<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Chi Squared Distribution</title> <link rel="stylesheet" href="../../../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> <link rel="home" href="../../../../index.html" title="Math Toolkit"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="cauchy_dist.html" title="Cauchy-Lorentz Distribution"> <link rel="next" href="exp_dist.html" title="Exponential Distribution"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../../../../../libs/libraries.htm">Libraries</a></td> <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> <td align="center"><a href="../../../../../../../../../more/index.htm">More</a></td> </tr></table> <hr> <div class="spirit-nav"> <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="exp_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section" lang="en"> <div class="titlepage"><div><div><h5 class="title"> <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 Squared Distribution</a> </h5></div></div></div> <p> </p> <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> <p> </p> <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> <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> <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<></a> <span class="special">></span> <span class="keyword">class</span> <span class="identifier">chi_squared_distribution</span><span class="special">;</span> <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> <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="Policies">Policy</a><span class="special">></span> <span class="keyword">class</span> <span class="identifier">chi_squared_distribution</span> <span class="special">{</span> <span class="keyword">public</span><span class="special">:</span> <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> <span class="comment">// Constructor: </span> <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> <span class="comment">// Accessor to parameter: </span> <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> <span class="comment">// Parameter estimation: </span> <span class="keyword">static</span> <span class="identifier">RealType</span> <span class="identifier">find_degrees_of_freedom</span><span class="special">(</span> <span class="identifier">RealType</span> <span class="identifier">difference_from_mean</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">sd</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">hint</span> <span class="special">=</span> <span class="number">100</span><span class="special">);</span> <span class="special">};</span> <span class="special">}}</span> <span class="comment">// namespaces </span></pre> <p> The Chi-Squared distribution is one of the most widely used distributions in statistical tests. If χ<sub>i</sub> are ν independent, normally distributed random variables with means μ<sub>i</sub> and variances σ<sub>i</sub><sup>2</sup>, then the random variable: </p> <p> <span class="inlinemediaobject"><img src="../../../../../equations/chi_squ_ref1.png"></span> </p> <p> is distributed according to the Chi-Squared distribution. </p> <p> The Chi-Squared distribution is a special case of the gamma distribution and has a single parameter ν that specifies the number of degrees of freedom. The following graph illustrates how the distribution changes for different values of ν: </p> <p> <span class="inlinemediaobject"><img src="../../../../../graphs/chi_squared_pdf.png" align="middle"></span> </p> <a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.member_functions"></a><h5> <a name="id1028218"></a> <a class="link" href="chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.chi_squared_dist.member_functions">Member Functions</a> </h5> <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> </pre> <p> Constructs a Chi-Squared distribution with <span class="emphasis"><em>v</em></span> degrees of freedom. </p> <p> Requires v > 0, otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>. </p> <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> </pre> <p> Returns the parameter <span class="emphasis"><em>v</em></span> from which this object was constructed. </p> <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> <span class="identifier">RealType</span> <span class="identifier">difference_from_variance</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">hint</span> <span class="special">=</span> <span class="number">100</span><span class="special">);</span> </pre> <p> Estimates the sample size required to detect a difference from a nominal variance in a Chi-Squared test for equal standard deviations. </p> <div class="variablelist"> <p class="title"><b></b></p> <dl> <dt><span class="term">difference_from_variance</span></dt> <dd><p> The difference from the assumed nominal variance that is to be detected: Note that the sign of this value is critical, see below. </p></dd> <dt><span class="term">alpha</span></dt> <dd><p> The maximum acceptable risk of rejecting the null hypothesis when it is in fact true. </p></dd> <dt><span class="term">beta</span></dt> <dd><p> The maximum acceptable risk of falsely failing to reject the null hypothesis. </p></dd> <dt><span class="term">variance</span></dt> <dd><p> The nominal variance being tested against. </p></dd> <dt><span class="term">hint</span></dt> <dd><p> An optional hint on where to start looking for a result: the current sample size would be a good choice. </p></dd> </dl> </div> <p> Note that this calculation works with <span class="emphasis"><em>variances</em></span> and not <span class="emphasis"><em>standard deviations</em></span>. </p> <p> The sign of the parameter <span class="emphasis"><em>difference_from_variance</em></span> is important: the Chi Squared distribution is asymmetric, and the caller must decide in advance whether they are testing for a variance greater than a nominal value (positive <span class="emphasis"><em>difference_from_variance</em></span>) or testing for a variance less than a nominal value (negative <span class="emphasis"><em>difference_from_variance</em></span>). If the latter, then obviously it is a requirement that <code class="computeroutput"><span class="identifier">variance</span> <span class="special">+</span> <span class="identifier">difference_from_variance</span> <span class="special">></span> <span class="number">0</span></code>, since no sample can have a negative variance! </p> <p> This procedure uses the method in Diamond, W. J. (1989). Practical Experiment Designs, Van-Nostrand Reinhold, New York. </p> <p> See also section on Sample sizes required in <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc232.htm" target="_top">the NIST Engineering Statistics Handbook, Section 7.2.3.2</a>. </p> <a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.non_member_accessors"></a><h5> <a name="id1028549"></a> <a class="link" href="chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.chi_squared_dist.non_member_accessors">Non-member Accessors</a> </h5> <p> All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor functions</a> that are generic to all distributions are supported: <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>, <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 Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>, <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>, <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>, <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>, <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>, <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>. </p> <p> (We have followed the usual restriction of the mode to degrees of freedom >= 2, but note that the maximum of the pdf is actually zero for degrees of freedom from 2 down to 0, and provide an extended definition that would avoid a discontinuity in the mode as alternative code in a comment). </p> <p> The domain of the random variable is [0, +∞]. </p> <a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.examples"></a><h5> <a name="id1028652"></a> <a class="link" href="chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.chi_squared_dist.examples">Examples</a> </h5> <p> Various <a class="link" href="../../stat_tut/weg/cs_eg.html" title="Chi Squared Distribution Examples">worked examples</a> are available illustrating the use of the Chi Squared Distribution. </p> <a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.accuracy"></a><h5> <a name="id1028679"></a> <a class="link" href="chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.chi_squared_dist.accuracy">Accuracy</a> </h5> <p> The Chi-Squared distribution is implemented in terms of the <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">incomplete gamma functions</a>: please refer to the accuracy data for those functions. </p> <a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.implementation"></a><h5> <a name="id1028703"></a> <a class="link" href="chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.chi_squared_dist.implementation">Implementation</a> </h5> <p> In the following table <span class="emphasis"><em>v</em></span> is the number of degrees of freedom of the distribution, <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span> is the probability, and <span class="emphasis"><em>q = 1-p</em></span>. </p> <div class="informaltable"><table class="table"> <colgroup> <col> <col> </colgroup> <thead><tr> <th> <p> Function </p> </th> <th> <p> Implementation Notes </p> </th> </tr></thead> <tbody> <tr> <td> <p> pdf </p> </td> <td> <p> 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 / 2, x / 2) / 2 </p> </td> </tr> <tr> <td> <p> cdf </p> </td> <td> <p> Using the relation: p = <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(v / 2, x / 2) </p> </td> </tr> <tr> <td> <p> cdf complement </p> </td> <td> <p> Using the relation: q = <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(v / 2, x / 2) </p> </td> </tr> <tr> <td> <p> quantile </p> </td> <td> <p> 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 / 2, p) </p> </td> </tr> <tr> <td> <p> quantile from the complement </p> </td> <td> <p> 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 / 2, p) </p> </td> </tr> <tr> <td> <p> mean </p> </td> <td> <p> v </p> </td> </tr> <tr> <td> <p> variance </p> </td> <td> <p> 2v </p> </td> </tr> <tr> <td> <p> mode </p> </td> <td> <p> v - 2 (if v >= 2) </p> </td> </tr> <tr> <td> <p> skewness </p> </td> <td> <p> 2 * sqrt(2 / v) == sqrt(8 / v) </p> </td> </tr> <tr> <td> <p> kurtosis </p> </td> <td> <p> 3 + 12 / v </p> </td> </tr> <tr> <td> <p> kurtosis excess </p> </td> <td> <p> 12 / v </p> </td> </tr> </tbody> </table></div> <a name="math_toolkit.dist.dist_ref.dists.chi_squared_dist.references"></a><h5> <a name="id1029298"></a> <a class="link" href="chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.chi_squared_dist.references">References</a> </h5> <div class="itemizedlist"><ul type="disc"> <li> <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm" target="_top">NIST Exploratory Data Analysis</a> </li> <li> <a href="http://en.wikipedia.org/wiki/Chi-square_distribution" target="_top">Chi-square distribution</a> </li> <li> <a href="http://mathworld.wolfram.com/Chi-SquaredDistribution.html" target="_top">Weisstein, Eric W. "Chi-Squared Distribution." From MathWorld--A Wolfram Web Resource.</a> </li> </ul></div> </div> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> <td align="left"></td> <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and Thijs van den Berg<p> Distributed under the Boost Software License, Version 1.0. (See accompanying 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>) </p> </div></td> </tr></table> <hr> <div class="spirit-nav"> <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="exp_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>