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  25 <div class="section math_toolkit_dist_dist_ref_dists_chi_squared_dist">
  26 <div class="titlepage"><div><div><h5 class="title">
  27 <a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist"></a><a class="link" href="chi_squared_dist.html" title="Chi Squared Distribution">Chi
  28           Squared Distribution</a>
  29 </h5></div></div></div>
  30 <p>
  31 </p>
  32 <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">chi_squared</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
  33 <p>
  34           </p>
  35 <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>
  36
  37 <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>
  38           <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a>   <span class="special">=</span> <a class="link" href="../../../policy/pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
  39 <span class="keyword">class</span> <span class="identifier">chi_squared_distribution</span><span class="special">;</span>
  40
  41 <span class="keyword">typedef</span> <span class="identifier">chi_squared_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">chi_squared</span><span class="special">;</span>
  42
  43 <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="Policies">Policy</a><span class="special">&gt;</span>
  44 <span class="keyword">class</span> <span class="identifier">chi_squared_distribution</span>
  45 <span class="special">{</span>
  46 <span class="keyword">public</span><span class="special">:</span>
  47    <span class="keyword">typedef</span> <span class="identifier">RealType</span>  <span class="identifier">value_type</span><span class="special">;</span>
  48    <span class="keyword">typedef</span> <span class="identifier">Policy</span>    <span class="identifier">policy_type</span><span class="special">;</span>
  49
  50    <span class="comment">// Constructor:</span>
  51    <span class="identifier">chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">i</span><span class="special">);</span>
  52
  53    <span class="comment">// Accessor to parameter:</span>
  54    <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  55
  56    <span class="comment">// Parameter estimation:</span>
  57    <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_degrees_of_freedom</span><span class="special">(</span>
  58       <span class="identifier">RealType</span> <span class="identifier">difference_from_mean</span><span class="special">,</span>
  59       <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span>
  60       <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span>
  61       <span class="identifier">RealType</span> <span class="identifier">sd</span><span class="special">,</span>
  62       <span class="identifier">RealType</span> <span class="identifier">hint</span> <span class="special">=</span> <span class="number">100</span><span class="special">);</span>
  63 <span class="special">};</span>
  64
  65 <span class="special">}}</span> <span class="comment">// namespaces</span>
  66 </pre>
  67 <p>
  68             The Chi-Squared distribution is one of the most widely used distributions
  69             in statistical tests. If &#967;<sub>i</sub> &#160; are &#957; &#160;
  70 independent, normally distributed random
  71             variables with means &#956;<sub>i</sub> &#160; and variances &#963;<sub>i</sub><sup>2</sup>, then the random variable:
  72           </p>
  73 <p>
  74             <span class="inlinemediaobject"><img src="../../../../../equations/chi_squ_ref1.png"></span>
  75           </p>
  76 <p>
  77             is distributed according to the Chi-Squared distribution.
  78           </p>
  79 <p>
  80             The Chi-Squared distribution is a special case of the gamma distribution
  81             and has a single parameter &#957; &#160; that specifies the number of degrees of freedom.
  82             The following graph illustrates how the distribution changes for different
  83             values of &#957;:
  84           </p>
  85 <p>
  86             <span class="inlinemediaobject"><img src="../../../../../graphs/chi_squared_pdf.png" align="middle"></span>
  87           </p>
  88 <h5>
  89 <a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.h0"></a>
  90             <span><a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.member_functions"></a></span><a class="link" href="chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.chi_squared_dist.member_functions">Member
  91             Functions</a>
  92           </h5>
  93 <pre class="programlisting"><span class="identifier">chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">v</span><span class="special">);</span>
  94 </pre>
  95 <p>
  96             Constructs a Chi-Squared distribution with <span class="emphasis"><em>v</em></span> degrees
  97             of freedom.
  98           </p>
  99 <p>
 100             Requires v &gt; 0, otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>.
 101           </p>
 102 <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
 103 </pre>
 104 <p>
 105             Returns the parameter <span class="emphasis"><em>v</em></span> from which this object was
 106             constructed.
 107           </p>
 108 <pre class="programlisting"><span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_degrees_of_freedom</span><span class="special">(</span>
 109    <span class="identifier">RealType</span> <span class="identifier">difference_from_variance</span><span class="special">,</span>
 110    <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span>
 111    <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span>
 112    <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">,</span>
 113    <span class="identifier">RealType</span> <span class="identifier">hint</span> <span class="special">=</span> <span class="number">100</span><span class="special">);</span>
 114 </pre>
 115 <p>
 116             Estimates the sample size required to detect a difference from a nominal
 117             variance in a Chi-Squared test for equal standard deviations.
 118           </p>
 119 <div class="variablelist">
 120 <p class="title"><b></b></p>
 121 <dl>
 122 <dt><span class="term">difference_from_variance</span></dt>
 123 <dd><p>
 124                   The difference from the assumed nominal variance that is to be
 125                   detected: Note that the sign of this value is critical, see below.
 126                 </p></dd>
 127 <dt><span class="term">alpha</span></dt>
 128 <dd><p>
 129                   The maximum acceptable risk of rejecting the null hypothesis when
 130                   it is in fact true.
 131                 </p></dd>
 132 <dt><span class="term">beta</span></dt>
 133 <dd><p>
 134                   The maximum acceptable risk of falsely failing to reject the null
 135                   hypothesis.
 136                 </p></dd>
 137 <dt><span class="term">variance</span></dt>
 138 <dd><p>
 139                   The nominal variance being tested against.
 140                 </p></dd>
 141 <dt><span class="term">hint</span></dt>
 142 <dd><p>
 143                   An optional hint on where to start looking for a result: the current
 144                   sample size would be a good choice.
 145                 </p></dd>
 146 </dl>
 147 </div>
 148 <p>
 149             Note that this calculation works with <span class="emphasis"><em>variances</em></span>
 150             and not <span class="emphasis"><em>standard deviations</em></span>.
 151           </p>
 152 <p>
 153             The sign of the parameter <span class="emphasis"><em>difference_from_variance</em></span>
 154             is important: the Chi Squared distribution is asymmetric, and the caller
 155             must decide in advance whether they are testing for a variance greater
 156             than a nominal value (positive <span class="emphasis"><em>difference_from_variance</em></span>)
 157             or testing for a variance less than a nominal value (negative <span class="emphasis"><em>difference_from_variance</em></span>).
 158             If the latter, then obviously it is a requirement that <code class="computeroutput"><span class="identifier">variance</span> <span class="special">+</span>
 159             <span class="identifier">difference_from_variance</span> <span class="special">&gt;</span>
 160             <span class="number">0</span></code>, since no sample can have a negative
 161             variance!
 162           </p>
 163 <p>
 164             This procedure uses the method in Diamond, W. J. (1989). Practical Experiment
 165             Designs, Van-Nostrand Reinhold, New York.
 166           </p>
 167 <p>
 168             See also section on Sample sizes required in <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc232.htm" target="_top">the
 169             NIST Engineering Statistics Handbook, Section 7.2.3.2</a>.
 170           </p>
 171 <h5>
 172 <a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.h1"></a>
 173             <span><a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.non_member_accessors"></a></span><a class="link" href="chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.chi_squared_dist.non_member_accessors">Non-member
 174             Accessors</a>
 175           </h5>
 176 <p>
 177             All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member
 178             accessor functions</a> that are generic to all distributions are supported:
 179             <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>,
 180             <a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>, <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>, <a class="link" href="../nmp.html#math.dist.hazard">Hazard
 181             Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>,
 182             <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>,
 183             <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>,
 184             <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>,
 185             <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>,
 186             <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>.
 187           </p>
 188 <p>
 189             (We have followed the usual restriction of the mode to degrees of freedom
 190             &gt;= 2, but note that the maximum of the pdf is actually zero for degrees
 191             of freedom from 2 down to 0, and provide an extended definition that
 192             would avoid a discontinuity in the mode as alternative code in a comment).
 193           </p>
 194 <p>
 195             The domain of the random variable is [0, +&#8734;].
 196           </p>
 197 <h5>
 198 <a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.h2"></a>
 199             <span><a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.examples"></a></span><a class="link" href="chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.chi_squared_dist.examples">Examples</a>
 200           </h5>
 201 <p>
 202             Various <a class="link" href="../../stat_tut/weg/cs_eg.html" title="Chi Squared Distribution Examples">worked examples</a>
 203             are available illustrating the use of the Chi Squared Distribution.
 204           </p>
 205 <h5>
 206 <a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.h3"></a>
 207             <span><a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.accuracy"></a></span><a class="link" href="chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.chi_squared_dist.accuracy">Accuracy</a>
 208           </h5>
 209 <p>
 210             The Chi-Squared distribution is implemented in terms of the <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">incomplete
 211             gamma functions</a>: please refer to the accuracy data for those functions.
 212           </p>
 213 <h5>
 214 <a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.h4"></a>
 215             <span><a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.implementation"></a></span><a class="link" href="chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.chi_squared_dist.implementation">Implementation</a>
 216           </h5>
 217 <p>
 218             In the following table <span class="emphasis"><em>v</em></span> is the number of degrees
 219             of freedom of the distribution, <span class="emphasis"><em>x</em></span> is the random
 220             variate, <span class="emphasis"><em>p</em></span> is the probability, and <span class="emphasis"><em>q =
 221             1-p</em></span>.
 222           </p>
 223 <div class="informaltable"><table class="table">
 224 <colgroup>
 225 <col>
 226 <col>
 227 </colgroup>
 228 <thead><tr>
 229 <th>
 230                     <p>
 231                       Function
 232                     </p>
 233                   </th>
 234 <th>
 235                     <p>
 236                       Implementation Notes
 237                     </p>
 238                   </th>
 239 </tr></thead>
 240 <tbody>
 241 <tr>
 242 <td>
 243                     <p>
 244                       pdf
 245                     </p>
 246                   </td>
 247 <td>
 248                     <p>
 249                       Using the relation: pdf = <a class="link" href="../../../special/sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>(v
 250                       / 2, x / 2) / 2
 251                     </p>
 252                   </td>
 253 </tr>
 254 <tr>
 255 <td>
 256                     <p>
 257                       cdf
 258                     </p>
 259                   </td>
 260 <td>
 261                     <p>
 262                       Using the relation: p = <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(v
 263                       / 2, x / 2)
 264                     </p>
 265                   </td>
 266 </tr>
 267 <tr>
 268 <td>
 269                     <p>
 270                       cdf complement
 271                     </p>
 272                   </td>
 273 <td>
 274                     <p>
 275                       Using the relation: q = <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(v
 276                       / 2, x / 2)
 277                     </p>
 278                   </td>
 279 </tr>
 280 <tr>
 281 <td>
 282                     <p>
 283                       quantile
 284                     </p>
 285                   </td>
 286 <td>
 287                     <p>
 288                       Using the relation: x = 2 * <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(v
 289                       / 2, p)
 290                     </p>
 291                   </td>
 292 </tr>
 293 <tr>
 294 <td>
 295                     <p>
 296                       quantile from the complement
 297                     </p>
 298                   </td>
 299 <td>
 300                     <p>
 301                       Using the relation: x = 2 * <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(v
 302                       / 2, p)
 303                     </p>
 304                   </td>
 305 </tr>
 306 <tr>
 307 <td>
 308                     <p>
 309                       mean
 310                     </p>
 311                   </td>
 312 <td>
 313                     <p>
 314                       v
 315                     </p>
 316                   </td>
 317 </tr>
 318 <tr>
 319 <td>
 320                     <p>
 321                       variance
 322                     </p>
 323                   </td>
 324 <td>
 325                     <p>
 326                       2v
 327                     </p>
 328                   </td>
 329 </tr>
 330 <tr>
 331 <td>
 332                     <p>
 333                       mode
 334                     </p>
 335                   </td>
 336 <td>
 337                     <p>
 338                       v - 2 (if v &gt;= 2)
 339                     </p>
 340                   </td>
 341 </tr>
 342 <tr>
 343 <td>
 344                     <p>
 345                       skewness
 346                     </p>
 347                   </td>
 348 <td>
 349                     <p>
 350                       2 * sqrt(2 / v) == sqrt(8 / v)
 351                     </p>
 352                   </td>
 353 </tr>
 354 <tr>
 355 <td>
 356                     <p>
 357                       kurtosis
 358                     </p>
 359                   </td>
 360 <td>
 361                     <p>
 362                       3 + 12 / v
 363                     </p>
 364                   </td>
 365 </tr>
 366 <tr>
 367 <td>
 368                     <p>
 369                       kurtosis excess
 370                     </p>
 371                   </td>
 372 <td>
 373                     <p>
 374                       12 / v
 375                     </p>
 376                   </td>
 377 </tr>
 378 </tbody>
 379 </table></div>
 380 <h5>
 381 <a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.h5"></a>
 382             <span><a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.references"></a></span><a class="link" href="chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.chi_squared_dist.references">References</a>
 383           </h5>
 384 <div class="itemizedlist"><ul class="itemizedlist" type="disc">
 385 <li class="listitem">
 386                 <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm" target="_top">NIST
 387                 Exploratory Data Analysis</a>
 388               </li>
 389 <li class="listitem">
 390                 <a href="http://en.wikipedia.org/wiki/Chi-square_distribution" target="_top">Chi-square
 391                 distribution</a>
 392               </li>
 393 <li class="listitem">
 394                 <a href="http://mathworld.wolfram.com/Chi-SquaredDistribution.html" target="_top">Weisstein,
 395                 Eric W. "Chi-Squared Distribution." From MathWorld--A Wolfram
 396                 Web Resource.</a>
 397               </li>
 398 </ul></div>
 399 </div>
 400 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 401 <td align="left"></td>
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 403       Lalande, Johan R&#229;de, Gautam Sewani, Thijs van den Berg and Benjamin Sobotta<p>
 404         Distributed under the Boost Software License, Version 1.0. (See accompanying
 405         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>)
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