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| <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="id1020125"></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="id1020456"></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="id1020559"></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="id1020586"></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="id1020610"></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="id1020906"></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, 2010 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> |
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