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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.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">&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>
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">&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>
34 <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>
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">&lt;&gt;</span> <span class="identifier">chi_squared</span><span class="special">;</span>
38
39<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>
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 &#967;<sub>i</sub> &#160; are &#957; &#160;
66independent, normally distributed random
67 variables with means &#956;<sub>i</sub> &#160; and variances &#963;<sub>i</sub><sup>2</sup>, then the random variable:
68 </p>
69<p>
70 <span class="inlinemediaobject"><img src="../../../../equations/chi_squ_ref1.svg"></span>
71 </p>
72<p>
73 is distributed according to the Chi-Squared distribution.
74 </p>
75<p>
76 The Chi-Squared distribution is a special case of the gamma distribution
77 and has a single parameter &#957; &#160; that specifies the number of degrees of freedom.
78 The following graph illustrates how the distribution changes for different
79 values of &#957;:
80 </p>
81<p>
82 <span class="inlinemediaobject"><img src="../../../../graphs/chi_squared_pdf.svg" align="middle"></span>
83 </p>
84<h5>
85<a name="math_toolkit.dist_ref.dists.chi_squared_dist.h0"></a>
86 <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
87 Functions</a>
88 </h5>
89<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>
90</pre>
91<p>
92 Constructs a Chi-Squared distribution with <span class="emphasis"><em>v</em></span> degrees
93 of freedom.
94 </p>
95<p>
96 Requires v &gt; 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
97 </p>
98<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>
99</pre>
100<p>
101 Returns the parameter <span class="emphasis"><em>v</em></span> from which this object was
102 constructed.
103 </p>
104<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>
105 <span class="identifier">RealType</span> <span class="identifier">difference_from_variance</span><span class="special">,</span>
106 <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span>
107 <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span>
108 <span class="identifier">RealType</span> <span class="identifier">variance</span><span class="special">,</span>
109 <span class="identifier">RealType</span> <span class="identifier">hint</span> <span class="special">=</span> <span class="number">100</span><span class="special">);</span>
110</pre>
111<p>
112 Estimates the sample size required to detect a difference from a nominal
113 variance in a Chi-Squared test for equal standard deviations.
114 </p>
115<div class="variablelist">
116<p class="title"><b></b></p>
117<dl class="variablelist">
118<dt><span class="term">difference_from_variance</span></dt>
119<dd><p>
120 The difference from the assumed nominal variance that is to be detected:
121 Note that the sign of this value is critical, see below.
122 </p></dd>
123<dt><span class="term">alpha</span></dt>
124<dd><p>
125 The maximum acceptable risk of rejecting the null hypothesis when
126 it is in fact true.
127 </p></dd>
128<dt><span class="term">beta</span></dt>
129<dd><p>
130 The maximum acceptable risk of falsely failing to reject the null
131 hypothesis.
132 </p></dd>
133<dt><span class="term">variance</span></dt>
134<dd><p>
135 The nominal variance being tested against.
136 </p></dd>
137<dt><span class="term">hint</span></dt>
138<dd><p>
139 An optional hint on where to start looking for a result: the current
140 sample size would be a good choice.
141 </p></dd>
142</dl>
143</div>
144<p>
145 Note that this calculation works with <span class="emphasis"><em>variances</em></span> and
146 not <span class="emphasis"><em>standard deviations</em></span>.
147 </p>
148<p>
149 The sign of the parameter <span class="emphasis"><em>difference_from_variance</em></span>
150 is important: the Chi Squared distribution is asymmetric, and the caller
151 must decide in advance whether they are testing for a variance greater
152 than a nominal value (positive <span class="emphasis"><em>difference_from_variance</em></span>)
153 or testing for a variance less than a nominal value (negative <span class="emphasis"><em>difference_from_variance</em></span>).
154 If the latter, then obviously it is a requirement that <code class="computeroutput"><span class="identifier">variance</span>
155 <span class="special">+</span> <span class="identifier">difference_from_variance</span>
156 <span class="special">&gt;</span> <span class="number">0</span></code>,
157 since no sample can have a negative variance!
158 </p>
159<p>
160 This procedure uses the method in Diamond, W. J. (1989). Practical Experiment
161 Designs, Van-Nostrand Reinhold, New York.
162 </p>
163<p>
164 See also section on Sample sizes required in <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc232.htm" target="_top">the
165 NIST Engineering Statistics Handbook, Section 7.2.3.2</a>.
166 </p>
167<h5>
168<a name="math_toolkit.dist_ref.dists.chi_squared_dist.h1"></a>
169 <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
170 Accessors</a>
171 </h5>
172<p>
173 All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
174 functions</a> that are generic to all distributions are supported:
175 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
176 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
177 <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>,
178 <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>,
179 <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>,
180 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
181 <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>,
182 <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>.
183 </p>
184<p>
185 (We have followed the usual restriction of the mode to degrees of freedom
186 &gt;= 2, but note that the maximum of the pdf is actually zero for degrees
187 of freedom from 2 down to 0, and provide an extended definition that would
188 avoid a discontinuity in the mode as alternative code in a comment).
189 </p>
190<p>
191 The domain of the random variable is [0, +&#8734;].
192 </p>
193<h5>
194<a name="math_toolkit.dist_ref.dists.chi_squared_dist.h2"></a>
195 <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>
196 </h5>
197<p>
198 Various <a class="link" href="../../stat_tut/weg/cs_eg.html" title="Chi Squared Distribution Examples">worked examples</a>
199 are available illustrating the use of the Chi Squared Distribution.
200 </p>
201<h5>
202<a name="math_toolkit.dist_ref.dists.chi_squared_dist.h3"></a>
203 <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>
204 </h5>
205<p>
206 The Chi-Squared distribution is implemented in terms of the <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">incomplete
207 gamma functions</a>: please refer to the accuracy data for those functions.
208 </p>
209<h5>
210<a name="math_toolkit.dist_ref.dists.chi_squared_dist.h4"></a>
211 <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>
212 </h5>
213<p>
214 In the following table <span class="emphasis"><em>v</em></span> is the number of degrees
215 of freedom of the distribution, <span class="emphasis"><em>x</em></span> is the random variate,
216 <span class="emphasis"><em>p</em></span> is the probability, and <span class="emphasis"><em>q = 1-p</em></span>.
217 </p>
218<div class="informaltable"><table class="table">
219<colgroup>
220<col>
221<col>
222</colgroup>
223<thead><tr>
224<th>
225 <p>
226 Function
227 </p>
228 </th>
229<th>
230 <p>
231 Implementation Notes
232 </p>
233 </th>
234</tr></thead>
235<tbody>
236<tr>
237<td>
238 <p>
239 pdf
240 </p>
241 </td>
242<td>
243 <p>
244 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
245 / 2, x / 2) / 2
246 </p>
247 </td>
248</tr>
249<tr>
250<td>
251 <p>
252 cdf
253 </p>
254 </td>
255<td>
256 <p>
257 Using the relation: p = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(v
258 / 2, x / 2)
259 </p>
260 </td>
261</tr>
262<tr>
263<td>
264 <p>
265 cdf complement
266 </p>
267 </td>
268<td>
269 <p>
270 Using the relation: q = <a class="link" href="../../sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(v
271 / 2, x / 2)
272 </p>
273 </td>
274</tr>
275<tr>
276<td>
277 <p>
278 quantile
279 </p>
280 </td>
281<td>
282 <p>
283 Using the relation: x = 2 * <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(v
284 / 2, p)
285 </p>
286 </td>
287</tr>
288<tr>
289<td>
290 <p>
291 quantile from the complement
292 </p>
293 </td>
294<td>
295 <p>
296 Using the relation: x = 2 * <a class="link" href="../../sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(v
297 / 2, p)
298 </p>
299 </td>
300</tr>
301<tr>
302<td>
303 <p>
304 mean
305 </p>
306 </td>
307<td>
308 <p>
309 v
310 </p>
311 </td>
312</tr>
313<tr>
314<td>
315 <p>
316 variance
317 </p>
318 </td>
319<td>
320 <p>
321 2v
322 </p>
323 </td>
324</tr>
325<tr>
326<td>
327 <p>
328 mode
329 </p>
330 </td>
331<td>
332 <p>
333 v - 2 (if v &gt;= 2)
334 </p>
335 </td>
336</tr>
337<tr>
338<td>
339 <p>
340 skewness
341 </p>
342 </td>
343<td>
344 <p>
345 2 * sqrt(2 / v) == sqrt(8 / v)
346 </p>
347 </td>
348</tr>
349<tr>
350<td>
351 <p>
352 kurtosis
353 </p>
354 </td>
355<td>
356 <p>
357 3 + 12 / v
358 </p>
359 </td>
360</tr>
361<tr>
362<td>
363 <p>
364 kurtosis excess
365 </p>
366 </td>
367<td>
368 <p>
369 12 / v
370 </p>
371 </td>
372</tr>
373</tbody>
374</table></div>
375<h5>
376<a name="math_toolkit.dist_ref.dists.chi_squared_dist.h5"></a>
377 <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>
378 </h5>
379<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
380<li class="listitem">
381 <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm" target="_top">NIST
382 Exploratory Data Analysis</a>
383 </li>
384<li class="listitem">
385 <a href="http://en.wikipedia.org/wiki/Chi-square_distribution" target="_top">Chi-square
386 distribution</a>
387 </li>
388<li class="listitem">
389 <a href="http://mathworld.wolfram.com/Chi-SquaredDistribution.html" target="_top">Weisstein,
390 Eric W. "Chi-Squared Distribution." From MathWorld--A Wolfram
391 Web Resource.</a>
392 </li>
393</ul></div>
394</div>
395<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
396<td align="left"></td>
397<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
398 Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
399 Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
400 Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
401 Distributed under the Boost Software License, Version 1.0. (See accompanying
402 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>)
403 </p>
404</div></td>
405</tr></table>
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