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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.f_dist"></a><a class="link" href="f_dist.html" title="F Distribution">F Distribution</a>
  28 </h4></div></div></div>
  29 <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">fisher_f</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
  30 <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>
  31
  32 <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>
  33           <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>
  34 <span class="keyword">class</span> <span class="identifier">fisher_f_distribution</span><span class="special">;</span>
  35
  36 <span class="keyword">typedef</span> <span class="identifier">fisher_f_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">fisher_f</span><span class="special">;</span>
  37
  38 <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>
  39 <span class="keyword">class</span> <span class="identifier">fisher_f_distribution</span>
  40 <span class="special">{</span>
  41 <span class="keyword">public</span><span class="special">:</span>
  42    <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
  43
  44    <span class="comment">// Construct:</span>
  45    <span class="identifier">fisher_f_distribution</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">i</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">j</span><span class="special">);</span>
  46
  47    <span class="comment">// Accessors:</span>
  48    <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>
  49    <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>
  50 <span class="special">};</span>
  51
  52 <span class="special">}}</span> <span class="comment">//namespaces</span>
  53 </pre>
  54 <p>
  55           The F distribution is a continuous distribution that arises when testing
  56           whether two samples have the same variance. If &#967;<sup>2</sup><sub>m</sub> and &#967;<sup>2</sup><sub>n</sub> are independent
  57           variates each distributed as Chi-Squared with <span class="emphasis"><em>m</em></span> and
  58           <span class="emphasis"><em>n</em></span> degrees of freedom, then the test statistic:
  59         </p>
  60 <div class="blockquote"><blockquote class="blockquote"><p>
  61             <span class="serif_italic">F<sub>n,m</sub> = (&#967;<sup>2</sup><sub>n</sub> / n) / (&#967;<sup>2</sup><sub>m</sub> / m)</span>
  62           </p></blockquote></div>
  63 <p>
  64           Is distributed over the range [0, &#8734;] with an F distribution, and has the
  65           PDF:
  66         </p>
  67 <div class="blockquote"><blockquote class="blockquote"><p>
  68             <span class="inlinemediaobject"><img src="../../../../equations/fisher_pdf.svg"></span>
  69
  70           </p></blockquote></div>
  71 <p>
  72           The following graph illustrates how the PDF varies depending on the two
  73           degrees of freedom parameters.
  74         </p>
  75 <div class="blockquote"><blockquote class="blockquote"><p>
  76             <span class="inlinemediaobject"><img src="../../../../graphs/fisher_f_pdf.svg" align="middle"></span>
  77
  78           </p></blockquote></div>
  79 <h5>
  80 <a name="math_toolkit.dist_ref.dists.f_dist.h0"></a>
  81           <span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.member_functions"></a></span><a class="link" href="f_dist.html#math_toolkit.dist_ref.dists.f_dist.member_functions">Member Functions</a>
  82         </h5>
  83 <pre class="programlisting"><span class="identifier">fisher_f_distribution</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">df1</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&amp;</span> <span class="identifier">df2</span><span class="special">);</span>
  84 </pre>
  85 <p>
  86           Constructs an F-distribution with numerator degrees of freedom <span class="emphasis"><em>df1</em></span>
  87           and denominator degrees of freedom <span class="emphasis"><em>df2</em></span>.
  88         </p>
  89 <p>
  90           Requires that <span class="emphasis"><em>df1</em></span> and <span class="emphasis"><em>df2</em></span> are
  91           both greater than zero, otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
  92           is called.
  93         </p>
  94 <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>
  95 </pre>
  96 <p>
  97           Returns the numerator degrees of freedom parameter of the distribution.
  98         </p>
  99 <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>
 100 </pre>
 101 <p>
 102           Returns the denominator degrees of freedom parameter of the distribution.
 103         </p>
 104 <h5>
 105 <a name="math_toolkit.dist_ref.dists.f_dist.h1"></a>
 106           <span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.non_member_accessors"></a></span><a class="link" href="f_dist.html#math_toolkit.dist_ref.dists.f_dist.non_member_accessors">Non-member
 107           Accessors</a>
 108         </h5>
 109 <p>
 110           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 111           functions</a> that are generic to all distributions are supported:
 112           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 113           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 114           <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>,
 115           <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>,
 116           <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>,
 117           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 118           <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>,
 119           <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>.
 120         </p>
 121 <p>
 122           The domain of the random variable is [0, +&#8734;].
 123         </p>
 124 <h5>
 125 <a name="math_toolkit.dist_ref.dists.f_dist.h2"></a>
 126           <span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.examples"></a></span><a class="link" href="f_dist.html#math_toolkit.dist_ref.dists.f_dist.examples">Examples</a>
 127         </h5>
 128 <p>
 129           Various <a class="link" href="../../stat_tut/weg/f_eg.html" title="F Distribution Examples">worked examples</a>
 130           are available illustrating the use of the F Distribution.
 131         </p>
 132 <h5>
 133 <a name="math_toolkit.dist_ref.dists.f_dist.h3"></a>
 134           <span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.accuracy"></a></span><a class="link" href="f_dist.html#math_toolkit.dist_ref.dists.f_dist.accuracy">Accuracy</a>
 135         </h5>
 136 <p>
 137           The normal distribution is implemented in terms of the <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">incomplete
 138           beta function</a> and its <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">inverses</a>,
 139           refer to those functions for accuracy data.
 140         </p>
 141 <h5>
 142 <a name="math_toolkit.dist_ref.dists.f_dist.h4"></a>
 143           <span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.implementation"></a></span><a class="link" href="f_dist.html#math_toolkit.dist_ref.dists.f_dist.implementation">Implementation</a>
 144         </h5>
 145 <p>
 146           In the following table <span class="emphasis"><em>v1</em></span> and <span class="emphasis"><em>v2</em></span>
 147           are the first and second degrees of freedom parameters of the distribution,
 148           <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span> is
 149           the probability, and <span class="emphasis"><em>q = 1-p</em></span>.
 150         </p>
 151 <div class="informaltable"><table class="table">
 152 <colgroup>
 153 <col>
 154 <col>
 155 </colgroup>
 156 <thead><tr>
 157 <th>
 158                   <p>
 159                     Function
 160                   </p>
 161                 </th>
 162 <th>
 163                   <p>
 164                     Implementation Notes
 165                   </p>
 166                 </th>
 167 </tr></thead>
 168 <tbody>
 169 <tr>
 170 <td>
 171                   <p>
 172                     pdf
 173                   </p>
 174                 </td>
 175 <td>
 176                   <p>
 177                     The usual form of the PDF is given by:
 178                   </p>
 179                   <div class="blockquote"><blockquote class="blockquote"><p>
 180                       <span class="inlinemediaobject"><img src="../../../../equations/fisher_pdf.svg"></span>
 181
 182                     </p></blockquote></div>
 183                   <p>
 184                     However, that form is hard to evaluate directly without incurring
 185                     problems with either accuracy or numeric overflow.
 186                   </p>
 187                   <p>
 188                     Direct differentiation of the CDF expressed in terms of the incomplete
 189                     beta function
 190                   </p>
 191                   <p>
 192                     led to the following two formulas:
 193                   </p>
 194                   <div class="blockquote"><blockquote class="blockquote"><p>
 195                       <span class="serif_italic">f<sub>v1,v2</sub>(x) = y * <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>(v2
 196                       / 2, v1 / 2, v2 / (v2 + v1 * x))</span>
 197                     </p></blockquote></div>
 198                   <p>
 199                     with y = (v2 * v1) / ((v2 + v1 * x) * (v2 + v1 * x))
 200                   </p>
 201                   <p>
 202                     and
 203                   </p>
 204                   <div class="blockquote"><blockquote class="blockquote"><p>
 205                       <span class="serif_italic">f<sub>v1,v2</sub>(x) = y * <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>(v1
 206                       / 2, v2 / 2, v1 * x / (v2 + v1 * x))</span>
 207                     </p></blockquote></div>
 208                   <p>
 209                     with y = (z * v1 - x * v1 * v1) / z<sup>2</sup>
 210                   </p>
 211                   <p>
 212                     and z = v2 + v1 * x
 213                   </p>
 214                   <p>
 215                     The first of these is used for v1 * x &gt; v2, otherwise the
 216                     second is used.
 217                   </p>
 218                   <p>
 219                     The aim is to keep the <span class="emphasis"><em>x</em></span> argument to <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>
 220                     away from 1 to avoid rounding error.
 221                   </p>
 222                 </td>
 223 </tr>
 224 <tr>
 225 <td>
 226                   <p>
 227                     cdf
 228                   </p>
 229                 </td>
 230 <td>
 231                   <p>
 232                     Using the relations:
 233                   </p>
 234                   <div class="blockquote"><blockquote class="blockquote"><p>
 235                       <span class="serif_italic">p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(v1
 236                       / 2, v2 / 2, v1 * x / (v2 + v1 * x))</span>
 237                     </p></blockquote></div>
 238                   <p>
 239                     and
 240                   </p>
 241                   <div class="blockquote"><blockquote class="blockquote"><p>
 242                       <span class="serif_italic">:p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(v2
 243                       / 2, v1 / 2, v2 / (v2 + v1 * x))</span>
 244                     </p></blockquote></div>
 245                   <p>
 246                     The first is used for v1 * x &gt; v2, otherwise the second is
 247                     used.
 248                   </p>
 249                   <p>
 250                     The aim is to keep the <span class="emphasis"><em>x</em></span> argument to <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a> well
 251                     away from 1 to avoid rounding error.
 252                   </p>
 253                 </td>
 254 </tr>
 255 <tr>
 256 <td>
 257                   <p>
 258                     cdf complement
 259                   </p>
 260                 </td>
 261 <td>
 262                   <p>
 263                     Using the relations:
 264                   </p>
 265                   <div class="blockquote"><blockquote class="blockquote"><p>
 266                       <span class="serif_italic">p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(v1
 267                       / 2, v2 / 2, v1 * x / (v2 + v1 * x))</span>
 268                     </p></blockquote></div>
 269                   <p>
 270                     and
 271                   </p>
 272                   <div class="blockquote"><blockquote class="blockquote"><p>
 273                       <span class="serif_italic">p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(v2
 274                       / 2, v1 / 2, v2 / (v2 + v1 * x))</span>
 275                     </p></blockquote></div>
 276                   <p>
 277                     The first is used for v1 * x &lt; v2, otherwise the second is
 278                     used.
 279                   </p>
 280                   <p>
 281                     The aim is to keep the <span class="emphasis"><em>x</em></span> argument to <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a> well
 282                     away from 1 to avoid rounding error.
 283                   </p>
 284                 </td>
 285 </tr>
 286 <tr>
 287 <td>
 288                   <p>
 289                     quantile
 290                   </p>
 291                 </td>
 292 <td>
 293                   <p>
 294                     Using the relation:
 295                   </p>
 296                   <div class="blockquote"><blockquote class="blockquote"><p>
 297                       <span class="serif_italic">x = v2 * a / (v1 * b)</span>
 298                     </p></blockquote></div>
 299                   <p>
 300                     where:
 301                   </p>
 302                   <div class="blockquote"><blockquote class="blockquote"><p>
 303                       <span class="serif_italic">a = <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>(v1
 304                       / 2, v2 / 2, p)</span>
 305                     </p></blockquote></div>
 306                   <p>
 307                     and
 308                   </p>
 309                   <div class="blockquote"><blockquote class="blockquote"><p>
 310                       <span class="serif_italic">b = 1 - a</span>
 311                     </p></blockquote></div>
 312                   <p>
 313                     Quantities <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>
 314                     are both computed by <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>
 315                     without the subtraction implied above.
 316                   </p>
 317                 </td>
 318 </tr>
 319 <tr>
 320 <td>
 321                   <p>
 322                     quantile
 323                   </p>
 324                   <p>
 325                     from the complement
 326                   </p>
 327                 </td>
 328 <td>
 329                   <p>
 330                     Using the relation:
 331                   </p>
 332                   <div class="blockquote"><blockquote class="blockquote"><p>
 333                       <span class="serif_italic">x = v2 * a / (v1 * b)</span>
 334                     </p></blockquote></div>
 335                   <p>
 336                     where
 337                   </p>
 338                   <div class="blockquote"><blockquote class="blockquote"><p>
 339                       <span class="serif_italic">a = <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>(v1
 340                       / 2, v2 / 2, p)</span>
 341                     </p></blockquote></div>
 342                   <p>
 343                     and
 344                   </p>
 345                   <div class="blockquote"><blockquote class="blockquote"><p>
 346                       <span class="serif_italic">b = 1 - a</span>
 347                     </p></blockquote></div>
 348                   <p>
 349                     Quantities <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>
 350                     are both computed by <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>
 351                     without the subtraction implied above.
 352                   </p>
 353                 </td>
 354 </tr>
 355 <tr>
 356 <td>
 357                   <p>
 358                     mean
 359                   </p>
 360                 </td>
 361 <td>
 362                   <p>
 363                     v2 / (v2 - 2)
 364                   </p>
 365                 </td>
 366 </tr>
 367 <tr>
 368 <td>
 369                   <p>
 370                     variance
 371                   </p>
 372                 </td>
 373 <td>
 374                   <p>
 375                     2 * v2<sup>2 </sup> * (v1 + v2 - 2) / (v1 * (v2 - 2) * (v2 - 2) * (v2 - 4))
 376                   </p>
 377                 </td>
 378 </tr>
 379 <tr>
 380 <td>
 381                   <p>
 382                     mode
 383                   </p>
 384                 </td>
 385 <td>
 386                   <p>
 387                     v2 * (v1 - 2) / (v1 * (v2 + 2))
 388                   </p>
 389                 </td>
 390 </tr>
 391 <tr>
 392 <td>
 393                   <p>
 394                     skewness
 395                   </p>
 396                 </td>
 397 <td>
 398                   <p>
 399                     2 * (v2 + 2 * v1 - 2) * sqrt((2 * v2 - 8) / (v1 * (v2 + v1 -
 400                     2))) / (v2 - 6)
 401                   </p>
 402                 </td>
 403 </tr>
 404 <tr>
 405 <td>
 406                   <p>
 407                     kurtosis and kurtosis excess
 408                   </p>
 409                 </td>
 410 <td>
 411                   <p>
 412                     Refer to, <a href="http://mathworld.wolfram.com/F-Distribution.html" target="_top">Weisstein,
 413                     Eric W. "F-Distribution." From MathWorld--A Wolfram
 414                     Web Resource.</a>
 415                   </p>
 416                 </td>
 417 </tr>
 418 </tbody>
 419 </table></div>
 420 </div>
 421 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 422 <td align="left"></td>
 423 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 424       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 425       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 426       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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 428         Distributed under the Boost Software License, Version 1.0. (See accompanying
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