
 357                   </p>
 358                 </td>
 359 <td>
 360                   <p>
 361                     Refer to, <a href="http://mathworld.wolfram.com/NoncentralF-Distribution.html" target="_top">Weisstein,
 362                     Eric W. "Noncentral F-Distribution." From MathWorld--A
 363                     Wolfram Web Resource.</a>, and to the <a href="http://reference.wolfram.com/mathematica/ref/NoncentralFRatioDistribution.html" target="_top">Mathematica
 364                     documentation</a>
 365                   </p>
 366                 </td>
 367 </tr>
 368 <tr>
 369 <td>
 370                   <p>
 371                     kurtosis and kurtosis excess
 372                   </p>
 373                 </td>
 374 <td>
 375                   <p>
 376                     Refer to, <a href="http://mathworld.wolfram.com/NoncentralF-Distribution.html" target="_top">Weisstein,
 377                     Eric W. "Noncentral F-Distribution." From MathWorld--A
 378                     Wolfram Web Resource.</a>, and to the <a href="http://reference.wolfram.com/mathematica/ref/NoncentralFRatioDistribution.html" target="_top">Mathematica
 379                     documentation</a>
 380                   </p>
 381                 </td>
 382 </tr>
 383 </tbody>
 384 </table></div>
 385 <p>
 386           Some analytic properties of noncentral distributions (particularly unimodality,
 387           and monotonicity of their modes) are surveyed and summarized by:
 388         </p>
 389 <p>
 390           Andrea van Aubel &amp; Wolfgang Gawronski, Applied Mathematics and Computation,
 391           141 (2003) 3-12.
 392         </p>
 393 </div>
 394 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 395 <td align="left"></td>
 396 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 397       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 398       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 399       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 400         Distributed under the Boost Software License, Version 1.0. (See accompanying
 401         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>)
 402       </p>
 403 </div></td>
 404 </tr></table>
 405 <hr>
 406 <div class="spirit-nav">
 407 <a accesskey="p" href="nc_chi_squared_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="nc_t_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
 408 </div>
 409 </body>
 410 </html>