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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.f_dist"></a><a class="link" href="f_dist.html" title="F Distribution">F Distribution</a> | |
28 | </h4></div></div></div> | |
29 | <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">fisher_f</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> | |
30 | <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> | |
31 | ||
32 | <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> | |
33 | <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 15. 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> | |
34 | <span class="keyword">class</span> <span class="identifier">fisher_f_distribution</span><span class="special">;</span> | |
35 | ||
36 | <span class="keyword">typedef</span> <span class="identifier">fisher_f_distribution</span><span class="special"><></span> <span class="identifier">fisher_f</span><span class="special">;</span> | |
37 | ||
38 | <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 15. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> | |
39 | <span class="keyword">class</span> <span class="identifier">fisher_f_distribution</span> | |
40 | <span class="special">{</span> | |
41 | <span class="keyword">public</span><span class="special">:</span> | |
42 | <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> | |
43 | ||
44 | <span class="comment">// Construct:</span> | |
45 | <span class="identifier">fisher_f_distribution</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">i</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">j</span><span class="special">);</span> | |
46 | ||
47 | <span class="comment">// Accessors:</span> | |
48 | <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom1</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> | |
49 | <span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom2</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> | |
50 | <span class="special">};</span> | |
51 | ||
52 | <span class="special">}}</span> <span class="comment">//namespaces</span> | |
53 | </pre> | |
54 | <p> | |
55 | The F distribution is a continuous distribution that arises when testing | |
56 | whether two samples have the same variance. If χ<sup>2</sup><sub>m</sub>   and χ<sup>2</sup><sub>n</sub>   are independent | |
57 | variates each distributed as Chi-Squared with <span class="emphasis"><em>m</em></span> and | |
58 | <span class="emphasis"><em>n</em></span> degrees of freedom, then the test statistic: | |
59 | </p> | |
60 | <p> | |
61 | F<sub>n,m</sub>   = (χ<sup>2</sup><sub>n</sub>   / n) / (χ<sup>2</sup><sub>m</sub>   / m) | |
62 | </p> | |
63 | <p> | |
64 | Is distributed over the range [0, ∞] with an F distribution, and has the | |
65 | PDF: | |
66 | </p> | |
67 | <p> | |
68 | <span class="inlinemediaobject"><img src="../../../../equations/fisher_pdf.svg"></span> | |
69 | </p> | |
70 | <p> | |
71 | The following graph illustrates how the PDF varies depending on the two | |
72 | degrees of freedom parameters. | |
73 | </p> | |
74 | <p> | |
75 | <span class="inlinemediaobject"><img src="../../../../graphs/fisher_f_pdf.svg" align="middle"></span> | |
76 | </p> | |
77 | <h5> | |
78 | <a name="math_toolkit.dist_ref.dists.f_dist.h0"></a> | |
79 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.member_functions"></a></span><a class="link" href="f_dist.html#math_toolkit.dist_ref.dists.f_dist.member_functions">Member Functions</a> | |
80 | </h5> | |
81 | <pre class="programlisting"><span class="identifier">fisher_f_distribution</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">df1</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">RealType</span><span class="special">&</span> <span class="identifier">df2</span><span class="special">);</span> | |
82 | </pre> | |
83 | <p> | |
84 | Constructs an F-distribution with numerator degrees of freedom <span class="emphasis"><em>df1</em></span> | |
85 | and denominator degrees of freedom <span class="emphasis"><em>df2</em></span>. | |
86 | </p> | |
87 | <p> | |
88 | Requires that <span class="emphasis"><em>df1</em></span> and <span class="emphasis"><em>df2</em></span> are | |
89 | both greater than zero, otherwise <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> | |
90 | is called. | |
91 | </p> | |
92 | <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom1</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> | |
93 | </pre> | |
94 | <p> | |
95 | Returns the numerator degrees of freedom parameter of the distribution. | |
96 | </p> | |
97 | <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">degrees_of_freedom2</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> | |
98 | </pre> | |
99 | <p> | |
100 | Returns the denominator degrees of freedom parameter of the distribution. | |
101 | </p> | |
102 | <h5> | |
103 | <a name="math_toolkit.dist_ref.dists.f_dist.h1"></a> | |
104 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.non_member_accessors"></a></span><a class="link" href="f_dist.html#math_toolkit.dist_ref.dists.f_dist.non_member_accessors">Non-member | |
105 | Accessors</a> | |
106 | </h5> | |
107 | <p> | |
108 | All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor | |
109 | functions</a> that are generic to all distributions are supported: | |
110 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, | |
111 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, | |
112 | <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>, | |
113 | <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>, | |
114 | <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>, | |
115 | <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, | |
116 | <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>, | |
117 | <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>. | |
118 | </p> | |
119 | <p> | |
120 | The domain of the random variable is [0, +∞]. | |
121 | </p> | |
122 | <h5> | |
123 | <a name="math_toolkit.dist_ref.dists.f_dist.h2"></a> | |
124 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.examples"></a></span><a class="link" href="f_dist.html#math_toolkit.dist_ref.dists.f_dist.examples">Examples</a> | |
125 | </h5> | |
126 | <p> | |
127 | Various <a class="link" href="../../stat_tut/weg/f_eg.html" title="F Distribution Examples">worked examples</a> | |
128 | are available illustrating the use of the F Distribution. | |
129 | </p> | |
130 | <h5> | |
131 | <a name="math_toolkit.dist_ref.dists.f_dist.h3"></a> | |
132 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.accuracy"></a></span><a class="link" href="f_dist.html#math_toolkit.dist_ref.dists.f_dist.accuracy">Accuracy</a> | |
133 | </h5> | |
134 | <p> | |
135 | The normal distribution is implemented in terms of the <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">incomplete | |
136 | beta function</a> and its <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">inverses</a>, | |
137 | refer to those functions for accuracy data. | |
138 | </p> | |
139 | <h5> | |
140 | <a name="math_toolkit.dist_ref.dists.f_dist.h4"></a> | |
141 | <span class="phrase"><a name="math_toolkit.dist_ref.dists.f_dist.implementation"></a></span><a class="link" href="f_dist.html#math_toolkit.dist_ref.dists.f_dist.implementation">Implementation</a> | |
142 | </h5> | |
143 | <p> | |
144 | In the following table <span class="emphasis"><em>v1</em></span> and <span class="emphasis"><em>v2</em></span> | |
145 | are the first and second degrees of freedom parameters of the distribution, | |
146 | <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span> is | |
147 | the probability, and <span class="emphasis"><em>q = 1-p</em></span>. | |
148 | </p> | |
149 | <div class="informaltable"><table class="table"> | |
150 | <colgroup> | |
151 | <col> | |
152 | <col> | |
153 | </colgroup> | |
154 | <thead><tr> | |
155 | <th> | |
156 | <p> | |
157 | Function | |
158 | </p> | |
159 | </th> | |
160 | <th> | |
161 | <p> | |
162 | Implementation Notes | |
163 | </p> | |
164 | </th> | |
165 | </tr></thead> | |
166 | <tbody> | |
167 | <tr> | |
168 | <td> | |
169 | <p> | |
170 | ||
171 | </p> | |
172 | </td> | |
173 | <td> | |
174 | <p> | |
175 | The usual form of the PDF is given by: | |
176 | </p> | |
177 | <p> | |
178 | <span class="inlinemediaobject"><img src="../../../../equations/fisher_pdf.svg"></span> | |
179 | </p> | |
180 | <p> | |
181 | However, that form is hard to evaluate directly without incurring | |
182 | problems with either accuracy or numeric overflow. | |
183 | </p> | |
184 | <p> | |
185 | Direct differentiation of the CDF expressed in terms of the incomplete | |
186 | beta function | |
187 | </p> | |
188 | <p> | |
189 | led to the following two formulas: | |
190 | </p> | |
191 | <p> | |
192 | f<sub>v1,v2</sub>(x) = y * <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>(v2 | |
193 | / 2, v1 / 2, v2 / (v2 + v1 * x)) | |
194 | </p> | |
195 | <p> | |
196 | with y = (v2 * v1) / ((v2 + v1 * x) * (v2 + v1 * x)) | |
197 | </p> | |
198 | <p> | |
199 | and | |
200 | </p> | |
201 | <p> | |
202 | f<sub>v1,v2</sub>(x) = y * <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a>(v1 | |
203 | / 2, v2 / 2, v1 * x / (v2 + v1 * x)) | |
204 | </p> | |
205 | <p> | |
206 | with y = (z * v1 - x * v1 * v1) / z<sup>2</sup> | |
207 | </p> | |
208 | <p> | |
209 | and z = v2 + v1 * x | |
210 | </p> | |
211 | <p> | |
212 | The first of these is used for v1 * x > v2, otherwise the | |
213 | second is used. | |
214 | </p> | |
215 | <p> | |
216 | The aim is to keep the <span class="emphasis"><em>x</em></span> argument to <a class="link" href="../../sf_beta/beta_derivative.html" title="Derivative of the Incomplete Beta Function">ibeta_derivative</a> | |
217 | away from 1 to avoid rounding error. | |
218 | </p> | |
219 | </td> | |
220 | </tr> | |
221 | <tr> | |
222 | <td> | |
223 | <p> | |
224 | cdf | |
225 | </p> | |
226 | </td> | |
227 | <td> | |
228 | <p> | |
229 | Using the relations: | |
230 | </p> | |
231 | <p> | |
232 | p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(v1 | |
233 | / 2, v2 / 2, v1 * x / (v2 + v1 * x)) | |
234 | </p> | |
235 | <p> | |
236 | and | |
237 | </p> | |
238 | <p> | |
239 | p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(v2 | |
240 | / 2, v1 / 2, v2 / (v2 + v1 * x)) | |
241 | </p> | |
242 | <p> | |
243 | The first is used for v1 * x > v2, otherwise the second is | |
244 | used. | |
245 | </p> | |
246 | <p> | |
247 | The aim is to keep the <span class="emphasis"><em>x</em></span> argument to <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a> well | |
248 | away from 1 to avoid rounding error. | |
249 | </p> | |
250 | </td> | |
251 | </tr> | |
252 | <tr> | |
253 | <td> | |
254 | <p> | |
255 | cdf complement | |
256 | </p> | |
257 | </td> | |
258 | <td> | |
259 | <p> | |
260 | Using the relations: | |
261 | </p> | |
262 | <p> | |
263 | p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibetac</a>(v1 | |
264 | / 2, v2 / 2, v1 * x / (v2 + v1 * x)) | |
265 | </p> | |
266 | <p> | |
267 | and | |
268 | </p> | |
269 | <p> | |
270 | p = <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a>(v2 | |
271 | / 2, v1 / 2, v2 / (v2 + v1 * x)) | |
272 | </p> | |
273 | <p> | |
274 | The first is used for v1 * x < v2, otherwise the second is | |
275 | used. | |
276 | </p> | |
277 | <p> | |
278 | The aim is to keep the <span class="emphasis"><em>x</em></span> argument to <a class="link" href="../../sf_beta/ibeta_function.html" title="Incomplete Beta Functions">ibeta</a> well | |
279 | away from 1 to avoid rounding error. | |
280 | </p> | |
281 | </td> | |
282 | </tr> | |
283 | <tr> | |
284 | <td> | |
285 | <p> | |
286 | quantile | |
287 | </p> | |
288 | </td> | |
289 | <td> | |
290 | <p> | |
291 | Using the relation: | |
292 | </p> | |
293 | <p> | |
294 | x = v2 * a / (v1 * b) | |
295 | </p> | |
296 | <p> | |
297 | where: | |
298 | </p> | |
299 | <p> | |
300 | a = <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a>(v1 | |
301 | / 2, v2 / 2, p) | |
302 | </p> | |
303 | <p> | |
304 | and | |
305 | </p> | |
306 | <p> | |
307 | b = 1 - a | |
308 | </p> | |
309 | <p> | |
310 | Quantities <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> | |
311 | are both computed by <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibeta_inv</a> | |
312 | without the subtraction implied above. | |
313 | </p> | |
314 | </td> | |
315 | </tr> | |
316 | <tr> | |
317 | <td> | |
318 | <p> | |
319 | quantile | |
320 | </p> | |
321 | <p> | |
322 | from the complement | |
323 | </p> | |
324 | </td> | |
325 | <td> | |
326 | <p> | |
327 | Using the relation: | |
328 | </p> | |
329 | <p> | |
330 | x = v2 * a / (v1 * b) | |
331 | </p> | |
332 | <p> | |
333 | where | |
334 | </p> | |
335 | <p> | |
336 | a = <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a>(v1 | |
337 | / 2, v2 / 2, p) | |
338 | </p> | |
339 | <p> | |
340 | and | |
341 | </p> | |
342 | <p> | |
343 | b = 1 - a | |
344 | </p> | |
345 | <p> | |
346 | Quantities <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>b</em></span> | |
347 | are both computed by <a class="link" href="../../sf_beta/ibeta_inv_function.html" title="The Incomplete Beta Function Inverses">ibetac_inv</a> | |
348 | without the subtraction implied above. | |
349 | </p> | |
350 | </td> | |
351 | </tr> | |
352 | <tr> | |
353 | <td> | |
354 | <p> | |
355 | mean | |
356 | </p> | |
357 | </td> | |
358 | <td> | |
359 | <p> | |
360 | v2 / (v2 - 2) | |
361 | </p> | |
362 | </td> | |
363 | </tr> | |
364 | <tr> | |
365 | <td> | |
366 | <p> | |
367 | variance | |
368 | </p> | |
369 | </td> | |
370 | <td> | |
371 | <p> | |
372 | 2 * v2<sup>2 </sup> * (v1 + v2 - 2) / (v1 * (v2 - 2) * (v2 - 2) * (v2 - 4)) | |
373 | </p> | |
374 | </td> | |
375 | </tr> | |
376 | <tr> | |
377 | <td> | |
378 | <p> | |
379 | mode | |
380 | </p> | |
381 | </td> | |
382 | <td> | |
383 | <p> | |
384 | v2 * (v1 - 2) / (v1 * (v2 + 2)) | |
385 | </p> | |
386 | </td> | |
387 | </tr> | |
388 | <tr> | |
389 | <td> | |
390 | <p> | |
391 | skewness | |
392 | </p> | |
393 | </td> | |
394 | <td> | |
395 | <p> | |
396 | 2 * (v2 + 2 * v1 - 2) * sqrt((2 * v2 - 8) / (v1 * (v2 + v1 - | |
397 | 2))) / (v2 - 6) | |
398 | </p> | |
399 | </td> | |
400 | </tr> | |
401 | <tr> | |
402 | <td> | |
403 | <p> | |
404 | kurtosis and kurtosis excess | |
405 | </p> | |
406 | </td> | |
407 | <td> | |
408 | <p> | |
409 | Refer to, <a href="http://mathworld.wolfram.com/F-Distribution.html" target="_top">Weisstein, | |
410 | Eric W. "F-Distribution." From MathWorld--A Wolfram | |
411 | Web Resource.</a> | |
412 | </p> | |
413 | </td> | |
414 | </tr> | |
415 | </tbody> | |
416 | </table></div> | |
417 | </div> | |
418 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
419 | <td align="left"></td> | |
420 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
421 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
422 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
423 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
424 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
425 | 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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