1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Chi Squared Distribution</title> 5<link rel="stylesheet" href="../../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../dists.html" title="Distributions"> 9<link rel="prev" href="cauchy_dist.html" title="Cauchy-Lorentz Distribution"> 10<link rel="next" href="empirical_cdf.html" title="Empirical Cumulative Distribution Function"> 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="cauchy_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="empirical_cdf.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.chi_squared_dist"></a><a class="link" href="chi_squared_dist.html" title="Chi Squared Distribution">Chi Squared 28 Distribution</a> 29</h4></div></div></div> 30<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</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">></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"><</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 21. 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<></a> <span class="special">></span> 35<span class="keyword">class</span> <span class="identifier">chi_squared_distribution</span><span class="special">;</span> 36 37<span class="keyword">typedef</span> <span class="identifier">chi_squared_distribution</span><span class="special"><></span> <span class="identifier">chi_squared</span><span class="special">;</span> 38 39<span class="keyword">template</span> <span class="special"><</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 40<span class="keyword">class</span> <span class="identifier">chi_squared_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">chi_squared_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">i</span><span class="special">);</span> 48 49 <span class="comment">// Accessor to parameter:</span> 50 <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> 51 52 <span class="comment">// Parameter estimation:</span> 53 <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_degrees_of_freedom</span><span class="special">(</span> 54 <span class="identifier">RealType</span> <span class="identifier">difference_from_mean</span><span class="special">,</span> 55 <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> 56 <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> 57 <span class="identifier">RealType</span> <span class="identifier">sd</span><span class="special">,</span> 58 <span class="identifier">RealType</span> <span class="identifier">hint</span> <span class="special">=</span> <span class="number">100</span><span class="special">);</span> 59<span class="special">};</span> 60 61<span class="special">}}</span> <span class="comment">// namespaces</span> 62</pre> 63<p> 64 The Chi-Squared distribution is one of the most widely used distributions 65 in statistical tests. If χ<sub>i</sub> are ν 66independent, normally distributed random 67 variables with means μ<sub>i</sub> and variances σ<sub>i</sub><sup>2</sup>, then the random variable: 68 </p> 69<div class="blockquote"><blockquote class="blockquote"><p> 70 <span class="inlinemediaobject"><img src="../../../../equations/chi_squ_ref1.svg"></span> 71 72 </p></blockquote></div> 73<p> 74 is distributed according to the Chi-Squared distribution. 75 </p> 76<p> 77 The Chi-Squared distribution is a special case of the gamma distribution 78 and has a single parameter ν that specifies the number of degrees of freedom. 79 The following graph illustrates how the distribution changes for different 80 values of ν: 81 </p> 82<div class="blockquote"><blockquote class="blockquote"><p> 83 <span class="inlinemediaobject"><img src="../../../../graphs/chi_squared_pdf.svg" align="middle"></span> 84 85 </p></blockquote></div> 86<h5> 87<a name="math_toolkit.dist_ref.dists.chi_squared_dist.h0"></a> 88 <span class="phrase"><a name="math_toolkit.dist_ref.dists.chi_squared_dist.member_functions"></a></span><a class="link" href="chi_squared_dist.html#math_toolkit.dist_ref.dists.chi_squared_dist.member_functions">Member 89 Functions</a> 90 </h5> 91<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> 92</pre> 93<p> 94 Constructs a Chi-Squared distribution with <span class="emphasis"><em>v</em></span> degrees 95 of freedom. 96 </p> 97<p> 98 Requires v > 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. 99 </p> 100<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> 101</pre> 102<p> 103 Returns the parameter <span class="emphasis"><em>v</em></span> from which this object was 104 constructed. 105 </p> 106<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> 107 <span class="identifier">RealType</span> <span class="identifier">difference_from_variance</span><span class="special">,</span> 108 <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> 109 <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> 110 <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">,</span> 111 <span class="identifier">RealType</span> <span class="identifier">hint</span> <span class="special">=</span> <span class="number">100</span><span class="special">);</span> 112</pre> 113<p> 114 Estimates the sample size required to detect a difference from a nominal 115 variance in a Chi-Squared test for equal standard deviations. 116 </p> 117<div class="variablelist"> 118<p class="title"><b></b></p> 119<dl class="variablelist"> 120<dt><span class="term">difference_from_variance</span></dt> 121<dd><p> 122 The difference from the assumed nominal variance that is to be detected: 123 Note that the sign of this value is critical, see below. 124 </p></dd> 125<dt><span class="term">alpha</span></dt> 126<dd><p> 127 The maximum acceptable risk of rejecting the null hypothesis when 128 it is in fact true. 129 </p></dd> 130<dt><span class="term">beta</span></dt> 131<dd><p> 132 The maximum acceptable risk of falsely failing to reject the null 133 hypothesis. 134 </p></dd> 135<dt><span class="term">variance</span></dt> 136<dd><p> 137 The nominal variance being tested against. 138 </p></dd> 139<dt><span class="term">hint</span></dt> 140<dd><p> 141 An optional hint on where to start looking for a result: the current 142 sample size would be a good choice. 143 </p></dd> 144</dl> 145</div> 146<p> 147 Note that this calculation works with <span class="emphasis"><em>variances</em></span> and 148 not <span class="emphasis"><em>standard deviations</em></span>. 149 </p> 150<p> 151 The sign of the parameter <span class="emphasis"><em>difference_from_variance</em></span> 152 is important: the Chi Squared distribution is asymmetric, and the caller 153 must decide in advance whether they are testing for a variance greater 154 than a nominal value (positive <span class="emphasis"><em>difference_from_variance</em></span>) 155 or testing for a variance less than a nominal value (negative <span class="emphasis"><em>difference_from_variance</em></span>). 156 If the latter, then obviously it is a requirement that <code class="computeroutput"><span class="identifier">variance</span> 157 <span class="special">+</span> <span class="identifier">difference_from_variance</span> 158 <span class="special">></span> <span class="number">0</span></code>, 159 since no sample can have a negative variance! 160 </p> 161<p> 162 This procedure uses the method in Diamond, W. J. (1989). Practical Experiment 163 Designs, Van-Nostrand Reinhold, New York. 164 </p> 165<p> 166 See also section on Sample sizes required in <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc232.htm" target="_top">the 167 NIST Engineering Statistics Handbook, Section 7.2.3.2</a>. 168 </p> 169<h5> 170<a name="math_toolkit.dist_ref.dists.chi_squared_dist.h1"></a> 171 <span class="phrase"><a name="math_toolkit.dist_ref.dists.chi_squared_dist.non_member_accessors"></a></span><a class="link" href="chi_squared_dist.html#math_toolkit.dist_ref.dists.chi_squared_dist.non_member_accessors">Non-member 172 Accessors</a> 173 </h5> 174<p> 175 All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor 176 functions</a> that are generic to all distributions are supported: 177 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, 178 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, 179 <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>, 180 <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>, 181 <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>, 182 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, 183 <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>, 184 <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>. 185 </p> 186<p> 187 (We have followed the usual restriction of the mode to degrees of freedom 188 >= 2, but note that the maximum of the pdf is actually zero for degrees 189 of freedom from 2 down to 0, and provide an extended definition that would 190 avoid a discontinuity in the mode as alternative code in a comment). 191 </p> 192<p> 193 The domain of the random variable is [0, +∞]. 194 </p> 195<h5> 196<a name="math_toolkit.dist_ref.dists.chi_squared_dist.h2"></a> 197 <span class="phrase"><a name="math_toolkit.dist_ref.dists.chi_squared_dist.examples"></a></span><a class="link" href="chi_squared_dist.html#math_toolkit.dist_ref.dists.chi_squared_dist.examples">Examples</a> 198 </h5> 199<p> 200 Various <a class="link" href="../../stat_tut/weg/cs_eg.html" title="Chi Squared Distribution Examples">worked examples</a> 201 are available illustrating the use of the Chi Squared Distribution. 202 </p> 203<h5> 204<a name="math_toolkit.dist_ref.dists.chi_squared_dist.h3"></a> 205 <span class="phrase"><a name="math_toolkit.dist_ref.dists.chi_squared_dist.accuracy"></a></span><a class="link" href="chi_squared_dist.html#math_toolkit.dist_ref.dists.chi_squared_dist.accuracy">Accuracy</a> 206 </h5> 207<p> 208 The Chi-Squared distribution is implemented in terms of the <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">incomplete 209 gamma functions</a>: please refer to the accuracy data for those functions. 210 </p> 211<h5> 212<a name="math_toolkit.dist_ref.dists.chi_squared_dist.h4"></a> 213 <span class="phrase"><a name="math_toolkit.dist_ref.dists.chi_squared_dist.implementation"></a></span><a class="link" href="chi_squared_dist.html#math_toolkit.dist_ref.dists.chi_squared_dist.implementation">Implementation</a> 214 </h5> 215<p> 216 In the following table <span class="emphasis"><em>v</em></span> is the number of degrees 217 of freedom of the distribution, <span class="emphasis"><em>x</em></span> is the random variate, 218 <span class="emphasis"><em>p</em></span> is the probability, and <span class="emphasis"><em>q = 1-p</em></span>. 219 </p> 220<div class="informaltable"><table class="table"> 221<colgroup> 222<col> 223<col> 224</colgroup> 225<thead><tr> 226<th> 227 <p> 228 Function 229 </p> 230 </th> 231<th> 232 <p> 233 Implementation Notes 234 </p> 235 </th> 236</tr></thead> 237<tbody> 238<tr> 239<td> 240 <p> 241 pdf 242 </p> 243 </td> 244<td> 245 <p> 246 Using the relation: pdf = <a class="link" href="../../sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>(v 247 / 2, x / 2) / 2 248 </p> 249 </td> 250</tr> 251<tr> 252<td> 253 <p> 254 cdf 255 </p> 256 </td> 257<td> 258 <p> 259 Using the relation: p = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(v 260 / 2, x / 2) 261 </p> 262 </td> 263</tr> 264<tr> 265<td> 266 <p> 267 cdf complement 268 </p> 269 </td> 270<td> 271 <p> 272 Using the relation: q = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(v 273 / 2, x / 2) 274 </p> 275 </td> 276</tr> 277<tr> 278<td> 279 <p> 280 quantile 281 </p> 282 </td> 283<td> 284 <p> 285 Using the relation: x = 2 * <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(v 286 / 2, p) 287 </p> 288 </td> 289</tr> 290<tr> 291<td> 292 <p> 293 quantile from the complement 294 </p> 295 </td> 296<td> 297 <p> 298 Using the relation: x = 2 * <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(v 299 / 2, p) 300 </p> 301 </td> 302</tr> 303<tr> 304<td> 305 <p> 306 mean 307 </p> 308 </td> 309<td> 310 <p> 311 v 312 </p> 313 </td> 314</tr> 315<tr> 316<td> 317 <p> 318 variance 319 </p> 320 </td> 321<td> 322 <p> 323 2v 324 </p> 325 </td> 326</tr> 327<tr> 328<td> 329 <p> 330 mode 331 </p> 332 </td> 333<td> 334 <p> 335 v - 2 (if v >= 2) 336 </p> 337 </td> 338</tr> 339<tr> 340<td> 341 <p> 342 skewness 343 </p> 344 </td> 345<td> 346 <p> 347 2 * sqrt(2 / v) == sqrt(8 / v) 348 </p> 349 </td> 350</tr> 351<tr> 352<td> 353 <p> 354 kurtosis 355 </p> 356 </td> 357<td> 358 <p> 359 3 + 12 / v 360 </p> 361 </td> 362</tr> 363<tr> 364<td> 365 <p> 366 kurtosis excess 367 </p> 368 </td> 369<td> 370 <p> 371 12 / v 372 </p> 373 </td> 374</tr> 375</tbody> 376</table></div> 377<h5> 378<a name="math_toolkit.dist_ref.dists.chi_squared_dist.h5"></a> 379 <span class="phrase"><a name="math_toolkit.dist_ref.dists.chi_squared_dist.references"></a></span><a class="link" href="chi_squared_dist.html#math_toolkit.dist_ref.dists.chi_squared_dist.references">References</a> 380 </h5> 381<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 382<li class="listitem"> 383 <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm" target="_top">NIST 384 Exploratory Data Analysis</a> 385 </li> 386<li class="listitem"> 387 <a href="http://en.wikipedia.org/wiki/Chi-square_distribution" target="_top">Chi-square 388 distribution</a> 389 </li> 390<li class="listitem"> 391 <a href="http://mathworld.wolfram.com/Chi-SquaredDistribution.html" target="_top">Weisstein, 392 Eric W. "Chi-Squared Distribution." From MathWorld--A Wolfram 393 Web Resource.</a> 394 </li> 395</ul></div> 396</div> 397<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 398<td align="left"></td> 399<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 400 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 401 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 402 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 403 Daryle Walker and Xiaogang Zhang<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>) 406 </p> 407</div></td> 408</tr></table> 409<hr> 410<div class="spirit-nav"> 411<a accesskey="p" href="cauchy_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="empirical_cdf.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a> 412</div> 413</body> 414</html> 415
