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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;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>
  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;15.&#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> &#160; and &#967;<sup>2</sup><sub>n</sub> &#160; 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 <p>
  61           F<sub>n,m</sub> &#160; = (&#967;<sup>2</sup><sub>n</sub> &#160; / n) / (&#967;<sup>2</sup><sub>m</sub> &#160; / m)
  62         </p>
  63 <p>
  64           Is distributed over the range [0, &#8734;] with an F distribution, and has the
  65           PDF:
  66         </p>
  67 <p>
  68           <span class="inlinemediaobject"><img src="../../../../equations/fisher_pdf.svg"></span>
  69         </p>
  70 <p>
  71           The following graph illustrates how the PDF varies depending on the two
  72           degrees of freedom parameters.
  73         </p>
  74 <p>
  75           <span class="inlinemediaobject"><img src="../../../../graphs/fisher_f_pdf.svg" align="middle"></span>
  76         </p>
  77 <h5>
  78 <a name="math_toolkit.dist_ref.dists.f_dist.h0"></a>
  79           <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>
  80         </h5>
  81 <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>
  82 </pre>
  83 <p>
  84           Constructs an F-distribution with numerator degrees of freedom <span class="emphasis"><em>df1</em></span>
  85           and denominator degrees of freedom <span class="emphasis"><em>df2</em></span>.
  86         </p>
  87 <p>
  88           Requires that <span class="emphasis"><em>df1</em></span> and <span class="emphasis"><em>df2</em></span> are
  89           both greater than zero, otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
  90           is called.
  91         </p>
  92 <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>
  93 </pre>
  94 <p>
  95           Returns the numerator degrees of freedom parameter of the distribution.
  96         </p>
  97 <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>
  98 </pre>
  99 <p>
 100           Returns the denominator degrees of freedom parameter of the distribution.
 101         </p>
 102 <h5>
 103 <a name="math_toolkit.dist_ref.dists.f_dist.h1"></a>
 104           <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
 105           Accessors</a>
 106         </h5>
 107 <p>
 108           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 109           functions</a> that are generic to all distributions are supported:
 110           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 111           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 112           <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>,
 113           <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>,
 114           <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>,
 115           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 116           <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>,
 117           <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>.
 118         </p>
 119 <p>
 120           The domain of the random variable is [0, +&#8734;].
 121         </p>
 122 <h5>
 123 <a name="math_toolkit.dist_ref.dists.f_dist.h2"></a>
 124           <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>
 125         </h5>
 126 <p>
 127           Various <a class="link" href="../../stat_tut/weg/f_eg.html" title="F Distribution Examples">worked examples</a>
 128           are available illustrating the use of the F Distribution.
 129         </p>
 130 <h5>
 131 <a name="math_toolkit.dist_ref.dists.f_dist.h3"></a>
 132           <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>
 133         </h5>
 134 <p>
 135           The normal distribution is implemented in terms of the <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">incomplete
 136           beta function</a> and its <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">inverses</a>,
 137           refer to those functions for accuracy data.
 138         </p>
 139 <h5>
 140 <a name="math_toolkit.dist_ref.dists.f_dist.h4"></a>
 141           <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>
 142         </h5>
 143 <p>
 144           In the following table <span class="emphasis"><em>v1</em></span> and <span class="emphasis"><em>v2</em></span>
 145           are the first and second degrees of freedom parameters of the distribution,
 146           <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span> is
 147           the probability, and <span class="emphasis"><em>q = 1-p</em></span>.
 148         </p>
 149 <div class="informaltable"><table class="table">
 150 <colgroup>
 151 <col>
 152 <col>
 153 </colgroup>
 154 <thead><tr>
 155 <th>
 156                   <p>
 157                     Function
 158                   </p>
 159                 </th>
 160 <th>
 161                   <p>
 162                     Implementation Notes
 163                   </p>
 164                 </th>
 165 </tr></thead>
 166 <tbody>
 167 <tr>
 168 <td>
 169                   <p>
 170                     pdf
 171                   </p>
 172                 </td>
 173 <td>
 174                   <p>
 175                     The usual form of the PDF is given by:
 176                   </p>
 177                   <p>
 178                     <span class="inlinemediaobject"><img src="../../../../equations/fisher_pdf.svg"></span>
 179                   </p>
 180                   <p>
 181                     However, that form is hard to evaluate directly without incurring
 182                     problems with either accuracy or numeric overflow.
 183                   </p>
 184                   <p>
 185                     Direct differentiation of the CDF expressed in terms of the incomplete
 186                     beta function
 187                   </p>
 188                   <p>
 189                     led to the following two formulas:
 190                   </p>
 191                   <p>
 192                     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
 193                     / 2, v1 / 2, v2 / (v2 + v1 * x))
 194                   </p>
 195                   <p>
 196                     with y = (v2 * v1) / ((v2 + v1 * x) * (v2 + v1 * x))
 197                   </p>
 198                   <p>
 199                     and
 200                   </p>
 201                   <p>
 202                     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
 203                     / 2, v2 / 2, v1 * x / (v2 + v1 * x))
 204                   </p>
 205                   <p>
 206                     with y = (z * v1 - x * v1 * v1) / z<sup>2</sup>
 207                   </p>
 208                   <p>
 209                     and z = v2 + v1 * x
 210                   </p>
 211                   <p>
 212                     The first of these is used for v1 * x &gt; v2, otherwise the
 213                     second is used.
 214                   </p>
 215                   <p>
 216                     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>
 217                     away from 1 to avoid rounding error.
 218                   </p>
 219                 </td>
 220 </tr>
 221 <tr>
 222 <td>
 223                   <p>
 224                     cdf
 225                   </p>
 226                 </td>
 227 <td>
 228                   <p>
 229                     Using the relations:
 230                   </p>
 231                   <p>
 232                     p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(v1
 233                     / 2, v2 / 2, v1 * x / (v2 + v1 * x))
 234                   </p>
 235                   <p>
 236                     and
 237                   </p>
 238                   <p>
 239                     p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(v2
 240                     / 2, v1 / 2, v2 / (v2 + v1 * x))
 241                   </p>
 242                   <p>
 243                     The first is used for v1 * x &gt; v2, otherwise the second is
 244                     used.
 245                   </p>
 246                   <p>
 247                     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
 248                     away from 1 to avoid rounding error.
 249                   </p>
 250                 </td>
 251 </tr>
 252 <tr>
 253 <td>
 254                   <p>
 255                     cdf complement
 256                   </p>
 257                 </td>
 258 <td>
 259                   <p>
 260                     Using the relations:
 261                   </p>
 262                   <p>
 263                     p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(v1
 264                     / 2, v2 / 2, v1 * x / (v2 + v1 * x))
 265                   </p>
 266                   <p>
 267                     and
 268                   </p>
 269                   <p>
 270                     p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(v2
 271                     / 2, v1 / 2, v2 / (v2 + v1 * x))
 272                   </p>
 273                   <p>
 274                     The first is used for v1 * x &lt; v2, otherwise the second is
 275                     used.
 276                   </p>
 277                   <p>
 278                     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
 279                     away from 1 to avoid rounding error.
 280                   </p>
 281                 </td>
 282 </tr>
 283 <tr>
 284 <td>
 285                   <p>
 286                     quantile
 287                   </p>
 288                 </td>
 289 <td>
 290                   <p>
 291                     Using the relation:
 292                   </p>
 293                   <p>
 294                     x = v2 * a / (v1 * b)
 295                   </p>
 296                   <p>
 297                     where:
 298                   </p>
 299                   <p>
 300                     a = <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>(v1
 301                     / 2, v2 / 2, p)
 302                   </p>
 303                   <p>
 304                     and
 305                   </p>
 306                   <p>
 307                     b = 1 - a
 308                   </p>
 309                   <p>
 310                     Quantities <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>
 311                     are both computed by <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>
 312                     without the subtraction implied above.
 313                   </p>
 314                 </td>
 315 </tr>
 316 <tr>
 317 <td>
 318                   <p>
 319                     quantile
 320                   </p>
 321                   <p>
 322                     from the complement
 323                   </p>
 324                 </td>
 325 <td>
 326                   <p>
 327                     Using the relation:
 328                   </p>
 329                   <p>
 330                     x = v2 * a / (v1 * b)
 331                   </p>
 332                   <p>
 333                     where
 334                   </p>
 335                   <p>
 336                     a = <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>(v1
 337                     / 2, v2 / 2, p)
 338                   </p>
 339                   <p>
 340                     and
 341                   </p>
 342                   <p>
 343                     b = 1 - a
 344                   </p>
 345                   <p>
 346                     Quantities <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span>
 347                     are both computed by <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>
 348                     without the subtraction implied above.
 349                   </p>
 350                 </td>
 351 </tr>
 352 <tr>
 353 <td>
 354                   <p>
 355                     mean
 356                   </p>
 357                 </td>
 358 <td>
 359                   <p>
 360                     v2 / (v2 - 2)
 361                   </p>
 362                 </td>
 363 </tr>
 364 <tr>
 365 <td>
 366                   <p>
 367                     variance
 368                   </p>
 369                 </td>
 370 <td>
 371                   <p>
 372                     2 * v2<sup>2 </sup> * (v1 + v2 - 2) / (v1 * (v2 - 2) * (v2 - 2) * (v2 - 4))
 373                   </p>
 374                 </td>
 375 </tr>
 376 <tr>
 377 <td>
 378                   <p>
 379                     mode
 380                   </p>
 381                 </td>
 382 <td>
 383                   <p>
 384                     v2 * (v1 - 2) / (v1 * (v2 + 2))
 385                   </p>
 386                 </td>
 387 </tr>
 388 <tr>
 389 <td>
 390                   <p>
 391                     skewness
 392                   </p>
 393                 </td>
 394 <td>
 395                   <p>
 396                     2 * (v2 + 2 * v1 - 2) * sqrt((2 * v2 - 8) / (v1 * (v2 + v1 -
 397                     2))) / (v2 - 6)
 398                   </p>
 399                 </td>
 400 </tr>
 401 <tr>
 402 <td>
 403                   <p>
 404                     kurtosis and kurtosis excess
 405                   </p>
 406                 </td>
 407 <td>
 408                   <p>
 409                     Refer to, <a href="http://mathworld.wolfram.com/F-Distribution.html" target="_top">Weisstein,
 410                     Eric W. "F-Distribution." From MathWorld--A Wolfram
 411                     Web Resource.</a>
 412                   </p>
 413                 </td>
 414 </tr>
 415 </tbody>
 416 </table></div>
 417 </div>
 418 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 419 <td align="left"></td>
 420 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 421       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 422       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 423       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 424         Distributed under the Boost Software License, Version 1.0. (See accompanying
 425         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>)
 426       </p>
 427 </div></td>
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